Download Chapter 8. Waveguides, Resonant Cavities, and Optical Fibers

Chapter 8. Waveguides, Resonant Cavities, and Optical
Fibers
A waveguide is a device used to carry electromagnetic waves from one place to another
without significant loss in intensity while confining them near the propagation axis. The most
common type of waveguides for radio waves and microwaves is a hollow metal pipe. Waves
propagate through the waveguide, being confined to the interior of the pipe. A representative
waveguide in the optical region is an optical fiber. Fiber-optic communication and a variety of
other applications exploit the extremely low attenuation and dispersion of silica-based optical
fibers in the optical communication band of 1.3-1.6 𝜇m.
8.1 Wave Propagation between Parallel Conducting Plates
We consider the propagation of electromagnetic waves in the region between two parallel,
perfectly conducting plates for 0 < 𝑥 < 𝑎. The wave propagates along the 𝑧-axis. We consider
the polarization state known as TE, transverse electric, for which the electric field is parallel to
the 𝑦-axis (see Fig 8.1), i.e., the electric field is expressed as 𝑬(𝐱, 𝑡) = 𝒆𝑦 𝐸(𝑥)𝑒 𝑖(𝑘𝑧 𝑧−𝜔𝑡) .
Fig 8.1
The electric field satisfies the wave equation,
∇2 𝐄(𝐱, 𝑡) =
1 ∂2 𝐄(𝐱, 𝑡)
𝑐 2 𝜕𝑡 2
(8.1)
therefore, a monochromatic wave of frequency 𝜔 satisfies
∂2
𝜔2
𝐸(𝑥)
+
(
− 𝑘𝑧2 ) 𝐸(𝑥) = 0
𝜕𝑥 2
𝑐2
(8.2)
Using the boundary condition, 𝐸(0) = 𝐸(𝑎) = 0, we obtain
𝐸(𝑥) = 𝐸0 sin (
𝑛𝜋
𝑥) ,
𝑎
𝑛 = 1,2,3, …
(8.3)
The electric field amplitude for the first two modes for 𝑛 = 1,2,
𝐸(𝑥, 𝑧) = 𝐸0 sin (
𝑛𝜋
𝑥) cos(𝑘𝑧 𝑧 − 𝜔𝑡)
𝑎
are shown in the contour plots of Fig 8.2.
1
(8.4)
Fig 8.2
Dispersion relation and cut-off frequency
The wave equation gives rise to the dispersion relation
2
𝑘𝑧,𝑛
𝜔 2 𝑛2 𝜋 2
= 2− 2
𝑐
𝑎
(8.5)
The dispersion of the first three modes, 𝑛 = 1,2,3, is shown in Fig 8.3.
Fig 8.3
The cut-off frequency for the lowest mode 𝑛 = 1 is defined as 𝜔𝑐 = 𝜋𝑐/𝑎. For 𝜔 < 𝜔𝑐 , 𝑘𝑧
becomes pure imaginary, and hence no wave is allowed to propagate in the waveguide. For 𝑛th mode, the cut-off frequency is 𝑛𝜔𝑐 . If the frequency of an electromagnetic wave fall in the
region, 𝑚𝜔𝑐 < 𝜔 < (𝑚 + 1)𝜔𝑐 , the waveguide can support 𝑚 modes for 𝑛 = 1,2, … 𝑚.
Phase velocity and group velocity
Based on the dispersion relation Eq. 8.4, the wave number can be expressed
𝑘𝑧,𝑛 =
1
√𝜔 2 − 𝑛2 𝜔𝑐2
𝑐
(8.6)
Using this relation, we obtain the phase velocity,
𝑣𝑝 =
𝜔
=
𝑘𝑧
𝑐
√1 −
2
𝑛2 𝜔𝑐2
𝜔2
(8.7)
𝑣𝑝 always exceeds the speed of light 𝑐. It, however, does not violate the postulate of the
relativity that no signal can propagate with a velocity greater than 𝑐. The radiation energy
propagates down the waveguide with the group velocity,
𝜕𝜔
𝜕𝑘𝑧 −1
𝑛2 𝜔𝑐2
𝑣𝑔 =
=(
) = 𝑐 √1 −
𝜕𝑘𝑧
𝜕𝜔
𝜔2
(8.8)
It is readily apparent that
𝑣𝑝 𝑣𝑔 = 𝑐 2
(8.9)
8.2 Waveguides
We consider electromagnetic waves propagating in a hollow, cylindrical waveguide (Fig 8.4).
We assume the boundary surfaces are perfect conductors.
Fig 8.4
The electric field and magnetic induction (𝐄 and 𝐁) both satisfy the wave equation (Eq. 8.1) in
vacuum. For monochromatic waves, i.e., waves of the form 𝑬(𝐱, 𝑡) = 𝐄(𝐱)𝑒 −𝑖𝜔𝑡 , the wave
equations become
𝜔2
𝜔2
(8.10)
∇2 𝐄(𝐱, 𝑡) + 2 𝐄(𝐱, 𝑡) = 0,
∇2 𝐁(𝐱, 𝑡) + 2 𝐁(𝐱, 𝑡) = 0
c
c
The electric and magnetic fields of a monochromatic wave travelling through the waveguide in
the positive direction of the z-axis have the generic form
𝐄(𝑥, 𝑦, 𝑧, 𝑡) = 𝐄0 (𝑥, 𝑦)𝑒 𝑖(𝑘𝑧−𝜔𝑡)
𝐁(𝑥, 𝑦, 𝑧, 𝑡) = 𝐁0 (𝑥, 𝑦)𝑒 𝑖(𝑘𝑧−𝜔𝑡)
(8.11)
where 𝐄0 = Ex 𝐞x + Ey 𝐞y + Ez 𝐞z and 𝐁0 = Bx 𝐞x + By 𝐞y + Bz 𝐞z. These harmonic fields must
satisfy the Maxwell equations:
𝜔
𝛁 ⋅ 𝐄 = 0 𝛁 × 𝐁 = −𝑖 2 𝐄
(8.12)
𝑐
𝛁 ⋅ 𝐁 = 0 𝛁 × 𝐄 = 𝑖𝜔𝐁
where all the quantities are complex functions of 𝐱.
Longitudinal (𝒛) and transverse (𝒙, 𝒚) fields
It is useful to separate the fields into components parallel to and transverse to the 𝑧 axis. The
Maxwell equations can be written out in terms of transverse and longitudinal components as
following:
3
𝛁 ⋅ 𝐄 = 0 and 𝛁 ⋅ 𝐁 = 0
𝜕𝐸𝑥 𝜕𝐸𝑦
𝜕𝐸𝑧 𝜕𝐵𝑥 𝜕𝐵𝑦
𝜕𝐵𝑧
+
=−
,
+
=−
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑥
𝜕𝑦
𝜕𝑧
(8.13)
𝜔
𝛁 × 𝐄 = 𝑖𝜔𝐁 and 𝛁 × 𝐁 = −𝑖 𝑐 2 𝐄
Longitudinal:
𝜕𝐸𝑦 𝜕𝐸𝑥
−
= 𝑖𝜔𝐵𝑧 ,
𝜕𝑥
𝜕𝑦
𝜕𝐸𝑧
,
𝜕𝑦
Transverse:
𝜕𝐸𝑧
𝑖𝑘𝐸
−
𝑖𝜔𝐵
=
,
𝑥
𝑦
{
𝜕𝑥
𝑖𝜔𝐵𝑥 + 𝑖𝑘𝐸𝑦 =
𝜕𝐵𝑦 𝜕𝐵𝑥
𝑖𝜔
−
= − 2 𝐸𝑧
𝜕𝑥
𝜕𝑦
𝑐
−
𝑖𝜔
𝜕𝐵𝑧
𝐸
+
𝑖𝑘𝐵
=
𝑥
𝑦
𝑐2
𝜕𝑦
𝑖𝜔
𝜕𝐵𝑧
𝑖𝑘𝐵𝑥 + 2 𝐸𝑦 =
𝑐
𝜕𝑥
(8.14)
(8.15)
Unlike propagation in free space, guided waves, in general, are not transverse, i.e., the
longitudinal components 𝐸𝑧 and 𝐵𝑧 do not vanish. It is evident from Eq. 8.15 that if 𝐸𝑧 and 𝐵𝑧
are known the transverse components of 𝐄 and 𝐁 are determined assuming the nonvanishing
of at least one of 𝐸𝑧 and 𝐵𝑧 . Then, the transverse fields are
𝐸𝑥 =
𝑖
𝜕𝐸𝑧
𝜕𝐵𝑧
(𝑘
+
𝜔
)
(𝜔/𝑐)2 − 𝑘 2
𝜕𝑥
𝜕𝑦
(8.16)
𝐸𝑦 =
𝑖
𝜕𝐸𝑧
𝜕𝐵𝑧
(𝑘
−𝜔
)
2
2
(𝜔/𝑐) − 𝑘
𝜕𝑦
𝜕𝑥
(8.17)
𝐵𝑥 =
𝑖
𝜕𝐵𝑧 𝜔 𝜕𝐸𝑧
(𝑘
−
)
2
2
(𝜔/𝑐) − 𝑘
𝜕𝑥 𝑐 2 𝜕𝑦
(8.18)
𝐵𝑦 =
𝑖
𝜕𝐵𝑧 𝜔 𝜕𝐸𝑧
(𝑘
+
)
2
2
(𝜔/𝑐) − 𝑘
𝜕𝑦 𝑐 2 𝜕𝑥
(8.19)
The wave equations for 𝐸𝑧 and 𝐵𝑧 :
𝜕2
𝜕2
𝜔2
[ 2 + 2 + 2 − 𝑘 2 ] 𝐸𝑧 (𝑥, 𝑦) = 0
𝜕𝑥
𝜕𝑦
𝑐
(8.20)
𝜕2
𝜕2
𝜔2
[ 2 + 2 + 2 − 𝑘 2 ] 𝐵𝑧 (𝑥, 𝑦) = 0
𝜕𝑥
𝜕𝑦
𝑐
(8.21)
and
Noting that these two equations are independent of each other, we can classify guided waves
into different types of modes. If 𝐸𝑧 = 0, we call the waves transverse electric (TE) modes.
Similarly, transverse magnetic (TM) modes have no longitudinal component of the magnetic
field, 𝐵𝑧 = 0. A TEM mode has neither electric nor magnetic field in the longitudinal direction. A
hollow waveguide, however, does not support TEM modes: the surface at a cross section is an
equipotential and hence the electric field vanishes inside.
4
Rectangular Waveguides
To get a sense of how waves propagate in a waveguide, we look into a metal tube of
rectangular shape (Fig 8.5).
Fig 8.5
TE modes
Suppose we are interested in TE modes. We obtain 𝐵𝑧 (𝑥, 𝑦) by solving the wave equation (Eq.
8.12) then using Eqs. 8.16, 17, 18, and 19, we obtain
𝐸𝑥 =
𝑖𝜔
𝜕𝐵𝑧
2
2
(𝜔/𝑐) − 𝑘 𝜕𝑦
(8.22)
𝐸𝑦 =
−𝑖𝜔
𝜕𝐵𝑧
(𝜔/𝑐)2 − 𝑘 2 𝜕𝑥
(8.23)
𝐵𝑥 =
𝑖𝑘
𝜕𝐵𝑧
(𝜔/𝑐)2 − 𝑘 2 𝜕𝑥
(8.24)
𝐵𝑦 =
𝑖𝑘
𝜕𝐵𝑧
2
2
(𝜔/𝑐) − 𝑘 𝜕𝑦
(8.25)
The general solution of Eq. 8.21 has the form,
𝐵𝑧 (𝑥, 𝑦) = [𝐴 sin(𝑘𝑥 𝑥) + 𝐵 cos(𝑘𝑥 𝑥)] ⋅ [𝐶 sin(𝑘𝑦 𝑦) + 𝐷 cos(𝑘𝑦 𝑦)]
(8.26)
where the coefficients 𝐴, 𝐵, 𝐶, and 𝐷, and the wavenumbers, 𝑘𝑥 and 𝑘𝑦 , are determined by
boundary conditions. Under the assumption that the metal is a perfect conductor,
electromagnetic waves vanish inside the material. Accordingly, the electric and the magnetic
fields satisfy the boundary conditions that the parallel components of the electric field and the
normal component of the magnetic field vanish at the interior surface, 𝐄∥ = 0 and 𝐁⊥ = 0.
Applying the boundary condition 𝐄∥ = 0 to Eqs. 8.22 and 8.23, we obtain
𝐵𝑧 (𝑥, 𝑦) = 𝐵0 cos (
𝑚𝜋
𝑛𝜋
𝑥) cos ( 𝑦) , 𝑚, 𝑛 = 0,1,2, …
𝑎
𝑏
Figure 8.6 shows the spatial profile of the field intensity for the low-order TE modes.
5
(8.27)
Fig 8.6. Field distribution of TE modes in a cross section of a rectangular waveguide
Inserting Eq. 8.27 into the wave equation, Eq. 8.21, we get the dispersion relation,
1
2
𝑘 = √𝜔 2 − 𝜔𝑚𝑛
𝑐
(8.28)
where the cut-off frequency
𝜔𝑚𝑛
= 𝜋𝑐 √
𝑚 2 𝑛2
+
𝑎2 𝑏 2
(8.29)
If 𝜔 < 𝜔𝑚𝑛 , the wave number 𝑘 is imaginary, then the wave attenuates exponentially as 𝑒 −|𝑘|𝑧 .
Therefore, the frequency of a travelling wave must be higher than the cut-off frequency. The
phase and the group velocities,
𝜔
𝑐
𝑣𝑝 = =
(8.30)
2 /𝜔 2
𝑘 √1 − 𝜔𝑚𝑛
𝑣𝑔 =
𝜕𝜔
2 /𝜔 2
= 𝑐√1 − 𝜔𝑚𝑛
𝜕𝑘
(8.31)
indicate that the waveguide is highly dispersive, especially, near the cut-off frequency.
8.3 Coaxial Transmission Line
A coaxial transmission line, consisting of a long straight wire surrounded by a cylindrical
conducting sheath, support TEM modes, i.e., 𝐸𝑧 = 0 and 𝐵𝑧 = 0.
Fig 8.7
In this case the wave equation yields the dispersion relation
𝑘=
𝜔
𝑐
(8.32)
so that the waves travel at speed of light 𝑐, and are nondispersive. The Maxwell’s equations
(Eq. 8.15) lead to
(8.33)
𝑐𝐵𝑦 = 𝐸𝑥 and 𝑐𝐵𝑥 = −𝐸𝑦
6
(so 𝐄 and 𝐁 are mutually perpendicular), and (together with 𝛁 ⋅ 𝐄 = 0 and 𝛁 ⋅ 𝐁 = 0):
𝜕𝐸𝑥 𝜕𝐸𝑦
+
= 0,
𝜕𝑥
𝜕𝑦
𝜕𝐵𝑥 𝜕𝐵𝑦
+
= 0,
𝜕𝑦
{ 𝜕𝑥
𝜕𝐸𝑦 𝜕𝐸𝑥
−
=0
𝜕𝑥
𝜕𝑦
𝜕𝐵𝑦 𝜕𝐵𝑥
−
=0
𝜕𝑥
𝜕𝑦
(8.34)
These take the form of the equations of electrostatics and magnetostatics in vacuum in two
dimensions. The solution with cylindrical symmetry can be obtained from the case of an infinite
line charge and an infinite straight current, respectively:
𝐴
𝐞𝑟
𝑟
{
𝐴
𝐁0 (𝑟, 𝜙) = 𝐞𝜙
𝑐𝑟
𝐄0 (𝑟, 𝜙) =
for a constant 𝐴. From Eq. 8.10,
𝐴
𝐄(𝑟, 𝑧, 𝑡) = 𝑟 𝑒 𝑖(𝑘𝑧−𝜔𝑡) 𝐞𝑟
𝐴
𝐁(𝑟, 𝑧, 𝑡) = 𝑐𝑟 𝑒 𝑖(𝑘𝑧−𝜔𝑡) 𝐞𝜙
(8.35)
(8.36)
8.4 Resonant Cavities
Another type of device closely related to waveguides and of considerable practical importance
is the cavity resonator. Cavity resonators can store energy in oscillating EM fields. Furthermore,
practical cavity resonators dissipate a fraction of the stored energy in each cycle of oscillation.
Rectangular resonator
The simplest cavity resonator is a rectangular parallelepiped with perfectly conducting walls (Fig
8.8). For such a cavity, the appropriate boundary conditions are that the parallel components of
the electric field and the normal component of the magnetic field vanish at the interior surface,
𝐄∥ = 0 and 𝐁⊥ = 0.
Fig 8.8
7
𝐄 and 𝐁 must satisfy the wave equations, Eq. 8.10; thus 𝐸𝑥 must satisfy
𝜕2
𝜕2
𝜕 2 𝜔2
[ 2 + 2 + 2 + 2 ] 𝐸𝑥 = 0
𝜕𝑥
𝜕𝑥
𝜕𝑧
𝑐
(8.37)
Since 𝐸𝑥 = 0 at 𝑦 = 0, 𝑏 and 𝑧 = 0, 𝑑, 𝐸𝑥 must have the form
𝐸𝑥 = 𝐸1 𝑓1 (𝑥) sin 𝑘𝑦 𝑦 sin 𝑘𝑧 𝑧
(8.38)
𝐸𝑦 = 𝐸2 𝑓2 (𝑦) sin 𝑘𝑥 𝑥 sin 𝑘𝑧 𝑧
(8.39)
𝐸𝑧 = 𝐸3 𝑓3 (𝑧) sin 𝑘𝑥 𝑥 sin 𝑘𝑦 𝑦
(8.40)
Similarly,
where
𝑘𝑥 =
𝑙𝜋
𝑚𝜋
𝑛𝜋
, 𝑘𝑦 =
, 𝑘𝑧 =
,
𝑎
𝑏
𝑑
𝑙, 𝑚, 𝑛 = 0,1,2, …
(8.41)
𝛁 ⋅ 𝐄 = 0 leads to the relation,
𝐸1
𝑑𝑓1
𝑑𝑓2
𝑑𝑓3
sin 𝑘𝑦 𝑦 sin 𝑘𝑧 𝑧 + 𝐸2
sin 𝑘𝑥 𝑥 sin 𝑘𝑧 𝑧 + 𝐸3
sin 𝑘𝑥 𝑥 sin 𝑘𝑦 𝑦 = 0 (8.42)
𝑑𝑥
𝑑𝑦
𝑑𝑧
This is accomplished if 𝑓1 = cos 𝑘𝑥 𝑥 , 𝑓2 = cos 𝑘𝑦 𝑦 , 𝑓3 = cos 𝑘𝑧 𝑧, and
𝑘𝑥 𝐸1 + 𝑘𝑦 𝐸2 + 𝑘𝑧 𝐸3 = 0
(8.43)
which is just the condition that 𝐤 is perpendicular to 𝐄. The wave equation leads to the discrete
resonant frequencies of the cavity,
𝜔2
𝑙 2 𝑚 2 𝑛2
2
2
2
2
= 𝑘𝑥 + 𝑘𝑦 + 𝑘𝑧 = 𝜋 ( 2 + 2 + 2 )
𝑐2
𝑎
𝑏
𝑑
(8.44)
Assuming 𝑎 < 𝑏 < 𝑑, the lowest mode has 𝑙 = 0, 𝑚 = 1, 𝑛 = 1 (TE011 mode):
𝜋
𝜋
𝐸𝑥 = 𝐸1 sin ( 𝑦) sin ( 𝑧), 𝐸𝑦 = 𝐸𝑧 = 0
𝑏
𝑑
(8.45)
Cavity quality factor Q: power losses in a cavity
We have found that resonant cavities of no loss have discrete resonance frequencies. In
practice there will not be a delta function singularity at resonance frequencies. Because of the
dissipation of energy in the cavity walls and in the dielectric filling the cavity, appreciable
excitations can occur over a narrow band around the resonance frequency. A measure of the
sharpness of response of the cavity to external excitation is the cavity quality factor Q, defined
as 2𝜋 times the ratio of the time-averaged energy stored in the cavity to the energy loss per
cycle:
𝑄 = 2𝜋
𝑈
𝑇0 (−
8
𝑑𝑈
)
𝑑𝑡
=
𝜔0 𝑈
𝑑𝑈
−
𝑑𝑡
(8.46)
where 𝑈 is the stored energy and 𝑇0 is the oscillation period at resonance. Then, we can write
an equation for the behavior of 𝑈 as a function of time:
𝑑𝑈
𝜔0
=−
𝑈
𝑑𝑡
𝑄
(8.47)
with solution
𝑈(𝑡) = 𝑈0 𝑒
𝜔
− 0𝑡
𝑄
(8.48)
If an energy 𝑈0 is initially stored in the cavity, it decays exponentially with a decay constant
𝜔0 /𝑄. This indicates that the oscillations of the fields in the cavity are damping as
𝐸(𝑡) = 𝐸0 𝑒
𝜔
− 0 𝑡 −𝑖(𝜔 +Δ𝜔)𝑡
2𝑄
0
𝑒
,𝑡 > 0
(8.49)
where a shift Δ𝜔 of the resonant frequency is allowed. The spectrum of the oscillating field is
obtained by Fourier transform:
∞
∞
𝜔
1
1
− 0𝑡
(8.50)
𝐸(𝜔) =
∫ 𝐸(𝑡)𝑒 𝑖𝜔𝑡 𝑑𝑡 =
∫ 𝐸0 𝑒 2𝑄 𝑒 𝑖(𝜔−𝜔0 −Δ𝜔)𝑡 𝑑𝑡
√2𝜋 −∞
√2𝜋 0
The integral leads to
𝐸0
1
(8.51)
𝐸(𝜔) =
𝜔
0
√2𝜋 −𝑖(𝜔 − 𝜔0 − Δ𝜔) +
2𝑄
and the energy spectrum
|𝐸0 |2
1
(8.52)
|𝐸(𝜔)|2 =
2
𝜔
2𝜋
(𝜔 − 𝜔0 − Δ𝜔)2 + ( 0 )
2𝑄
The resonance shape, shown in Fig. 8.9, has a full width at half maximum (FWHM)
Γ=
𝜔0
𝑄
(8.53)
Q=
𝜔0
Γ
(8.54)
thus, the Q of cavity can be determined by
Fig 8.9 The resonance line
shape is a Lorentzian.
9
8.5 Optical Fibers
An optical fiber is a dielectric waveguide that operates at optical frequencies. It confines
electromagnetic waves within its surfaces and guides the light in a direction parallel to its axis.
Optical fibers are commonly made of silica (SiO2), which is highly transparent in the optical
band between 0.8 and 1.7 𝜇m.
Fig 8.10 Optical fiber
attenuation spectrum.
Figure 8.10 illustrates the attenuation spectrum for an ensemble of fiber optic cable material
types. The three principal windows of operation, propagation through a cable, are indicated.
These correspond to wavelength regions where attenuation is low and matched to the ability of
a transmitter to generate light efficiently and a receiver to carry out detection. Hence, the
lasers deployed in optical communications typically operate at or around 850 nm (first
window), 1310 nm (second window), and 1550 nm (third and fourth windows).
Fig 8.11 Schematic of an optical
fiber structure.
The basic structure of an optical fiber is shown in Fig 8.11. A circular solid core of refractive
index 𝑛1 is surrounded by a cladding having a slightly lower refractive index 𝑛0 < 𝑛1 :
Δ=
𝑛12 − 𝑛02
𝑛0
≅ 1−
→ 𝑛0 = 𝑛1 (1 − Δ)
2
𝑛1
2𝑛1
10
(8.55)
The parameter Δ is called the core-cladding index difference or simply the index difference.
Values of 𝑛0 are chosen such that Δ is nominally 0.01. Since 𝑛0 < 𝑛1 , electromagnetic energy at
optical frequencies is made to propagate along the fiber waveguide through internal reflection
at the core-cladding interface.
Transmission via optical fibers falls into two classes: multimode or single-mode transmission.
Schematics of single-mode and multimode step-index optical fibers are illustrated in Fig. 8.12. A
few typical sizes of fibers are also given.
Fig 8.12. Single-mode and multimode step-index optical fibers
Single-mode propagation
Consider EM waves propagating along a cylindrical fiber shown in Fig. 8.13. The EM fields of a
monochromatic wave (see Eq. 8.11) are expressed as
𝐄(𝐱, 𝑡) = 𝐄0 (𝑟, 𝜙)𝑒 𝑖(𝑘𝑧−𝜔𝑡)
𝐁(𝐱, 𝑡) = 𝐁0 (𝑟, 𝜙)𝑒 𝑖(𝑘𝑧−𝜔𝑡)
(8.56)
in terms of a cylindrical coordinate system (𝑟, 𝜙, 𝑧). The wave number 𝑘 is the 𝑧-component of
the propagation vector and will be determined by the boundary conditions on the EM fields at
the core cladding interface.
Fig 8.13. Cylindrical coordinate system
used for analyzing EM wave propagation
in an optical fiber.
11
Substituting Eq. 8.56 into Maxwell’s curl equations, we can express the transverse components
in terms of the longitudinal components, 𝐸𝑧 and 𝐵𝑧 (see Eqs. 8.16-8.19):
𝐸𝑟 =
𝐸𝜙 =
𝐵𝑟 =
𝑖
𝜔 2 /𝑣 2
−
−
𝑘2
𝑖
𝜔 2 /𝑣 2
𝑖
𝜔 2 /𝑣 2
𝐵𝜙 =
−
−
(8.57)
𝑘 𝜕𝐸𝑧
𝜕𝐵𝑧
(
−𝜔
)
𝑟 𝜕𝜙
𝜕𝑟
(8.58)
𝜕𝐵𝑧
𝜔 𝜕𝐸𝑧
− 2
)
𝜕𝑟 𝑣 𝑟 𝜕𝜙
(8.59)
𝑘 𝜕𝐵𝑧 𝜔 𝜕𝐸𝑧
(
+
)
𝑟 𝜕𝜙 𝑣 2 𝜕𝑟
(8.60)
(𝑘
𝑘2
𝑖
𝜔 2 /𝑣 2
𝜕𝐸𝑧 𝜔 𝜕𝐵𝑧
+
)
𝜕𝑟
𝑟 𝜕𝜙
(𝑘
𝑘2
𝑘2
where 𝑣 = 𝑐/𝑛 is the speed of light in the dielectric medium of 𝑛. The wave equations of 𝐸𝑧
and 𝐵𝑧 in the cylindrical coordinates are
𝜕 2 𝐸𝑧 1 𝜕𝐸𝑧 1 𝜕 2 𝐸𝑧
𝜔2
+
+
+
(
− 𝑘 2 ) 𝐸𝑧 = 0
𝜕𝑟 2 𝑟 𝜕𝑟 𝑟 2 𝜕𝜙 2
𝑣2
(8.61)
𝜕 2 𝐵𝑧 1 𝜕𝐵𝑧 1 𝜕 2 𝐵𝑧
𝜔2
+
+
+ ( 2 − 𝑘 2 ) 𝐵𝑧 = 0
𝜕𝑟 2 𝑟 𝜕𝑟 𝑟 2 𝜕𝜙 2
𝑣
(8.62)
Fig 8.14. Guided modes of an optical fiber
In contrast to the case of ideal metallic guides, coupling of 𝐸𝑧 and 𝐵𝑧 are in general required by
the boundary conditions. That is, there is no separation into purely TE or TM modes. General
solutions for 𝐸𝑧 and 𝐵𝑧 are expressed as the linear superposition of Bessel functions,
12
𝐴 𝐽𝑙 (𝛾𝑙𝑚 𝑟)𝑒 𝑖𝑙𝜙 , 𝑟 < 𝑎
and
𝐵 𝐾𝑙 (𝛽𝑙𝑚 𝑟)𝑒
𝑖𝑙𝜙
(8.63)
, 𝑟>𝑎
where 𝛾 2 = 𝜔2 /𝑣12 − 𝑘 2 and 𝛽 2 = 𝜔2 /𝑣02 − 𝑘 2 . Matching boundary conditions at 𝜌 = 𝑎 leads
to an eigen value equation for the various modes. The field distributions of several guided
modes of an optical fiber are shown in Fig. 8.14.
Multimode propagation
Since the wavelength of the light, ~1 𝜇m, is significantly smaller than the core diameter (50200 𝜇m) of a multi-mode optical fiber core, the ideas of geometrical optics apply. The interface
between core and cladding can be treated as locally flat. If the angle of incidence 𝜃𝑖 of a ray
𝑛
originating within the core is greater than 𝜃𝑐 = sin−1 (𝑛0 ), the critical angle for total internal
1
reflection, the ray will continue to be confined and to propagate. Propagation occurs for rays
𝜋
𝜋
with 𝜃 = − 𝜃𝑖 < 𝜃𝑚𝑎𝑥 = − 𝜃𝑐 = cos−1 (𝑛0 /𝑛1 ). Typically Δ ≤ 0.01, thus
2
2
𝜃𝑚𝑎𝑥 ≈ √2Δ ≤ 0.14 radian = 8∘
(8.64)
Fig 8.15. Longitudinal section of the core,
showing meriodional propagating rays
with complementary angles of incidence
𝝅
𝜽 < 𝜽𝒎𝒂𝒙 = − 𝜽𝒄 = 𝐜𝐨𝐬 −𝟏 (𝒏𝟎 /𝒏𝟏 ).
𝟐
Number of propagating modes
We can estimate the number of propagating modes using simple phase-space arguments. The
transverse wave number 𝑘⊥ ≈ 𝑘𝜃 is limited by 𝜃 < 𝜃𝑚𝑎𝑥 . 2D phase-space number density is
𝑑𝑁 = 2 ⋅
𝑑𝑆𝑘
𝑑2𝑘
= 2 ⋅ 𝜋𝑎2
1
(2𝜋)2
( 2)
𝜋𝑎
(8.65)
where the factor 2 is for two polarization states, 𝑑𝑆𝑘 is the wave-number area element, and
1/𝜋𝑎2 is the unit area of the phase space which is the inverse of the cross-section area. With
𝑑 2 𝑘 = 2𝜋𝑘⊥ 𝑑𝑘⊥ = 2𝜋𝑘 2 𝜃𝑑𝜃, we have
𝑁 ≈ 𝑎2 𝑘 2 ∫
𝜃𝑚𝑎𝑥
𝜃 𝑑𝜃 ≈
0
1
1
2
(𝑘𝑎√2Δ) = 𝑉 2
2
2
(8.66)
Where 𝑉 ≡ 𝑘𝑎√2Δ, called the fiber parameter. Typical numbers are 𝜆 = 1 𝜇m, 𝑎 =25-100 𝜇m,
𝑛1 ≅ 1.4 (𝑘𝑎 ≈300-600), and Δ ≈ 0.005, leading to 𝑁 ≈450-1800.
13