Carleton University

1
Week 2 topics.

• Review of optical wave propagation in basic
  dielectric guiding structures planar, channel on
  flat substrates, and circular/elliptical., mode
  profiles, mode orthogonality, polarization,
  confinement.
• In the following references are made to the
  textbooks of Agrawal (Ag) and Okamoto (Ok)

2
Waveguide modes, summary

• Electromagnetic plane waves in infinite media
  (Ag1.3.1 and Ok1.3-1.4)
• Ray optics description of waveguiding (Ag1.1.1
  and Ok1.1-1.2)
• Confinement through total internal reflection
• Requirement for periodic behavior (i.e.
  propagating wave) along waveguide axis imposes
  restrictions on allowed propagation angles for
  rays inside waveguide.
• Wave optics description of waveguiding (Ag4.1
  and Ok2.1)
• Slab waveguide confined in x, infinite in y and z
• If the field oscillates in time at a fixed
  frequency w/2p,
• can we find wave-like solutions along z of the
  form exp j(wt-bz)
• Allows Maxwells equations to be decoupled into
• TE modes where only Ey,Hx, Hz are non-zero
• TM modes where only Hy,Ex, Ez are non-zero
• Careful with neglecting terms in ?n2 while
  deriving the wave equation.
• Find general solutions for waves in each of the
  regions forming the waveguide
• Impose boundary conditions Fields must decay to
  zero at infinite distance from core, tangential
  fields must be continuous at all interfaces.
• This yields a transcendental equation for b, with
  m (0,1,2,) as a parameter, as a function of the
  waveguide structure (sizes, refractive indices).
• Replacing b in the general solutions for each
  region fully specifies the fields, up to one
  constant parameter which must be determined by
  comparison with the total power carried by the
  mode.

3
Plane waves in infinite media

• Monochromatic plane waves in space have time
  varying electric and magnetic fields with a
  temporal dependence of the form exp j(wt)
• And a spatial dependence as follows
• exp j(kr)
• E and H fields are vectors aligned somewhere in
  the plane perpendicular to the direction of k
• Where w is the radial frequency (2pf) and k is
  the wavevector (whose magnitude is
  kwn/cnk02pn/l).
• All the fields components are related by
  Maxwells equations
• Such waves are non-physical because they carry
  infinite power.

4
Photonic components

• Need to confine and direct optical energy in
  desired direction(s)
• Need to control the amplitude and phase of
  optical waves in order to realise certain
  functions.
• ? Light pipes
• Metal walls are lossy over long distances (10000
  bounces/cm at 200 GHz in single mode tube)
• Guiding can be obtained in dielectric structures
  with minimal loss (records around 0.1 dB/km of
  loss)
• Mathematical description of guidance will be made
  in 2D for clarity (real 3D structures are just
  mathematically more complex).

5
Snells Law of refraction

• n0 sin f0 n1 sinf1
• When f0 reaches 90o, then f1 is the critical
  angle for total internal reflection and is equal
  to
• fc sin-1(n0/n1)
• If you have managed to get light in a core of
  higher index n1 than its surrounding, light can
  propagate inside the core as long as f1 gt fc

f0
n0
n1
f1
n0
6
Light guidance by total internal refraction in
dielectric structure

• The maximum value of ? for TIR in the core is
  such that
• sin ? (n12 n02)½ n1 (2D)½ ? for low D
  cores
• The parameter (sin ?) is called the numerical
  aperture.
• is a measure of guidance (n12 n02)/(2n12) ?
  (n1 n0)/n1
• For telecom SMF, the NA is 0.13 and D is 0.4

7
But what is a guided mode?

• It is an optical wave that propagates down the
  axis of a structure with a definite periodicity
  and a finite amount of power.
• As a result
• It has a well defined propagation constant in one
  direction
• It has a well-defined (invariant) intensity
  pattern in the plane perpendicular to the
  direction of propagation
• (constant at least over a few wavelengths
  distance)

8
Ray optic approach to waveguide resonanceimagine
a truncated plane wave bouncing inside a high
index dielectric through TIR
Phase fronts must be continuous while bouncing up
and down This imposes an additional condition on
the allowable ray angles
9
Matching the phases for the two paths
This ensures that the waves are in phase as they
propagate along z. The phase shift F appearing
in the equation is occurring because the wave is
totally reflected. It is called a GOOS-HÄNCHEN
shift and it depends on the polarization of the
light relative to the plane of incidence.
Only those ray angles that satisfy this equation
are allowed inside the guide. For a given wg (a,
n1,n0) and wavelength (l 2p/k), there is only
a fixed number of solutions (one per value of m).
NB here k k0
10
Goos-Hänchen shift upon TIR

• For light polarized perpendicular to the plane of
  incidence (TE), the amplitude reflection
  coefficient is complex
• r (n1sin? j(n12 cos2? n02))/ (n1sin? -
  j(n12 cos2? n02))
• r exp(-jF) (i.e. total reflection but with a
  phase shift)
• FTE -2 tan-1(2D/sin2? 1)½
• For TM polarization, the shift is different
• FTM -2 tan-1((n1/n0)2(2D/sin2? 1)½)

NB GH shifts are usually defined as half those
given here
11
TE and TM modes

• Only exactly true for 2D geometries (see next
  lecture)
• TE there are NO Electric field components of
  the guided wave along the mode propagation
  direction (z). For the slab described above the
  only electric field component is along the y
  axis it is tangential and must be continuous.
• TM there are NO Magnetic field components of the
  guided wave along the mode propagation direction
  (z). For the slab described above the only
  magnetic field component is along the y axis it
  is tangential and must be continuous.
• TE and TM waves are intuitively obvious from the
  ray picture.

12
Matching the phases for the two paths
Only those ray angles that satisfy this equation
are allowed inside the guide. For a given
waveguide (a, n1,n0) and wavelength (l 2p/k),
there is only a fixed number of solutions (one
per value of m). Recall that F depends on
polarization, hence solutions are different for
TE and TM
NB here k k0
13
Definition of the mode propagation constant or
effective index b
x
kn1k0
p
z
?
b Neffk0((n1k0)2 p2)½
By definition, in the waveguide core, b is the
projection along z of the wave-vector and p is
the projection along x. Since the allowed angles
form a discrete set, so do the effective indices
these are waveguide modes. Obviously, the
maximum possible value of Neff is n1,
14
What about the fields outside the core?
kn0k0
x
b
kn1k0
z
p
?
b
Tangential electric fields must be continuous at
dielectric interfaces. Therefore, the fields
located outside the core must have the same z
dependence or the same b. When TIR occurs, b is
larger than k0n0, (check with Snells
law) therefore there is no propagating wave
outside the core (no possible wave-vector
direction whose projection along z would equal b).
15
What happens when ? increases to the point where
TIR is lost ?
x
kn0k0
b
z
kn1k0
p
?
b
At some point, b becomes smaller than k0n0, and a
real wave can propagate. Light is no longer
confined to the waveguide. The mathematical
minimum value of Neff for guided waves is n0
16
Normalized parameters to simplify mathematics
n0 or nc

• Define the normalized frequency
• V ak0(n12-n02)½
• The normalized effective index
• b (Neff2 n02)/(n12 n02)
• ? b is bound between 0 and 1
• The asymmetry parameter d
• d (n02 nc2)/(n12 n02) (with nc lt n0 lt n1)
• This is the more general case where the core is
  surrounded by 2 different materials.
• d 0 when the waveguide is symmetric.

2a
n1
n0
17
Re-writing the dispersion equation in terms of
the normalized parameters instead of angles (Agr.
p.147)

• TE modes (E field perpendicular to plane of
  incidence)
• 2V(1-b)½ mp tan-1(b/(1-b))½
  tan-1((bd)/(1-b))½
• TM modes (H field perpendicular to plane of
  incidence)
• 2V(1-b)½ mp tan-1((n12/n02)(b/(1-b))½)
  tan-1((n12/nc2)(bd)/(1-b))½)
• In both cases, fix a, l, n1, n0, nc and solve for
  b.

18
Slab waveguide dispersion diagram (TE)
Agrawal p. 148)
19
Mode Cut-off frequency (or wavelength)

• From the diagram
• For given V, Neff decreases with increasing m
  (rays become less grazing at the core cladding
  interface).
• The number of allowed modes increases with V.
• A mode of index m becomes cut-off (light
  escapes into cladding) when b 0.
• Cut-off frequency Vm(TE) mp/2 ½ tan-1d½
• Cut-off frequency Vm(TM) mp/2 ½
  tan-1((n12/nc2)d½)

20
Allowed values of effective indices
n1
m0
m1
m2
ns
nc
x
For a mode to be confined to the core layer
without power leakage outside, the effective
index must be larger than the refractive indices
in those layers. In addition, the effective index
cannot be larger than the refractive index of
the core. This is also true in channel
waveguides and fibers.
21
Constructing the Field shape of modes
Field amplitudes
E -1 E 1
Okamoto page 5
Approximate picture Goos-Hänchen shift not
pictured.
22
Resulting Field shape of modes
ns
nc
ncore
Okamoto p. 24
Note the asymmetry of the field shape when the
substrate and cover have different refractive
indices. Ray picture breaks down. Need wave
optics calculations
23
Experimentally measured slab mode profiles (1)
24
Experimentally measured slab mode profiles (2)
25
What about more complicated guiding structures?
Index
Ray optics approach is difficult to apply here.
Much easier to use wave optics and Maxwells
Equations solutions.
Transverse Position
Note this waveguide profile includes
fabrication tolerances?