10.6 Arrayed Waveguide Devices
425
• The lens distributes the entering optical power among the different waveguides
in the grating array in region 3.
• Adjacent waveguides of the grating array in region 3 differ in path length by a
precise length L. The path length differences L can be chosen such that all
input wavelengths emerge at point 4 with different phase delays = 2πn c L/λ.
• The second lens in region 5 refocuses the light from all the grating array
waveguides onto the output slab waveguide array in region 6.
• Thus each wavelength is focused into a different output waveguide in region
From the phase-matching condition, the light emitted from the output channel
waveguides must satisfy the grating equation
n s d sin θ + n c L = mλ
(10.59)
where the integer m is the diffraction order of the grating.
Focusing is achieved by making the path length difference L between adjacent
array waveguides, measured inside the array, to be an integer multiple of the central
design wavelength of the demultiplexer
L = m
λ c
n c
(10.60)
where λ c is the central wavelength in vacuum; that is, it is defined as the pass
wavelength for the path from the center input waveguide to the center output
waveguide.
To determine the channel spacing, it is necessary to find the angular dispersion.
This is defined as the incremental lateral displacement of the focal spot along the
image plane per unit frequency change, and is found by differentiating Eq. (10.59)
with respect to frequency. Doing so, and considering the result in the vicinity of θ =
0, yields
dθ
dv
= −
mλ
2
n s cd
n g
n c
(10.61)
where the group index of the grating array waveguide is defined as
n g = n c − λ
dn c
dλ
(10.62)
In terms of frequency, the channel spacing ν is
v =
x
L f
dθ
dv
−1
=
x
L f
n s cd
mλ 2
n c
n g
(10.63)
or, in terms of wavelength,
425
• The lens distributes the entering optical power among the different waveguides
in the grating array in region 3.
• Adjacent waveguides of the grating array in region 3 differ in path length by a
precise length L. The path length differences L can be chosen such that all
input wavelengths emerge at point 4 with different phase delays = 2πn c L/λ.
• The second lens in region 5 refocuses the light from all the grating array
waveguides onto the output slab waveguide array in region 6.
• Thus each wavelength is focused into a different output waveguide in region
From the phase-matching condition, the light emitted from the output channel
waveguides must satisfy the grating equation
n s d sin θ + n c L = mλ
(10.59)
where the integer m is the diffraction order of the grating.
Focusing is achieved by making the path length difference L between adjacent
array waveguides, measured inside the array, to be an integer multiple of the central
design wavelength of the demultiplexer
L = m
λ c
n c
(10.60)
where λ c is the central wavelength in vacuum; that is, it is defined as the pass
wavelength for the path from the center input waveguide to the center output
waveguide.
To determine the channel spacing, it is necessary to find the angular dispersion.
This is defined as the incremental lateral displacement of the focal spot along the
image plane per unit frequency change, and is found by differentiating Eq. (10.59)
with respect to frequency. Doing so, and considering the result in the vicinity of θ =
0, yields
dθ
dv
= −
mλ
2
n s cd
n g
n c
(10.61)
where the group index of the grating array waveguide is defined as
n g = n c − λ
dn c
dλ
(10.62)
In terms of frequency, the channel spacing ν is
v =
x
L f
dθ
dv
−1
=
x
L f
n s cd
mλ 2
n c
n g
(10.63)
or, in terms of wavelength,