Page 161 - Fiber Bragg Gratings

138 Chapter 4 Theory of Fiber Bragg Gratings

there is no intersection with the outer or inner circles. There is a gap in
the spectrum, in which no phase matching is possible. At some point the
arrowhead meets the vertical dotted line for the radiation modes on line
a, and phase matching to the radiation modes begins. This will couple to
the lowest-order modes. With an infinite cladding, free-space radiation
mode phase matching occurs. As the wavelength becomes even shorter,
the angle of the radiation modes increases, and only when the vector 6
meets the n clad circle is radiation mode coupling at an angle of ff b. After
this point, the angle of the radiation mode increases beyond ff b. We now
note that the change in the mode index is 8n from the RH side of the
figure, so that we can calculate the wavelength at which the radiation
loss starts to occur.
Figure 4.6 shows the phase-matching diagram for coupling to the
guided and radiation modes and fields with a tilted grating, known as
side-tap-grating (STG, also see Chapter 6). This grating has a period
similar to Bragg gratings but does not have its grating planes normal to
the fiber axis, and it is tilted at an angle, 6 g. The diagram specifically
deals with the case of coupling to counterpropagating fields.
In the first interaction with n g, we have Bragg reflection at A Bragg.
We assume that the grating angle d g = 0, and that when the wavelength
s rt
is tuned, the effective index of the mode is n ^ at the point indicated by
a on Fig. 4 6, so that mathematically, this is simply phase matching to a
mode with the cladding index as

where start indicates the wavelength at which the radiation mode coupling
begins.
s
rt
Rearranging and using the approximation n eff ^ n *^ «* n^ agg , it
follows that

Therefore, radiation loss begins at a wavelength slightly shorter than the
Bragg wavelength, governed by the ratio in the brackets in Eq. (4.2.32).
For example, in a fiber with a large core-cladding index difference with
a tightly confined Bragg wavelength (1550 nm) mode (n eff = 1.475), the
start wavelength will be at —1537 nm, some 13 nm away.