Long-Period Fiber Gratings as Band-Rejection Filters

βcl 2 + κ2 = ω2 n 2 cl
(2)
c 2
where n cl is the refractive index of the cladding. Assuming a
simple step-index profile, the axially symmetric cladding
modes that are nonzero at the origin are described by Bessel
function of order zero, namely, J 0 (κ a cl ) where a cl is the
cladding radius. The difference in the wavelengths at which
the guided mode couples to the different cladding modes, δλ,
correspond to the separation of the zeros of the Bessel function
[13]. Since the difference in the zeroes of J 0 (x), denoted by
δx 0 , can be approximated by δx 0 ≈ π [14], the δλ’s will correspond
to the condition κ a cl = πp where p is an integer. The
expression for δλ is obtained in an indirect manner, as follows.
We first find the separation in wavelength between the
pth-cladding mode and λ cut . Using (1) and (2), we arrive at the
expression
λ p λ 2 cut
(λ p − λ cut ) ≈
8n cl (n eff − n cl ) . p2
a 2 (3)
cl
where n eff is the effective index of the guided LP 01 mode
(β 01 = 2πn eff /λ). In deriving (3), we have assumed that the
effective index of the fundamental mode at λ p and λ cut is the
same. For an AT&T DSF, this approximation leads to an error
< 2%. For the first few cladding modes one can further simplify
the expression by assuming that λ p and λ cut are in close
proximity. The wavelength separation between the pth and the
(p + 1)th-mode can then be approximated by
λ 3 cut + 1)
δλ p, p+1 ≈
.(2p
8n cl (n eff − n cl ) a 2 . (4)
cl
As an example, we consider a DSF with a long-period grating
with the following properties: = 550 μm, λ cut = 1.41 μm
and predict the wavelength separation between the first two
cladding modes δλ 1 to be 65 nm. Our experiments show that
the separation of wavelengths between the first and second
modes, δλ 1 for the above example is 57 nm.
The spectral dependence of the grating transmission can be
approximately determined by using expressions derived in the
literature [15] for codirectional mode-coupling. The ratio of
power coupled into the nth-cladding mode to the initial power
contained in the guided LP 01 [ mode √ is then given ] by [16]
( ) 2
sin 2 κ
P (n)
g L 1 + δ
κ
cl
(L)
g
P 01 (0) =
where δ is the detuning parameter
δ = 1 2
( ) 2
(5a)
1 + δ
κ g
{
β 01 − β (n)
cl
− 2π }
(5b)
κ g is the coupling constant for the grating and L is the grating
length. The coupling constant κ g is proportional to the UVinduced
index change and is typically increased to maximize the