Long-Period Fiber Gratings as Band-Rejection Filters

Principle of Operation
Consider a single-mode fiber with the propagation constant of
the fundamental mode, LP 01 , denoted by β 01 and the propagation
constants of the cladding modes given by β (n)
cl , where
the superscript denotes the order of the mode. The relative
positions of the propagation constants (for a given
ω, ω = ω 1 ) are shown in Figure 1. The hatched region
extending from 0 to n cl ω/c represents the continuum of radiation
modes that exist for an infinite cladding. We restrict this
analysis to a purely rectangular index modulation along the
fiber with the periodic index structure being perpendicular to
the fiber axis. These assumptions exclude blazed gratings and
as a result modal overlap conditions dictate that the fundamental
guided mode can couple only to those cladding modes
that are azimuthally symmetric with a central peak at r = 0.
The ordinals of β (n)
cl reflect this assumption.
In order to couple from a forward propagating guided
mode to backward propagating (guided or radiation) modes
the phase matching K-vector is large, thus requiring a shortperiod
grating. Examples of these gratings are Bragg-reflectors
(for coupling to a back-propagating guided mode, denoted
by −β 01 ) or blazed/tilted gratings (for coupling to backpropagating
radiation modes).
The phase matching condition between the guided mode
and the forward propagating cladding modes is given by
β 01 − β (n)
cl
= 2π
(1)
fiber at a specific wavelength. We then obtain a set of periodicities
(n) that will meet the phase-matching condition given
by (1). This step is then repeated for several different wavelengths;
the resulting plot of coupling-wavelength versus
grating-period is shown for an AT&T dispersion-shifted fiber
(DSF) in Figure 2. One can choose a grating period such that
mode coupling takes place at any desired wavelength. Further,
the choice of ’s also allows the designer to vary the separation
between two cladding modes, denoted by δλ.
The AT&T DSF was modeled by using an exact refractive
index profile in the mode-parameter calculation program
developed by Lenahan [10]. The method of solution
involved a finite element approach that reduced Maxwell’s
equations to standard eigenvalue equations and an eigensystem
package (EISPACK) computed the desired propagation
parameters for the guided mode. The cladding mode propagation
constants were calculated using the eigenvalue equa-
Broadband
Source
PC
Fiber
KrF Laser
Fiber
Optical
Spectrum
Analyzer
Figure 3: Experimental setup for long-period grating fabrication.
AM: Amplitude mask. PC: Polarization controller.
AM
where is the grating periodicity required to couple
the fundamental mode to the nth-cladding mode. In
this case, the phase matching vector is short resulting
in a long , typically on the order of hundreds of
microns. For another ω, (ω = ω 2 where ω 2 ω 1 ) one
can visualize the K-vector of the grating coupling the
guided mode to the edge of the radiation mode continuum.
This optical frequency ω cut corresponds to a
minimum wavelength, λ cut , at which the grating can
couple the fundamental mode to the radiation
modes. This parameter is used to describe grating
behavior in the subsequent analysis.
Theory
One can now predict the wavelengths at which
mode-coupling will be enabled by a particular grating
period. The first step in our modeling approach
involves the calculation of the propagation constants
of the guided and the various cladding modes of a