21-06-2012, 02:39 PM
Theory of Fiber Optical Bragg Grating- Revisited
Theory of Fiber Optical Bragg Grating- Revisited.pdf (Size: 84.47 KB / Downloads: 63)
INTRODUCTION
Fiber Bragg gratings represent an important element in the emerging fields of optical communications and optical
sensing. Despite its vast usefulness, the device is comparatively simple. Typically a dielectric cylinder of index n1,
usually referred as core, is surrounded by a concentric dielectric cylinder of index n2. The two refractive indices obey the
relation n1 > n2. In such an arrangement, since the field decays exponentially inside region of index n2, practically no
field exists outside of region 2. In its simplest form a fiber Bragg grating consists of a periodic modulation of the
refractive index in a core of a single mode optical fiber, where the phase fronts are perpendicular to the fiber’s
longitudinal axis and with grating planes having a constant period. Because of its intrinsic simple physical nature, the
theory behind Bragg grating is equally simple. Light, guided along the core of an optical fiber, is scattered by each
grating plane. If the Bragg condition is satisfied, the contributions of reflected light from each grating plane add
constructively in the backward direction to form a back reflected peak with center wavelength defined by the grating
period. A considerable amount of theoretical work1-11 has been reported with various approaches giving reasonable
results in predicting the reflectivity as a function of wavelength. The theoretical works fall into two categories, the matrix
transfer function method and the coupled mode theory. In the matrix transfer function, the Bragg grating is simulated as
an alternating stratified medium having index of refraction of n and n+Δn and weak guiding approximation is assumed,
i.e., the difference of index refraction between the core and cladding is ignored, which some times is called the scalar
wave approximation. In this approximation, the modes are transverse. When the grating is periodic, a closed solution for
the reflectivity as a function of scanning wavelength is obtained. However when the grating periodicity is slightly off as
happens in a non-uniform strain measurement, the reflected spectrum can only be obtained numerically12. The coupled
mode method assumes the grating serves as a perturbation which couples power between forward and backward moving
modes with a x (optical axis) dependence amplitude. This method also assumes that the amplitudes do not change
abruptly and the second derivative of amplitudes with respect to x are dropped resulting in the amplitudes being
represented by a first order linear differential equation with constant coefficients. Both approaches provide reasonable
results.
EVALUATION OF REFLECTIVITY USING TRANSFER MATRIX METHOD
Following references13-14closely, we outline the simple results of the expressions for light propagating in a Bragg grating.
Under the assumption of a scalar wave approximation (weakly guiding approximation) the electric and magnetic field
amplitudes are assumed to be perpendicular to the propagation direction and both satisfy the simple plane wave equation.
When the plane wave travels through regions of different index of refraction, the wave number adopts the local index and
part of the wave gets reflected back with different amplitude.
NON-UNIFORM GRATING
Non-uniform grating can exist either by design or arise from circumstance, such as in a structure non-uniform strain
measurement. Since the “period” has lost its meaning, a compact expression for reflectivity can not be obtained. On the
other hand, one can easily carry out the transfer function manipulation grating by grating by numerical procedure. As
previously reported, a pre-described non-uniform strain can be suggested, including linear, sinusoidal and random
distributions. Interesting results are obtained depending upon the regime of strain. The simulations indicate that for nonuniform
strain, the multi-peak spectra occur when the strain has reached the order of 10-2 or greater. If the strain is small,
say less than 10-3, non-uniformity is not an important issue; all the reflected spectra would give a sharp peak and
uniquely determine the strain. However when the strain increases to the order of 10-2, the spectrum is broadened and
splits into multiple peaks. Finally, when the strain increases beyond 10-2 for a non-uniform grating, the reflected signals
can be completely lost, which has been observed in some experiments19. These phenomena can be understood in a
qualitative sense, i.e., each grating plane defines a reflected and selected wavelength. And when the grating distance is
constant, all the reflected waves contribute constructively, creating a strong peak uniquely defining the grating distance.
But when the successive grating distances are off slightly, each distance selects a slightly shifted wavelength. Therefore
each back-scattered wave contributes non-coherently and multi-peak spectra are produced.
CONCLUSION
We have revisited the transfer matrix formulation once again, but also have gained some new understanding which was
not well known before: (1) The transfer matrix is easy to implement and involves less approximations and in our opinion
is more physically transparent than the coupled mode theory. (2) From the mathematical point of view, we understand
how the maximum reflectivity is obtained; it is not the individual layer that produces the phase change of π/2, equivalent
to say the individual layer width equal to λ/4n’ or λ/4n. The important fact is the sum of the phase changes in the
individual double layer has to be π or a multiple of π, and each layer can be slightly off from π/2. The reflectivity will
decrease if the phase shift k’r is moving away from odd multiple of π/2 because of the quantity sin(k’r) in Eq. (16). To
gain maximum reflectivity, each layer must have a phase path length of odd multiples of π/2 which has some practical
implication, namely we can inscribe less points on the fiber to achieve the same results. For example, in the n region we
can set l=3λ0/4n. This has been demonstrated in numerical simulation. (4) We have gained the understanding of the role
of number of double layers N.