Download Fiber Bragg Gratings

Properties and implementation of acousto-optic superlattice tunable filters in fibers
Supriyo Sinha and Karel “Double K” Edward Urbanek
Ginzton Labs, Stanford University, Stanford, CA 94305
I.
II.
III.
IV.
V.
VI.
VII.
VIII.
IX.
X.
XI.
XII.
Introduction
Physics of the Acousto-optic Effect
Introduction to Optical Fiber Properties
Fiber Bragg Grating Modelling
Superlattice Theory
Modeling of AOSTFs
Simulation
Past Experimental Results
Comparison with Other Filter Designs
Future Directions
Conclusions
Acknowledgements/References
I. Introduction
The attraction of acousto-optic tunable filters in
fiber lies in the ability to avoid coupling losses
of a comparable waveguide or external device. It
also makes sense to create a device of this sort in
a fiber due to the fact that the efficiency is
significantly improved with smaller crosssection.
There has been little success in
fabricating an all-fiber acousto-optic filter with
narrow linewidth due to the fact that the
linewidth is inversely proportional to the length
of the device, and the acoustic power is
attenuated along the fiber. We will investigate
the physical limits of these devices and show the
promise of a superlattice acousto-optic
modulator.
periodic perturbation. For resonant coupling to
occur, the beat length of the two optical
wavelengths in the fiber must match the acoustic
wavelength.
The photoelastic effect in glass couples the
mechanical strain caused by an acoustic wave to
the optical index of refraction as illustrated in the
following equation
 1
ij    2   pijkl Skl  p 
 n ij
Examining the coefficients of the strain optic
tensor, pijkl, for the material in question yields
insight into the magnitude of the photoelastic
effect. Isotropic fused silica has the following
strain-optic tensor:
0
0
0
 0.121 0.270 0.270

 0.270 0.121 0.270

0
0
0


 0.270 0.270 0.121

0
0
0


0
0
0.12895
0
0
 0

 0

0
0
0
0.12895
0

 0
0
0
0
0
0.12895 

Skl(p) is the strain field from an acoustic wave
and is given by
Skl ( p)  s(ks z  st )  so cos  ks z  st 
II. Physics of the Acousto-optic Effect
An acoustic wave in the fiber core periodically
strains the material, which creates a periodic
change in refractive index. This, of course, can
act like a diffraction grating. If the Bragg
condition is satisfied for a given set of optical
wavelengths and acoustic wavelengths, i.e.:
2
 k   m  l  
  
then coupling occurs between the two optical
waves. Here, the propagation constants of the
kth and mth modes are given by k and m
respectively,  is the acoustic wavelength in the
guide, and l is the Fourier component of the
where (ksz-st) is the optical phase, ks is the
acoustic wave vector,  is the acoustic
wavelength, s is the angular frequency and so is
the peak strain value.
The peak strain caused by a guided acoustic
wave with area, A, power Ps, in a medium with
Young’s modulus E and acoustic group velocity
vgs is solved as
s0   2 Ps / EA gs 
1/ 2
Using equations (??) and (??), and solving for the
index of refraction for light propagating along
the fiber (in the z-direction) gives:
1
nx  n0  n03  0.270  S sin   st  K s z 
2
 x  C
The acousto-optic grating propagates through the
fiber at speed of /K.


 


A  z    t e
i  t    z 
d
0
where ξμt and ξρt are radial transverse field
distributions. All of the modes, both guided and
radiation, are orthogonal.
The transverse
component of the magnetic field is given as
Ht 
 0 r
0
 

ez z  t 
The electric and magnetic fields must satisfy the
wave equation and are subject to the boundary
conditions imposed by the waveguide. The
solution is given by (ref)
 r  cos  
 x  C J     

 a   sin  
H y  neff
0
x
0
in the core where Jμ is the J-Bessel function In
the cladding, the fields take the form
2 a

2 a
2
2
ncore
 nclad
2
n 2  ncore
clad

2  2  2
  n  nclad
neff  nclad b  core
  nclad
1  l 
i  t    z 
A  z   t e
 c.c.



2  1

0

0 x
where Kμ is the K-Bessel function. Several
normalized parameters have been introduced
above which are defined below.
A basic understanding of the modes in an optical
fiber is beneficial to understanding the
superlattice effect. The modes in an optical fiber
consist of guided modes and radiation modes.
An arbitrary electrical field can be decomposed
into a sum of transverse guided mode
amplitudes, Aμ(z), and a continuum of radiation
mode amplitudes Aρ(z) with propagation
constants of βμ and βρ respectively.
The
decomposition is represented below
Et 
 r  cos  
K   

K   
 a   sin  
H y  neff
(here we will show some “back of the envelope”
calculations for various fiber sizes and acoustic
powers)
III. Introduction to Optical Fiber Properties
J 
b
 
  1
 
2
2
The normalization constant Cμ can be solved for
using Poynting vector relationship which states
that the power carried by the mode is |Aμ|2.
1/ 2

0 /  0
2w 
C 


av  neff  e J  1    J  1    
where eμ= 2 for the fundamental mode (μ=0),
otherwise eμ = 1. Matching the fields at the core
and cladding yields the following eigenvalue
equation which can be solved to yield the
propagation constants of the desired mode.

J  1  u 
J  u 
w
K  1  w 
K  w
IV. Fiber Bragg Grating Modeling
Fiber Bragg gratings (FBGs) have been an
enabling technology for several applications in
fiber optics such as distributed feedback lasers
(ref), wavelength filters (ref), pulse compression
(ref) and optical sensors (ref). In the most
general sense, fiber Bragg gratings are
perturbations in the dielectric tensor of the fiber
waveguide that possess some degree of
periodicity. Their fabrication is an extensively
studied area but one that is not overly relevant to
the understanding of the physics of the
superlattices. The interested reader is referred to
??guys?? (ref) for a comprehensive discussion of
the material science and fabrication behind
FBGs.
Several methods have been developed in the late
20th century to accurately describe and model
electromagnetic wave propagation across
periodic structures. The most popular approach
is coupled mode theory (ref), in which the
coupling between the eigenmodes of the
unperturbed waveguide is represented by a set of
differential equations. This method has the
advantage that grating characteristics such as
reflection efficiency and bandwidth can easily be
solved. However, since coupled mode theory
assumes that the eigenmodes do not change in
the presence of the grating, the solution is an
approximation.
A second method uses Bloch wave analysis on
the grating structure allowing for exact solutions,
since Block waves are the natural modes of
periodic media. The Bloch treatment is more
invovled than coupled mode theory, but it is
more appropriate for more complex structures
such as superlattices or when insight into
dispersion or microstructures characteristics is
required. Bloch wave analysis can be extended
to aperiodic structures using numerical
techniques. In addition to coupled mode theory
and Bloch mode theory, other purely numerical
techniques based on matrix methods (Ref,ref) are
often employed in computer algorithms that
break the grating down into suitably small
pieces, model each representative piece with it’s
own matrix and then multiply them all together
to obtain the transfer function of the entire
structure. More physics oriented approaches also
exist such as the WKB (ref), Hamiltonian (ref)
and variational (ref), but they are not commonly
used.
In this paper, we will discuss only coupled mode
theory and Bloch wave theory. The former will
be used throughout the paper to model the basic
characteristics of a simple periodic Bragg
grating. The latter will be used to analyze how
this Bragg grating behaves when modulated by a
coarse grating such as an acoustic wave. This
hybrid approach provides the easiest path to
understanding the physics of superlattices in
fibers.
Coupled Wave Theory
In section ??, we have derived the field
distributions in the fiber. The effects of the
dielectric tensor perturbation on the amplitudes
and phases of the eigenmodes must now be
included. We begin once again with the wave
equation.
2 E
2 P
 E  0 0 2   2
t
t

The response of the polarization in the medium
can be decomposed into an unperturbed term and
a grating term.
P  Punpert  P grating
where
Punpert       E  .
1
Substituting the eigenmodes of the system solved
for in section ?? in the above equation results in
an extremely complicated equation. Neglecting
coupling to radiation modes in the structure and
using
the
slowly
varying
envelope
approximation (SVEA) on the field allows the
wave equation to be simplified to
 

A
 1

z
  i  
 t e 
i t   z 
2


 c.c.  0 2 Pgating ,t
t

Multiplying by the conjugate of the eigenmode
and then using the orthogonality property of
eigenmodes, we arrive at


A it   z 
e
 c.c.
 2i0

z
 1 


can be transferred between the two eigenmodes.
The strength of the interaction is given by the
mode overlap between the refractive index
profile and the mode field which is represented
in eqn ?? as the transverse integral on the right
hand side. This requires that for non-zero
coupling between a symmetric mode and an antisymmetric mode requires a transverse refractive
index profile that is asymmetric. The second
important quantity results from the principle of
conservation of momentum. This dictates that
for efficient energy transfer, the phase constants
must be same between the two coupling modes
on either side of the equation. The resulting
detuning constant, ??delta_b?, given by
 l
  0
2
*
Pgrating ,t t dxdy
t 2
This is a general wave propagation equation
which can be used to describe coupling between
all the guided modes of the fiber, both codirectional and counter-propagating (the signs of
the ??beta?? will be opposite for the eigenmodes
for counter-propagating.
Next, the more specific case of a periodic
modulation of the index of refraction is
examined. The total polarization is given by
      
P     r  1    z   E
2 N

is zero for maximum energy transfer.
Assuming the perturbation of the index of
refraction is small, we can write
Now, we have the tools to examine the coupling
between two identical counterpropagating
modes. This is the situation that we will be
examining in detail in this paper. In this case
(4.2.10) reduces to
  z   2n n  z 
where

B

 v

 n  z   n 1  ei [  2 N /   z   z  ]  c.c. 
 2

z
Substituting 4.2.17 and 4.2.16 into 4.2.14, we
obtain
n i [  2 N /   z   z  ]

Ppert  2n 0  n 
e
 c.c.
2


 E

 i  z   z 
 i dc B  i ac A e 
where we have introduced some new notation to
yield some more physical insight. The derivative
of ??phi versus distance defines a chirp in the
grating. The DC coupling constant ??Kdc??
arises from any change in the average refractive
index of the mode and is given by
 dc  n 0  n t t *dxdy
We can now substitute 4.2.20 into 4.2.13 to
obtain an expression that describes the coupling
between the various eigenmodes in the presence
of a periodic refractive index modulation.
 l

 2i


1

A
0
z
e
i  t    z 

2
 c.c.   0 2 {2n 0  n 
t

n i [  2 N /   z   z  ]
e
 c.c.
2

 E 
 t
*
}dxdy
There are two important quantities in this
expression that determine how efficiently energy
Absorption, gain or scatter loss is incorporated
into the magnitude and sign of the imaginary part
of ??Kdc.
The ac coupling constant is
 ac  n 0 
n

t t *dxdy   dc
2
2
The amplitude of the driving mode can be also
written in a similar manner to obtain the second
of the two coupled differential equations.
A
i  z   z  
*
 i dc A  i ac B e 
z
Solving the coupled equations is a simple
eigenvalue problem. After using the boundary
conditions that there is no driving field incident
from the z = L and no driven field at the input
incident at z = 0 (where L is the size of the
grating), we obtain an amplitude reflection coefficient ρ given as
 ac sinh  L 
S (0)


R(0)  sinh  L   i cosh  L 
where
d  z  
1
   dc    

2
dz 

max  B 1  

n 

n 
where the Bragg wavelength is given the phase
matching condition.


neff ,  neff ,
Coupled mode theory can also yield the grating’s
bandwidth. The bandwidth, defined as the size
of the central lobe of the reflectivity spectrum,
(and found by setting the argument αL in
(4.3.11) equal to π) is solved straightforwardly as
2 
2
 neff L
( ac L) 2   2
   ac   2
2
The power reflection co-efficient is simply the
magnitude squared of 4.3.11
   ac
2
sinh 2  L 
2
 ac cosh 2  L    2
2
When |κac| < δ, amplitude and power reflection
co-efficients become

 ac sin( L)
i cos( L)   sin( L)
   ac
2
2
sin 2  L 
 2   ac cos2  L 
2
For a uniform grating there is no chirp and the
peak reflectivity occurs when δ = 0 and has a
value of
  tanh 2  ac L 
Bloch Wave Analysis
The coupled mode approach is a one step
approach to solving the grating problem. By
contrast, the Bloch method uses two stages.
First, the dispersion relation is solved to obtain
the eigenmodes in the periodic structure.
Second, the eigenmodes that are excited by the
plane wave spectrum are identified.
We begin with Floquet’s theorem which states
that for a chosen forward wave vector kf, only
wave vectors spaced by integral multiples of the
grating vector are permitted as diffracted wave
vectors. For full accuracy, partial waves from all
wave vectors should be included in any analysis
of a periodic medium. However, it can be
assumed that only two of these vectors are
resonantly coupled, those at kf and kf – K. This
assumption breaks down at small Bragg angles
when higher Fourier harmonics can no longer be
neglected, but this case is not considered in this
paper.
2
As we are dealing with identical counterpropagating modes, the transverse integral in
4.3.5 can be assumed be unity, yielding a peak
reflectivity wavelength of
A Bloch wave in a grating structure can be
written as
E ( z, t ) 


1
 ik z
i ( K k f ) z
V f e f  Vbe
eit  c.c.
2
where Vf and Vb are the field amplitudes for the
forward and backward waves respectively. The
partial wave with the larger amplitude is referred
to as the dominant wave vector. Once again we
are analyzing a linear grating with a permittivity
modulation given by
 r  z    r   r cos  Kz 
for
z0
where εr is the relative permittivity of the
unperturbed fiber. Maxwell’s equations are used
with (??2) and (??above equation) to yield a
homogeneous system of equations with
eigenvalues kf and eigenvectors {Vf, Vb}. Using
assumption that kf and ko are close enough in
value to justify following approximation
k 2f  k02  2k0  k f  k0 
we have
  2k0  K 
Thus the solution of (??5) produces
independent Bloch waves for each ξ.
eigenvalues can either be real or complex.
traveling waves, the eigenvalues are real
given by
  V f   0 

     Vb   0 
where coupling constant κ is given as
  Mk0 / 4

M   m   / 1   0 
1
1

where the kf+ eigenvalue corresponds to a
traveling wave with a group velocity in the –z
direction. The evanescent eigenvalues are given
by
     K / 2  j
1 2
where kf+ is the wave decaying in the –z
direction. We have introduced a parameter δ,
defined as
   / 2  4 1  B /   / M
which distinguishes between the traveling wave
and evanescent wave regimes. When δ2 > 1,
traveling waves are obtained; when δ2 > 1,
evanescent waves are obtained.
Below is the dispersion map for the grating.
and dephasing parameter, ξ, is given as
two
The
For
and
k f     k0     K / 2 ( / 2)(1  1/  2 )1/ 2
(kf
 


2n0
4 n
   B   2 0  B   
c
B
Figure 1(a)
When δ > 1, the dominant wave-vector is shorter
than ko, the Bloch wave is called a fast wave and
this case appears on the blue-shifted branch of
the Bragg condition. In this case, the energy in
the optical Bloch wave is redistributed in regions
of low refractive index. Similarly, slow waves in
which dominant wave-vector is larger than ko
appear on the red-shifted branch. For slow
waves δ < 1, and the energy is primarily
distributed in regions of high refractive index.
Solving for the group velocity yields an
interesting physical insight into the Bloch wave.
If we approximate the change in coupling
constant with respect to frequency as zero, we
solve for the group velocity as
1
 k 
g   f  
  
regions in the grating with zero optical energy
density.
Consequently, no power flow is
possible implying that the group velocity must
tend to zero at stop band and the grating traps the
optical energy. Physically, it is clear why the
real component of kf remains unchanged in the
stop band. In the stop band, there is no dominant
wavevector, thus both would have to increase or
decrease equally for any index change
experienced by the Bloch wave. Floquet’s
theorem requires that neither wave-vector’s real
component may change in the stop band, thus
only the imaginary part may change.
Further properties given by Bloch wave analysis
such as higher-order dispersion will not be
discussed here. See (ref??) for a more thorough
treatment of Bloch theory applied to periodic
structures.
 c / n0  1  1/  2 
1/ 2
This implies that the two-waves forming the
Bloch wave are bound together by the grating,
establishing a coherent group of discrete waves
possessing the same group velocity.
The
intensity profile across grating is obtained for the
evanescent and propagating case as
I e  1  cos  Kz  arccos    ,  2  1
I p  1  1/   cos  Kz  , 2  1
respectively. Clearly in the stop band, the fringe
visibility is unity. This means that there are
Superlattice
reflection

sideband:
efficiency
 s


  tanh 2  LJ m 

into
mth
2 Ps
EA gs




Figure shows zero coupling in unperturbed Fiber
Bragg grating in (a). In (b), an acoustic wave
allows coupling between forward and backwards
traveling Bloch waves.
If we look at the dispersion relation, we see the
Bragg condition relates the momentum of the
counter propagating optical waves to the acoustic
wave.
Figure. Frequency wave-vector diagram for a
Bragg grating. In A, a forward traveling acoustic
wave vector couples a forward-traveling Bloch
wave to a downshifted backwards traveling
Bloch wave. In case B, a backwards traveling
Bloch wave couples a forward-traveling Bloch
wave into an anamalously downshifted backward
Bloch wave.