2. VERY SIMPLE AND EFFECTIVE METHOD OF OPTICAL FIBRES
SIMULATION IS 2D FREQUENCY-DOMAIN FINITE DIFFERENCE
METHOD (FDM)
FDM REDUCES THE PROBLEM TO EIGENVALUES AND
EIGENVECTORS CALCULATING FOR REGULAR SPARSE
MATRIX OF SIMPLE CONSTRUCTION
GNU OCTAVE CONTAINS THE BUILT-IN EXTREMELY
EFFECTIVE ARPACK SOFT FOR THIS CASE. THESE
CALCULATIONS ARE VERY QUICK!
THE ONLY RESTRICTION IS TO AVOID THE DOUBLE CYCLES
“FOR” WHILE CALCULATING THE 2D REFRACTIVE INDEX
PROFILE MATRIX, WHICH IS EASY ENOUGH IMPLEMENTING
THE “MESHGRID” FUNCTION
4. THIS IS THE STANDARD EIGENVALUE PROBLEM FOR 5-DIAGONAL
SYMMETRIC SPARSE MATRIX M OF DIMENSION D×D, WHERE D = Nx×Ny
REQUIRED SPARSE
MATRIX M IS
REPRESENTED AS
M=Mdiag+Moff1+Moff1'
+Moff2+Moff2'
where
M1=reshape(c,D,1);
Mdiag = sparse(1:D,1:D,M1,D,D)
q=zeros(Nx,Ny); q(1:Nx-1,:)=a; Q=reshape(q,D,1); Moff1= Q(1:D-1);
d=b.*ones(D-Nx,1); Moff2=sparse(Nx+1:D,1:D-Nx,d,D,D);
5. FOR FIRST m MODES, EIGENPAIRS COULD BE FOUND AS FOLLOWS:
[w,s]=eigs(M,m,𝒌 𝟐
𝒏 𝟏
𝟐
),
WHERE n1 IS THE CORE (THE HIGHEST) INDEX FOR STRAIGHT FIBRE
MODES PROPAGATION CONSTANTS ARE
β1 = s(1,1)1/2
, β2 = s(2,2)1/2
, …, βm = s(m,m)1/2
.
EFFECTIVE INDICES ARE
n1,eff = β1/k, n2,eff = β2/k, …, nm,eff = βm/k.
THE FIELD OF THE MODE WITH NUMBER “m” IS
E(m) = reshape(w(:,m),Nx,Ny).
(surf(x,y,E(m)))
6. TYPE OF SMF-28 FIBRE (Δn = 0.005, 2ρ = 8.5 microns,
POWER-LAW CORE PROFILE WITH α = 8)
REFRACTIVE INDEX PROFILE
(USE “SURF” INSTEAD OF “MESH” FOR BETTER QUALITY OF 3D PLOTS)
ELAPSED TIME IS 0.766 SEC (Nx = Ny = 600)
8. SECOND MODE FIELD (λ = 1.2 microns)
ELAPSED TIME IS 23 SEC (Nx = Ny = 600)
9. BEND LOSS
ELAPSED TIME IS ~ 0.8 SEC (Nx = Ny = 600).
BENT FIBRE SHOULD BE DESCRIBED BY SUPERMODES
METHOD [1-3], INTRODUCED IN [4-6] FOR STRAIGHT
ISOTROPIC FIBRES
REFRACTIVE INDEX
PROFILES OF
STRAIGHT
AND
BENT
FIBRE WITH THE
DIAMETER 25 MM
10. ACTUALLY, THE “FUNDAMENTAL MODE” BEND LOSS
IS THE LOSS OF SOME SUPERMODE OF BENT FIBRE
WITH THE FIELD RESEMBLING THE FIELD OF THE
STRAIGHT FIBRE FUNDAMENTAL MODE [1-3].
THIS SUPERMODE MAY BE NOT A FUNDAMENTAL
SUPERMODE OF BENT FIBRE. IT COULD BE REFERRED
AS “FIRST SELECTED SUPERMODE” OR SSM-I [1].
THE SSM-II IS ALSO SHOULD BE INTRODUCED AS THE
ONE WITH PARAMETERS CLOSE TO THAT OF THE
SECOND MODE OF STRAIGHT FIBRE
THE LOSS OF SSM-II YIELDS THE ESTIMATION OF THE
SINGLE MODE REGIME EFFECTIVENESS
11. EIGENVALUE s0(1,1) OF THE FUNDAMENTAL MODE OF
STRAIGHT FIBRE IS THE INITIAL APPROXIMATION FOR SSM-I:
[w,s] = eigs(M,1,s0(1,1)).
IF NORMALISED OVERLAP INTEGRAL OF SSM-I FIELD AND OF
THE FUNDAMENTAL MODE OF STRAIGHT FIBRE IS LARGER
THAN 0.4, THEN SSM-I IS FOUND. OTHERWISE, ONE SHOULD
DIOUBLE THE CALCULATED SUPERMODES NUMBER AND
CALCULATE
[w,s] = eigs(M,2,s0(1,1)),
[w,s] = eigs(M,4,s0(1,1)),
[w,s] = eigs(M,8,s0(1,1)),
…
UNTIL THE NORMALISED OVERLAP INTEGRAL WITH ONE OF
THE CALCULATED SUPERMODES (SSM-I) WILL BE LARGER
THAN 0.4. THE SAME IS RIGHT FOR SECOND MODE OF
STRAIGHT FIBRE AND FOR CORRESPONDING SSM-II.
12. FOR LIGHT ABSORPTION, A COMPLEX INDEX RING IS
INTRODUCED IN THE CALADDING (THUS, SSM-I AND SSM-II
HAVE COMPLEX PARAMETERS)
2D PROFILE WITH ABSORBING RING, STRAIGHT FIBRE
(LEFT) AND BENT FIBRE (RIGHT)
Im VALUE IS TAKEN TOO LARGE HERE FOR ILLUSTRATIVE
PURPOSES. REAL PART OF REFRACTIVE INDEX PROFILE IS
MODIFIED AS Re2
n – Im2
n [1, 5]
13. WHAT IS THE VALUE OF IMAGINARY PART OF INDEX (Im) TO BE
TAKEN FOR SIMULATIONS? IT SHOULD BE FOUND BY
CALCUATIONS, CHECKING SEVERAL VALUES OF Im
BEND LOSS GRAPH OF FUNDAMENTAL MODE DEPENDING ON Im
FOR 10-MM DIAMETER BEND AND 15-MICRON ABSORBING LAYER
THUS, BEND LOSS IS ~ 24 DB/TURN, WHICH CORRESPONDS WELL
TO EXPERIMENTAL DATA FROM LITERATURE
MAXIMAL LOSS
SHOULD BE TAKEN
Nx = Ny = 600 (LOSS =
23.858 DB/TURN)
Nx = Ny = 300 (LOSS =
23.447 DB/TURN)
0.001 Im 0.01
100
10
14. THE SSM-I FIELD OF BENT FIBRE
ELAPSED TIME IS 61.75 SEC (Nx = Ny = 600)
RIPPLE IS THE ENERGY LEAKAGE INTO THE ABSORBING LAYER
DUE TO THE FIBRE BENDING LEADING TO BEND LOSS
15. THE SAME PROCEDURE IS VALUABLE FOR SSM-II LOSS UNDER
CUTOFF. IT AGREES WELL WITH STANDARD CUTOFF
WAVELENGTH ESTIMATION
λ=1.2 MICRONS (LOSS < 0.01 DB/M) AND λ=1.27 micron (25 DB/M)
ELAPSED TIME IS 90.7 SEC ELAPSED TIME IS 119.134 SEC
(Nx = Ny = 600)
17. BEND LOSS DEPENDENCE ON FIBRE ORIENTATION WITH
RESPECT TO THE BENDING PLANE
θ (DEG)
0 20 40 60 80 90
25
20
15
10
5
0
18. SECOND MODE FIELD SQUEZZED BY THE STRESS RODS
LOSS = 5.3 DB/M
STRESS RODS
19. THIRD MODE FIELD SQUEZZED BY THE STRESS RODS
LOSS = 53 DB/M
STRESS RODS
20. THE SIMPLEST MODEL OF REFRACTIVE INDEX MODIFICATION
BY STRESS FIELDS IN PANDA FIBRES IS DESCRIBED IN [3]
δnx δny
THESE PROFILES ARE ADDED TO INITIAL INDEX PROFILE
YIELDING TWO REFRACTIVE INDEX PROFILES FOR X- AND Y-
POLARISED LIGHT. X- AND Y-POLARISED MODES CALCULATION
METHODS IN THIS CASE ARE COMPLETELY THE SAME AS ABOVE
ELAPSED TIME IS 8.7 SEC (Nx = Ny = 600)
22. THE ABOVE METHOD IS UNIVERSAL, BECAUSE IT IS EASILY
APPLICAPLE IN THE SAME FASCION TO THE FIBRES WITH
DIP IN THE CORE, OR TO MORE COMPLICATED
REFRACTIVE INDEX PROFILES, SUCH AS DEPRESSED-CLAD
(W) FIBRES [1-6], DOUBLE DEPRESSED CLAD FIBRES [1],
TRENCH FIBRES, ISOTROPIC OR ANISOTROPIC, AN SO ON.
THE ONLY DIFFERENCE BETWEEN ALL THESE CASES IS
THE REFRACTIVE INDEX PROFILE MATRIX.
23. REFERENCES
[1] KURBATOV A.M., KURBATOV R.A. OPTICAL FIBER TECHNOLOGY, 32, 2016
(http://www.slideshare.net/KurbatovRoman/polarisation-maintaining-fibre-with-pure-silica-
core-and-two-depressed-claddings-for-fibre-optic-gyroscope )
[2] KURBATOV A.M., KURBATOV R.A. OPTICAL ENGINEERING, 52(3), 2013
(http://opticalengineering.spiedigitallibrary.org/article.aspx?articleid=1667262)
[3] KURBATOV A.M., KURBATOV R.A. OPTICAL AND QUANTUM ELECTRONICS, 48,
PAPER 439, 2016
(http://www.slideshare.net/KurbatovRoman/comparative-theoretical-study-of-polarising-
panda-type-and-microstructured-fibres-for-fibreoptic-gyroscope )
[4] FRANCOIS P.L., VASSALLO C. APPLIED OPTICS, 22, 1983
[5] HENRY W.M., LOVE J.D. OPTICAL AND QUANTUM ELECTRONICS, 25, 1993
[6] BESLEY J., LOVE J.D. PROC. IEE, 144, 1997
KURBATOV ROMAN, EMAIL: ROMUALD75@MAIL.RU
