Method and apparatus for modeling the modal properties of optical waveguides
View Patent ↗A method and apparatus models one or more electromagnetic field modes of a waveguide. The method includes calculating a first matrix having a plurality of elements and having a first bandwidth using a refractive index profile of the waveguide. The plurality of elements of the first matrix represents an action of Maxwell's equations on a transverse magnetic field within the waveguide. The method further includes rearranging the plurality of elements of the first matrix to form a second matrix having a second bandwidth smaller than the first bandwidth. The method further includes shifting the second matrix and inverting the shifted second matrix to form a third matrix. The method further includes calculating one or more eigenvalues or eigenvectors of the third matrix corresponding to one or more modes of the waveguide.
1. A method for modeling one or more electromagnetic field modes of a waveguide, the method comprising:
calculating, using a processor, a first matrix using a refractive index profile of the waveguide, the first matrix comprising a plurality of elements and having a first bandwidth, the plurality of elements of the first matrix stored in computer memory and representing an action of Maxwell's equations on a transverse magnetic field within the waveguide;
rearranging the plurality of elements of the first matrix stored in computer memory to form a second matrix having a second bandwidth smaller than the first bandwidth;
shifting the second matrix and inverting the shifted second matrix to form a third matrix;
calculating one or more eigenvalues or eigenvectors of the third matrix corresponding to one or more modes of the waveguide; and
calculating one or more propagation losses corresponding to the one or more modes of the waveguide using boundary conditions which comprise surrounding a simulation domain of the method by an absorbing material.
2. The method of claim 1 , wherein the waveguide comprises a photonic-bandgap fiber.
3. The method of claim 1 , wherein the waveguide comprises an air-core photonic-bandgap fiber.
4. The method of claim 1 , wherein the refractive index profile is translation invariant along a longitudinal axis of the waveguide.
5. The method of claim 1 , wherein the refractive index profile is digitized.
6. The method of claim 1 , wherein the waveguide comprises an air-core photonic-bandgap fiber having a longitudinal axis, an air core and a cladding structure comprising air holes and intervening membranes, and the refractive index profile has a digitized air core and cladding structure in a planar cross-section which is perpendicular to the longitudinal axis.
7. The method of claim 6 , wherein the refractive index profile is sampled over an area corresponding to a minimum cell that is a fundamental component of the cladding structure.
8. The method of claim 1 , wherein calculating the first matrix comprises defining boundary conditions.
9. The method of claim 1 , wherein calculating the first matrix comprises discretizing an eigenvalue equation satisfied by the transverse magnetic field, the discretizing being performed using Maxwell's equations as expressed by
∇
→
×
E
→
=
-
∂
B
→
∂
t
,
∇
→
×
H
→
=
∂
D
→
∂
t
,
and
∇
→
·
H
→
=
0.
10. The method of claim 9 , wherein discretizing the eigenvalue equation further comprises index-averaging over each discretization pixel straddling an air-core boundary.
11. The method of claim 9 , wherein the waveguide comprises an air-core photonic-bandgap fiber having a longitudinal axis, an air core and a cladding structure comprising air holes and intervening membranes, and wherein discretizing the eigenvalue equation comprises sampling two components of the transverse magnetic field in a planar cross-section of the waveguide.
12. The method of claim 11 , wherein sampling the two components of the transverse magnetic field is performed over an area corresponding to a minimum cell that is a fundamental component of the cladding structure.
13. The method of claim 1 , wherein calculating the first matrix comprises sampling a linear operator corresponding to the action of Maxwell's equations on a transverse magnetic field within the waveguide.
14. The method of claim 1 , wherein inverting the shifted second matrix comprises using an LU decomposition.
15. The method of claim 1 , wherein calculating one or more eigenvalues or eigenvectors of the third matrix comprises calculating only a selected number of the largest eigenvalues of the third matrix.
16. The method of claim 1 , wherein calculating one or more eigenvalues or eigenvectors of the third matrix comprises using finite-difference or finite-element calculations.
17. The method of claim 1 , wherein the waveguide comprises a photonic-bandgap fiber having a bandgap and calculating one or more eigenvalues or eigenvectors of the third matrix comprises finding one or more modes of the photonic-bandgap fiber, the one or more modes comprising at least one core mode, surface mode, ring mode, or bulk mode.
18. The method of claim 1 , wherein the first matrix comprises a first diagonal of elements each having its two indices equal to one another, and wherein the plurality of elements comprises non-zero elements having non-zero values, the first bandwidth being a maximum distance between the non-zero elements and the first diagonal of the first matrix.
19. The method of claim 1 , further comprising displaying information regarding the one or more propagation losses to model the one or more electromagnetic field modes of the waveguide.
20. A non-transitory tangible computer-readable medium having instructions stored thereon which cause a general-purpose computer to perform a method for modeling one or more electromagnetic field modes of a waveguide, the method comprising:
calculating a first matrix using a refractive index profile of the waveguide, the first matrix comprising a plurality of elements and having a first bandwidth, the plurality of elements of the first matrix stored in computer memory and representing an action of Maxwell's equations on a transverse magnetic field within the waveguide;
rearranging the plurality of elements of the first matrix stored in computer memory to form a second matrix having a second bandwidth smaller than the first bandwidth;
shifting the second matrix and inverting the shifted second matrix to form a third matrix;
calculating one or more eigenvalues or eigenvectors of the third matrix corresponding to one or more modes of the waveguide; and
calculating one or more propagation losses corresponding to the one or more modes of the waveguide using boundary conditions which comprise surrounding a simulation domain of the method by an absorbing material.
21. A computer system for modeling one or more electromagnetic field modes of a waveguide, the computer system comprising:
means for calculating a first matrix having a first bandwidth using a refractive index profile of the waveguide, the first matrix comprising a plurality of elements representing an action of Maxwell's equations on a transverse magnetic field within the waveguide;
means for rearranging the plurality of elements of the first matrix to form a second matrix having a second bandwidth smaller than the first bandwidth;
means for shifting the second matrix and inverting the shifted second matrix to form a third matrix;
means for calculating one or more eigenvalues or eigenvectors of the third matrix corresponding to one or more modes of the waveguide; and
means for calculating one or more propagation losses corresponding to the one or more modes of the waveguide using boundary conditions which comprise surrounding a simulation domain of the modeling by an absorbing material.