Method and apparatus for modeling the modal properties of optical waveguides

1. A method for modeling one or more electromagnetic field modes of a waveguide, the method comprising:

sampling a two-dimensional cross-section of the waveguide; and

using a computer to perform the steps comprising:

calculating a first matrix using the sampled two-dimensional cross-section of the waveguide, the first matrix comprising a plurality of elements and having a first bandwidth, the plurality of elements of the first matrix representing an action of Maxwell's equations on a transverse magnetic field within the waveguide;

rearranging the plurality of elements of the first matrix 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 mode propagation losses corresponding to the one or more modes of the waveguide.

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 waveguide has a refractive index profile which is translation invariant along a longitudinal axis of the waveguide.

5. The method of claim 1 , wherein sampling the two-dimensional cross-section of the waveguide comprises digitizing a refractive index profile of the waveguide.

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 sampling the two-dimensional cross-section of the waveguide comprises digitizing the air core and cladding structure in a planar cross-section which is perpendicular to the longitudinal axis.

7. The method of claim 6 , wherein sampling the two-dimensional cross-section of the waveguide is performed 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 near the bandgap.

18. A computer-readable medium having instructions stored thereon which cause a general-purpose computer to perform the method of claim 1 .

19. A computer system for modeling one or more electromagnetic field modes of a waveguide, the computer system comprising:

a processor comprising:

means for sampling a two-dimensional cross-section of the waveguide;

means for calculating a first matrix having a first bandwidth using the sampled two-dimensional cross-section 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 mode propagation losses corresponding to the one or more modes of the waveguide.