7/29/2019 2013.02.15-ECE595E-L16

1/19

ECE 595, Section 10Numerical Simulations

Lecture 16: Applications of the Beam

Propagation Method

Prof. Peter Bermel

February 15, 2013

7/29/2019 2013.02.15-ECE595E-L16

2/19

Outline

Recap from Wednesday

BPM Mode Solver

Vectorial BPM Applications: Waveguide

Photonic Crystal Fiber

2/15/2013 ECE 595, Prof. Bermel

7/29/2019 2013.02.15-ECE595E-L16

3/19

BPM Mode Solver

Can extend BPM method to solve for modes,by propagating in the imaginary direction

First, drop all derivatives in BPM equation:

Second, write down next step in z:

Third, substitute special value ofz:

2/15/2013 ECE 595, Prof. Bermel

7/29/2019 2013.02.15-ECE595E-L16

4/19

BPM Mode Solver

Since z initially unknown, assume largest

index possible, and decrease it as needed

Will eventually converge to correct answer

and effective refractive index:

Can use Gram-Schmidt normalization

procedure to find higher-order modes:

2/15/2013 ECE 595, Prof. Bermel

7/29/2019 2013.02.15-ECE595E-L16

5/19

VBPM on a Waveguide: Problem

Description

Cross section defined above; = 1.3 m

Propagation along z is semi-infinite

Must grid space with first-order triangularelements in cross-sectional plane; choose PML toreduce reflections to 10-100

Will vary z for maximum effectiveness

2/15/2013 ECE 595, Prof. Bermel

=3.20

=3.26

7/29/2019 2013.02.15-ECE595E-L16

6/19

VBPM on a Waveguide

Fundamental mode is calculated accurately

with 12,800 first-order triangular elements

2/15/2013 ECE 595, Prof. Bermel

7/29/2019 2013.02.15-ECE595E-L16

7/19

VBPM on a Waveguide

Propagation step size inZ, known as Z, should

equal transverse dimensions for best accuracy

2/15/2013 ECE 595, Prof. Bermel

7/29/2019 2013.02.15-ECE595E-L16

8/19

VBPM on a Waveguide: Longitudinal

Imaginary Propagation

With optimal step size, can solve the fundamentalmode of both polarizations in a pretty modestnumber of steps!

2/15/2013 ECE 595, Prof. Bermel

7/29/2019 2013.02.15-ECE595E-L16

9/19

VBPM on a Waveguide: Accuracy

Accuracy of calculation of waveguide coupling

length as a function of mesh divisions N

2/15/2013 ECE 595, Prof. Bermel

7/29/2019 2013.02.15-ECE595E-L16

10/19

VBPM on a Waveguide

Accuracy of coupling length as a function of

Z saturates below one wavelength

2/15/2013 ECE 595, Prof. Bermel

7/29/2019 2013.02.15-ECE595E-L16

11/19

VBPM on a Photonic Crystal Fiber

Originally conceived of by P.J. Russell

Confines light to core without total internal reflection!

2/15/2013 ECE 595, Prof. Bermel

7/29/2019 2013.02.15-ECE595E-L16

12/19

VBPM on a PhC Fiber

Effective index vs. PhC period

2/15/2013 ECE 595, Prof. Bermel

7/29/2019 2013.02.15-ECE595E-L16

13/19

VBPM on a PhC Fiber

Hy field distributions for the fundamental TEmodes

2/15/2013 ECE 595, Prof. Bermel

7/29/2019 2013.02.15-ECE595E-L16

14/19

VBPM on a PhC Fiber

Confinement loss decreases sharply as period increases

2/15/2013 ECE 595, Prof. Bermel

7/29/2019 2013.02.15-ECE595E-L16

15/19

VBPM on a PhC Fiber

Variation of the effective mode area with PhC

period

2/15/2013 ECE 595, Prof. Bermel

7/29/2019 2013.02.15-ECE595E-L16

16/19

VBPM on a PhC Fiber

Effective index increases modestly with increasing

period , indicating increased mode confinement

2/15/2013 ECE 595, Prof. Bermel

7/29/2019 2013.02.15-ECE595E-L16

17/19

VBPM on a PhC Fiber

2/15/2013 ECE 595, Prof. Bermel

Calculated dispersion relation (effective indexversus wavelength) for a PhC Fiber

7/29/2019 2013.02.15-ECE595E-L16

18/19

VBPM on a PhC Fiber

2/15/2013 ECE 595, Prof. Bermel

Obtained dispersion = 2/2 from earlier data

Note modest changes in parameters flip sign ofD

7/29/2019 2013.02.15-ECE595E-L16

19/19

Next Class

Is on Monday, Feb. 18

Next time, we shall finish theapplications of BPM, and possibly coverother FEM applications

Recommended reading: Obayya,Chapter 3

2/15/2013 ECE 595, Prof. Bermel