SPIE Photonics West, San Francisco, January 24, 2012
The light path tree algorithm for nonsequential field tracing
Michael Kuhn, LightTrans VirtualLab UG, Jena, Germany.
Frank Wyrowski, Friedrich Schiller University, Jena, Germany.
Christian Hellmann, LightTrans GmbH, Jena, Germany.
Concept of Field Tracing
Source
Detector
system is divided into subdomains
• Per subdomain
– Consider harmonic fields and solve approximate or rigorous
Maxwell‘s equations.
• System solution
– Continuity conditions of fields along boundaries.leads to a
Maxwell solution in R³.
Maxwell Equations
• Maxwell equations in frequency domain for
isotropic media and linear matter
• Summarized notation for 6 field components
The Input-Output Problem
detector
boundary
source
boundary
sequentiel
• Formally a system is subdivided into subdomains.
• Input: source field on the source boundary.
• Output: resulting field on the detector boundary.
Light Path Diagram in VirtualLab™ 5
The Input-Output Problem
detector
boundary
source
boundary
sequentiel
non-sequentiel
•
•
•
•
Formally a system is subdivided into subdomains.
Input: source field on the source boundary.
Output: resulting field on the detector boundary.
We look for a formulation that represents the Maxwell
problem in R³ for the unknown boundary fields
Example: Field Tracing Through a System
Source
Component 1
Component 2
• In Field Tracing all the steps (arrows) are
associated with fields, not just rays.
Detector
Local Propagation Operators
• Local free space
operator: maps
output fields to input
field
• Local component
operator: maps input
fields to output fields
Equilibrium Equation
• The continuity of the field at boundary j requires
to fulfill the equilibrium equation:
• The set of equilibrium equations for all j have to
be solved for the fields
.
Matrix Equations
Iterative Update Algorithm
• k-th iterate defined by truncated sum
• formula is to be optimized in order
to minimize the number of local
Maxwell problems to be solved.
Initial step
Update formula: Each iteration
step requires once the
application of the operator (CP).
Light Path Tree
• Nodes
– Type-(i): associated
with auxiliary vector W
– Type-(ii): associated
with solution vector V
• Connections
– Type-(i): associated
with operator C
– Type(ii): associated
with operator P
Light Path Tree
• The tree describes the
solution of the field
tracing problem for a
system in R³.
• Each tree level
corresponds to one
summation step.
• Optimal, since no
Maxwell problem is
solved twice.
• Tree can be generated
by a pilot ray algorithm.
Summation
Index k
Generation of the Tree: Pilot Ray Algorithm
Generation of the Tree: Pilot Ray Algorithm
Detector
Source
Component 1
Component 2
Stopping Criterion for Tree Generation
• At the current node we compute the relative
power of the updates with respect to the source
field
• The generation is stopped when the relative
power is below some threshold
Convergence Results
• Consider a sequence of plates with different
refractive index n in air.
source
target
n
air
n
air
air
• The threshold δ is set to 0.01. The number of
iterations to reach convergence is displayed.
Convergence Results: 2 and 4 plates
2 plates, n= 1.5
8 iterations required
2 plates, n= 3.0
4 plates, n= 1.5
13 iterations required 17 iterations required
Simulation Results: Single Plate
source
target
n
air
Input Field:
Plane Wave
air
Compute transmission
efficiency for varying
thickness of the plate and
varying wavelength of the
Dispersion Fused Silica source.
Efficiency Results: Single Plate
• Used SPW (spectrum of plane waves) operator for free
space.
• Use TEA (thin element approximation) for component
propagation through interfaces.
• Vary thickness of the plate between 1µm and 2µm.
• Vary wavelength between 400nm and 600nm.
Efficiency Results: Single Plate
Transmission Efficiency vs. Thickness.
Transmission Efficiency vs. Wavelength.
Comparison Field Tracing vs. FMM
Sequential Field Tracing
• Under the conditions
– Single source plane
– No bifurcations (e.g. only transmission, no reflection)
– Tracing sequence ordered by index
the matrix P simplifies to the form
Sequential Field Tracing
• The input-output problem simplifies to
with a free choice for the local propagation
operators.
Source
Detector
Article: Introduction to Field Tracing
Published in Special Issue of
Journal of Modern Optics on
Computational Optics and Photonics
Editors: Carsten Rockstuhl and Frank Wyrowski
March 2011, Volume 58, Numbers 5-6
Conclusions
• Non-sequential field
tracing algorithm has
been developed (not yet
available in VirtualLab).
• Field tracing solution is
well defined as solution
of the Maxwell problem
in R³.
• Convergence analysis
possible.
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