The project's timespan is 01.01.2023- 31.03.2026, whereas WIAS starts on
01.04.2023. The main aims of the project are modeling and design (FBH+WIAS),
as well as fabrication and characterization (FBH) of novel high-power Photonic
Crystal Surface Emitting Laser (PCSEL) devices. This will be achieved by
combining the expertise of the cooperation partners in semiconductor laser
modeling, design, fabrication, and characterization.

Semiconductor diode lasers are small, efficient, and relatively cheap devices
used in many modern applications. Multiple applications require emission
powers exceeding several ten Watts from a single diode and up to a few
kiloWatts from a combined laser system. In this project, we consider novel
Photonic Crystal Surface-Emitting Lasers (PCSELs), which, in contrast to
conventional high-power edge-emitting broad-area lasers (BALs), are capable of
emitting high power (up to 80 W at the moment
[Inoue 2023])
beams of nearly perfect quality in the
(*z*) direction, perpendicular to the (*x/y*) plain of
active material. The critical part of PCSELs, enabling an efficient coupling
of within the active layer generated optical fields and their redirection
along the *z* axis, is a properly constructed 2-dimensional photonic crystal
layer. In simple cases, this PC layer can be vertically homogeneous or consist
of several vertically homogeneous layers (e.g., three layers as shown in
Figure 1).
In many more general cases, when the borders of
the PC features are not vertical, the whole PC layer still can be reasonably
well approximated by multiple very thin vertically homogeneous layers.

The initial model to be considered and integrated numerically [Inoue 2019]
is derived from Maxwell equations and is a 1 (time)+2 (space) dimensional
system of PDEs for complex optical fields
*u(t,x,y)=(u ^{+},u^{-})^{T}*,

Several significant challenges arising when treating the above-stated problems
are a nontrivial construction of the implicitly defined
*4×4* field
coupling matrix
* C*, requiring a solution of the Helmholtz problem
and multiple integrations of the calculated mode profile with different
separately constructed exponentially growing and decaying Green's functions,
as well as simulations and (spectral) analysis of large discrete problems
relating up to several million variables in large-emission-area
(large

In the frame of the PCSELence project, we have developed analytic-expression-based
algorithms for the construction of the field coupling matrix * C*
entering model equations (1) and (3). Our algorithms bypass computer arithmetic-induced problems
when dealing with large and small exponentials; they are fast and exact, in contrast
to approximative approaches or procedures based on numerical integration methods
[1,2,3].
An efficient calculation of leading optical modes in PCSELs is crucial when
looking for suitable heterostructures including size and configuration of
lateral photonic crystal layers. Our algorithms and the numerical solver can be
also explored to consider PCSELs with multiple photonic crystal layers and large emission
areas.
Such calculations are beneficial when looking for PCSEL designs
with low thresholds and good main mode gain separation, which is crucial when
seeking a single-mode high-quality emission.
Moreover, our solver can also be used to predict an intrinsic linewidth if the stable
single-mode operation of PCSEL is ensured [4].

There are at least two sources of errors in calculating the eigenvalues of the
spectral problem.
One source of these errors is related to the nature of the coupling
matrix * C*, which, in general, is defined as an infinite sum of
submatrices, each constructed using specific Fourier coefficients of the
refractive index expansion in the PC layer. In practical calculations, one has
to truncate this sum, keeping only about

The numerical discretization of the problem induces another source for errors in calculated leading eigenvalues. In general, numerical approaches do not allow the finding of all (an infinite number!) eigenvalues of the continuous spectral problem. Fortunately, in practical applications, only several or several-tenths of eigenvalues are essential, and these eigenvalues can be found with pretty high precision when using relatively coarse numerical meshes. We also apply the higher-order numerical schemes to improve the accuracy: see Figs. 2d,e, presenting a study of numerical errors of two leading eigenvalues in dependence on the mesh's roughness and the scheme's precision order. Panel c of the same figure shows the time required to calculate five leading eigenvalues. For more details, see Ref. [2].

Fig. 3 shows an example of the calculated spectra of a particular PCSEL
device.
In panel (a),
we calculate the most important modes of the discretized spectral
problem (3) (small light-blue dots),
compare their eigenvalues to those of two limit-case problems (large full
triangles and red squares or rhombs), and inspect spatial distributions of five main modes
at the right-hand side of the panel. These simulations are performed for
fixed *L= *1 mm.
It is crucial to minimize
the imaginary part of *Ω* (i.e., to reduce the threshold gain and
bias current required to achieve the lasing), increase the separation
between *ImΩ* of
two principal modes (which is needed for achieving a single-mode lasing),
and minimize the contribution of mode intensity at the lateral borders of PCSEL
(which allows to reduce field losses *α _{e}* at these boundaries).
Panel (b) of the same figure shows changes of above-mentioned characteristics
with increasing

- [Inoue 2023]
T. Inoue et al.,
“Self-evolving photonic crystals for ultrafast photonics,”
*Nat Commun***14**:50, 2023 - [Inoue 2019]
T. Inoue et al.,
“Comprehensive analysis of photonic-crystal surface-emitting lasers via time-dependent
three-dimensional coupled-wave theory,”
*Phys. Rev. B***99**:035308, 2019 - [Liang 2012]
Y. Liang et al.,
“Three-dimensional coupled-wave analysis for square-lattice photonic
crystal surface emitting lasers with transverse-electric polarization: finite-size effects,”
*Optics Express***20**(14):15945, 2012

- [1]
M. Radziunas, E. Kuhn, H. Wenzel, B. King, and P. Crump,
“Calculation of optical modes in large emission area photonic crystal surface-emitting lasers,”
in
*Proceedings of the 23th Int. Conf. on Numerical Simulation of Optoelectronic Devices (NUSOD 2023)*, Turin, Italy, September 18-21, pp. 89-90, 2023, DOI: 10.1109/NUSOD59562.2023.10273475. pdf file. - [2]
M. Radziunas, E. Kuhn, H. Wenzel,
“Solving a spectral problem for large-area photonic crystal surface-emitting lasers,”
WIAS Preprint,
(3059), 2023.
- [3]
M. Radziunas, E. Kuhn, H. Wenzel, B. King, and P. Crump
“Optical mode calculation in large-area photonic crystal surface-emitting lasers,“
in
*IEEE Photonics Journal*,**16**(2):0601209, 2024. DOI: 10.1109/JPHOT.2024.3380532. - [4] H. Wenzel, E. Kuhn, B. King, P. Crump, and M. Radziunas, “Theory of the linewidth-power product of photonic-crystal surface-emitting lasers,” (arXiv:2402.11246 [physics.optics]), 2024.

For further information please contact

**Dr. Mindaugas Radziunas**

Weierstrass-Institute for Applied

Analysis and Stochastics

Mohrenstrasse 39

10117 Berlin

Tel.: (030) 20372-441

Fax : (030) 2044975

E-mail: radziunas@wias-berlin.de

WWW: http://www.wias-berlin.de