# LDSL-tool

## Longitudinal Dynamics in multisection Semiconductor Lasers

**Developed by:**

**Content:**

- Introduction
- Mathematical models and their analysis
- Applications, examples, comparisons with experiments
- Areas of self pulsations in parameter plane
- Frequency potential of multisection lasers
- Sampling of pulses and jitter
- Locking of pulsating laser to external modulated signal
- Excitability of lasers
- Filtering of output at certain wavelength
- Small signal modulation response
- Direct modulated lasers
- Simulations of complex ring laser devices

- Related publications
- Contact

### Introduction

LDSL-tool is a software for simulation and analysis of the (L)ongitudinal (D)ynamics in multisection (S)emiconductor (L)asers. This software is based on Traveling Wave (PDE) equations describing the propagation of optical fields along the longitudinal direction of the laser, which are nonlinearly coupled with the ordinary differential equations for carrier densities and polarization functions. LDSL-tool not only integrates the PDE model equations but also allows the analysis of the dynamics of longitudinal modes and the building of reduced ODE models based on a finite number of modes. After showing good qualitative and quantitative agreement between basic Traveling Wave and Mode Approximation models, the reduced models can be analyzed with well-known tools for bifurcation analysis, such as AUTO. Such different possibilities, together with some data post-processing routines, make our software a powerful tool suited for the simulation and analysis of different dynamic effects in semiconductor lasers.

**Motivation**-
Multisection semiconductor lasers seem to be key elements in optical communication systems. Depending on their structure and operational conditions, such lasers can demonstrate rich dynamics. Some of these dynamical regimes, such as, e.g., high-frequency self-pulsations can be applied for all optical signal regeneration. A deeper study of the underlying nonlinear processes and optimization of such lasers is still strongly required.

**Structure of the software tool**-
A deep understanding of nonlinear dynamics demonstrated by semiconductor lasers is very useful when designing lasers for specific purposes. Our software LDSL-tool is used to investigate and to design lasers that exhibit various nonlinear effects such as self pulsations, chaos, hysteresis, mode switching, excitability, and synchronization to an external signal frequency (see, e.g. WIAS Preprints 516, 597, 712, 713, 809, 849, 866, 1039, 1149, 1513, 1579, 1584, 1981, 2011, 2261, 2438, and 2604)

This software solves models of different complexity, ranging from partial differential equation (PDE) to reduced ordinary differential equation (ODE) systems. PDE models are based on the Traveling wave (TW) equations for counter-propagating optical fields, and ODE models are given by the mode approximation (MA) of the TW model.

In certain cases our software allows to analyse the mode dynamics of PDE systems and to compare the solutions demonstrated by TW model and reduced MA models. After showing good qualitative and quantitative agreement between basic TW and low dimensional MA models, the obtained system of ODE's can be analyzed with well known tools for bifurcation analysis such as AUTO.

**A variety of laser devices**-
Besides the above-mentioned multisection semiconductor lasers, our software allows for considering a variety of coupled laser devices, including straight multisection and ring lasers. Namely, we represent the considered laser device as a set of differently joined components with negligible lateral and transversal dimensions, assume that the field dynamics within each part of such device is governed by a pair of mutually coupled 1 (time) + 1 (space)-dimensional traveling wave equations, and describe the relations between optical fields at the junctions of different sections of the device by the field transmission-reflection conditions given by the user-defined complex-valued matrices.

*Sections*,*Junctions*, and*Optical injections*are the main building blocks of the laser devices considered by LDSL-tool (see WIAS-Preprints 1315, 2261).

### Mathematical models and their analysis

Our basic mathematical model is based on Traveling Wave equations for optical fields coupled with ordinary differential equations for carrier densities and polarization functions. Under certain assumptions, our software is able to build and analyze low-dimensional ODE models based on mode approximations. We have also introduced some limited possibilities to trace and analyze stationary states of a "full" Traveling Wave model.

**Traveling wave model**-
To resolve the longitudinal distribution and dynamics of the carrier density

*n*(*z*,*t*), the counter-propagating optical fields*ψ(z,t)=(ψ*and polarization functions^{+}(z,t), ψ^{-}(z,t))^{T}*p(z,t)=(p*in each part of the multi-section semiconductor laser or coupled laser system we use the Traveling Wave (TW) model:^{+}(z,t), p^{-}(z,t))^{T}Once considering quantum-dot lasers, we introduce one or two additional rate equations to desribe carrier transitions between carrier reservoir, ground- and excited- states of the quantum dots, see WIAS preprints 1506, 1579, 1584.

Straightforward integration of these equations can immediately give us, e.g., field output at laser facets and variation of mean carrier densities in time or field/carrier density distributions at some fixed time layer:

**Field decomposition into longitudinal modes**-
To get deeper information about the structure of optical fields, we are solving the spectral problem of the Traveling Wave model and finding the decomposition of optical field/polarization into modal components. Here, we consider slowly varying carrier densities as parameters and solve the spectral problem for each instant distribution of

*n*(*z*,*t*):Frequently, this field decomposition improves our understanding of lasers' non-stationary behavior. This approach properly indicates the modes which govern the complicated behavior of the laser and shows much more details than usual spectra of the optical field:

More details about the calculation of optical modes and the field decomposition into modal components can be found, e.g., in WIAS Preprints 712, 939, 2011, and 2261.

**Reduced models based on mode approximations**-
After restricting mode expansion to

*q*leading modes and substituting it to field/polarization equations in our TW model, one becomes*q*ordinary differential equations describing the evolution of complex amplitudes of optical modes:These ordinary differential equations and the equations for carrier densities can be solved and analyzed instead of the TW model. If selecting a sufficient number of leading modes, the solutions of the traveling wave model and mode approximation systems are in perfect agreement:

For more details on such reduced Mode Approximation model, see, e.g., WIAS Preprint 713, 1149, and 2261.

**Bifurcation analysis of reduced models**-
After showing good qualitative and quantitative agreement between basic Traveling Wave and Mode Approximation models, the reduced models can be analyzed with well-known tools for bifurcation analysis, such as AUTO:

For more details, see, e.g., WIAS Preprints 713, 985, 1149, 2261.

**Two parameter bifurcation diagrams**-
One can also perform a two-parameter bifurcation analysis of the mode approximation systems. In this case, the bifurcations are represented as curves in the two-parameter domain. These curves define the stability borders of different attractors in the considered system.

For more details, see, e.g., WIAS-Preprints 985, 866, and 2261.

**Tracing stationary states of the simplified TW model**-
Under similar assumptions needed to derive Mode Approximation systems, we can also trace stationary states of the "full" TW model by changing some parameters and analyzing their stability. For the representation of such results for a three-section laser with one active section, we use similar diagrams as are used for the analysis of "external cavity modes located along ellipses" in the Lang-Kobayashi model of lasers with external feedback:

For more details see, e.g., WIAS-Preprints 985, 1981, 1513, 2261, and 2961.

**Calculation of the stationary states of the general TW model**-
Calculation of stable and unstable steady states in the general TW model accounting for longitudinal distributions of carriers and nonlinear gain compression is much more involved since, in this case, instead of dealing with only a few scalar algebraic equations, we have to solve a system of algebraic and functional (z-dependent) equations, which can hardly be resolved on the functional level. In this case, we rely on numerical discretization, which allows functional equations to be replaced with several hundred algebraic equations, which can then be resolved using Newton iterations. Knowledge of these stationary states is needed, e.g., when performing the linewidth estimation of the laser emission. For more details, see WIAS-Preprint 2961.

### Applications, examples, comparisons with experiments

Besides the already mentioned analysis, the LDSL tool can also be applied for automatic loop computations, tuning selected laser parameters, and recording some of the most specific characteristics of the dynamical behavior of model equations. In this manner, we can locate regions of different stable dynamical laser behavior in parameter space.

**Areas of self pulsations in parameter plane**-
The LDSL tool automatically scans parameters to look for high-frequency self-pulsations with good extinction. In the following figure, a three-section DFB laser is considered. Phase and detuning parameters represent field phase shift due to current injection into the passive middle section and detuning between Bragg wavelengths of two DFB sections, respectively.

**Frequency potential of multisection lasers**-
Scanning of the same parameters as before for the three-section DFB laser with two active equally pumped DFB sections. The figure below shows areas of parameter plane where high frequency self pulsations with good and bad extinction ratio can be observed.

More details in WIAS Preprint 809 .

**Sampling of pulses and jitter**-
To characterize the quality of "noisy" self-pulsations demonstrated by lasers, we sample the pulsating output field with its mean frequency. Different projections of the sampled output give useful characteristics of the laser.

More details can be found in WIAS Technical Report 2 and WIAS Preprint 809 .

**Locking of pulsatinglaser to external modulated signal**-
LDSL tool can also be used to analyze the synchronization of SP to external optical or electrical signals. In this case, we sample our output signal with the period of the external modulated signal. In the case of synchronization, an open eye is seen in the eye diagram, and the pulse drifts along a horizontal line in the pulse drift diagram. Otherwise, the eye is closed, and the pulse drifts out from the fixed position.

**Filtering of output at certain wavelength**-
When we model optical injection or two or more well-separated optical modes define the laser dynamics, the high-frequency beating is seen in the temporal trace of the output signal. To distinguish the contribution of one or another wavelength in the total signal, one can apply filters, which in frequency or wavelength domains are described by a Lorentzian function and in the time domain can be given as a solution of an ordinary differential equation.

**Excitability of lasers**-
The excitability of DFB lasers with integrated passive delay section is realized by injecting short optical pulses. In this case, the theoretical study of model equations has allowed us to predict and realize excitability in experiments.

More details in WIAS Preprint 712 .

**Small signal modulation response**-
To perform a small signal analysis of the laser operating at the cw state, we apply small-amplitude periodic current modulation at fixed different frequencies and, after some long transient, estimate the amplitude of the resulting output. Alternatively, the same result can be achieved much faster by finding the Fourier transform of the transient output power after a delta-function-like perturbation of the current injection.

**Direct modulated lasers**-
Directly modulated semiconductor lasers are of great interest in laser applications for optical data transmission systems. Here, we demonstrate the required performance of the DFB laser with an integrated external cavity at a current modulation with 40 Gb/s PRBS. This modulation rate ~4 times exceeds the usual relaxation oscillation frequency of the considered laser with vanishing feedback.

**Simulations of complex ring laser devices**-
The concept of differently interconnected sections and junctions allows for modeling rather complicated multisection semiconductor lasers. One such nontrivial configuration is a semiconductor ring laser with four separate branches of the filtered optical feedback, see panel (a). The multi-channel feedback scheme of this laser admits a fast switching between steady states determined by the resonances of the ring laser and the wavelengths of the activated filtering channels. Colored frames in panel (a) represent different types of device sections. Namely, we distinguish here the amplifying sections (light red), where the field and carrier dynamics are governed by the full TW model, and two kinds of passive sections, where gain and refractive index functions are set to zero, allowing to ignore the carrier rate equations at all, whereas the propagating field experiences phase change and losses (blue) and, additionally, well-pronounced filtering of the optical frequencies (yellow). The notations of all sections in the section indexes are made according to the cardinal directions. For more details, see see WIAS-Preprints 2261, 2438.

**Estimation of the spectral linewidth**-
Based on the single-mode approximation of the TW wave model, see WIAS preprint 2838, we can estimate the spectral linewidth and some other important parameters of the laser with a steady-state (continuous wave) emission. The model for the linewidth is based on the field expansion into the optical modes, accounting for the effects of nonlinear gain compression, gain dispersion, and longitudinal spatial hole burning in multi-section cavity structures.

### Related publications

All publications listed below discussed different structures of multisection semiconductor lasers and were supported by simulations of the LDSL tool.

**Analysis of the Traveling Wave model**-
- M. Radziunas, "Calculation of steady states in dynamical semiconductor laser models," Optical and Quantum Electronics
**55**, 121, 2023. WIAS Preprint, (2961), 2022. - M. Radziunas, ''Longitudinal modes of multisection edge-emitting and ring semiconductor lasers'', Optical and Quantum Electronics
**47**(6), pp. 1319-1325, 2015. WIAS-Preprint, (2011). - M. Lichtner, M. Radziunas, L. Recke, ''Well posedness, smooth dependence and center manifold reduction for a semilinear hyperbolic system from laser dynamics'',
*Mathematical Methods in Applied Sciences***30**(8), pp. 931-960, 2007. - M. Radziunas, H.-J. Wünsche, B. Krauskopf, M. Wolfrum, " External cavity modes in Lang-Kobayashi and traveling wave models", in
*SPIE Proceedings Series*,**(6184)**, art. no. 61840X, 2006. WIAS-Preprint, (1111). - M. Radziunas, ''Numerical bifurcation analysis of traveling wave model of multisection semiconductor lasers'', Physica D,
**213(1)**, pp. 98-112, 2006. WIAS-Preprint, (985). - M. Radziunas, H.-J. Wünsche, ''Multisection Lasers: Longitudinal Modes and their Dynamics'', in
*Optoelectronic Devices - Advanced Simulation and Analysis*, pp. 121-150, ed. J. Piprek, Springer Verlag, New York, 2005. ISBN: 0-387-22659-1 WIAS-Preprint, (939). - J. Sieber, M. Radziunas, K. Schneider, ''Dynamics of multisection semiconductor lasers'',
*Math. Model. Anal.***9(1)**, pp. 51-66, 2004. pdf file.

- M. Radziunas, "Calculation of steady states in dynamical semiconductor laser models," Optical and Quantum Electronics
**Modeling and simulations of multisection lasers**-
- H. Wenzel, M. Kantner, M. Radziunas, U. Bandelow, "Semiconductor Laser Linewidth Theory Revisited," Appl. Sci.
**11**(13), 6004, 2021. WIAS Preprint, (2838). - M. Radziunas, D.J. Little, and D.M. Kane, "Numerical study of optical feedback coherence in semiconductor laser dynamics," Optics Letters,
**44**(17), pp. 4207-4210, 2019. WIAS-Preprint (2604). - M. Radziunas, ''Traveling wave modeling of nonlinear dynamics in multisection semiconductor laser diodes'', Chapter 31 in J. Piprek (Ed.), Handbook of Optoelectronic Device Modeling and Simulation: Lasers, Modulators, Photodetectors, Solar Cells, and Numerical Methods, Vol. 2, pp. 153-182, CRC Press, 2017. WIAS-Preprint (2261).
- M. Radziunas, A.G. Vladimirov, E.A. Viktorov, G. Fiol, H. Schmeckebier, D. Bimberg, ''Strong pulse asymmetry in quantum-dot mode-locked semiconductor lasers'', Appl. Phys. Lett.
**98**, art. no. 031104, 2011. WIAS-Preprint, (1579). - M. Radziunas, A.G. Vladimirov, E. Viktorov, ''Traveling wave modeling, simulation and analysis of quantum-dot mode-locked semiconductor lasers'', in
*SPIE Proceedings Series*,**(7720)**, art. no. 77200X, 2010. WIAS-Preprint, (1506). - M. Radziunas, ''Traveling wave modeling of semiconductor ring lasers'', in
*SPIE Proceedings Series*,**(6997)**, art. no. 69971B, 2008. WIAS-Preprint, (1315). - T. Perez, M. Radziunas, H.-J. Wünsche, C.R. Mirasso, F. Henneberger, ''Synchronization properties of two coupled multisection semiconductor lasers emitting chaotic light'', Phot. Techn. Lett.,
**18**(20), pp. 2135-2137, 2006. - M. Radziunas, H.-J. Wünsche, ''Multisection Lasers: Longitudinal Modes and their Dynamics'', in
*Optoelectronic Devices - Advanced Simulation and Analysis*, pp. 121-150, ed. J. Piprek, Springer Verlag, New York, 2005. ISBN: 0-387-22659-1 WIAS-Preprint, (939), 2004. - N. Korneyev, M. Radziunas, H.-J. Wünsche, F. Henneberger, ''Mutually injecting semiconductor lasers: simulations for short and zero delay'', in
*SPIE Proceedings Series*,**(5452)**, pp. 63-70, 2004. pdf file. - H.-J. Wünsche, M. Radziunas, S. Bauer, O. Brox, B. Sartorius, "Simulation of Phase-Controlled Mode-Beating Lasers", IEEE J Selected Topics of Quantum Electron.
**9(3)**, pp. 857-864, 2003. WIAS-Preprint, (809), 2003. - N. Korneyev, M. Radziunas, H.-J. Wünsche, F. Henneberger, ''Bifurcations of a DFB Laser with Short Optical Feedback: Numerical Experiment'', in
*SPIE Proceedings Series*,**(4986)**, pp. 480-489, 2003. pdf file. - M. Radziunas, H.-J. Wünsche, ''LDSL: a tool for simulation and analysis of longitudinal dynamics in multisection semiconductor laser'', in
*Proceedings of 2nd International Conference on Numerical Simulations of Optoelectronic Devices (NUSOD-02)*, Zürich, pp. 26-27, 2002. pdf file. - M. Radziunas, H.-J. Wünsche, ''Dynamics of multisection DFB semiconductor laser: traveling wave and mode approximation models'', in
*SPIE Proceedings Series*,**(4646)**, pp. 27-37, 2002. WIAS-Preprint 713 . - M. Radziunas, ''Sampling techniques applicable for the characterization of the quality of self pulsations in semiconductor lasers'', WIAS-Technical Report, (2), 2002.
- U. Bandelow,M. Radziunas, J. Sieber, M. Wolfrum, "Impact of gain dispersion on the spatio-temporal dynamics of multisection lasers", IEEE J Quantum Elect.
**37(2)**, pp. 183-188, 2001. WIAS-Preprint 597 . - U. Bandelow, M. Radziunas, V. Tronciu, H.-J. Wünsche, F. Henneberger, ''Tailoring the dynamics of diode lasers by dispersive reflectors'', in
*SPIE Proceedings Series*,**(3944)**, pp. 536-545, 2000. pdf file.

- H. Wenzel, M. Kantner, M. Radziunas, U. Bandelow, "Semiconductor Laser Linewidth Theory Revisited," Appl. Sci.
**Theory versus experiments**-
- M. Krüger, V.Z. Tronciu, A. Bawamia, Ch. Kürbis, M. Radziunas, H. Wenzel, A. Wicht, A. Peters, G. Tränkle, "Improving the spectral performance of extended cavity diode lasers using angled-facet laser diode chips," Appl. Phys. B
**125**, 66 (12pp), 2019. - M. Khoder, M. Radziunas, V.Z. Tronciu, G. Verschaffelt, "Study of wavelength switching time in tunable semiconductor micro-ring lasers: experiment and travelling wave description," OSA Continuum,
**1**(4), pp. 1226-1240, 2018. - V. Tronciu, H. Wenzel, M. Radziunas, M. Reggentin, J. Wiedmann, A. Knigge, "Investigation of red-emitting distributed Bragg reflector lasers by means of numerical simulations", IET Optoelectronics,
**12**(5), 228-232, 2018. - M. Radziunas, M. Khoder, V. Tronciu, J. Danckaert, G. Verschaffelt, ''Semiconductor ring laser with filtered optical feedback: traveling wave description and experimental validation,'' J. Opt. Soc. Am. B 35(2), 380-390, 2018. WIAS-Preprint (2438).
- V.Z. Tronciu, M. Radziunas, Ch. Kürbis, H. Wenzel, A. Wicht, ''Numerical and experimental investigations of micro-integrated external cavity diode lasers'', Optical and Quantum Electronics
**47**(6), pp. 1459-1464, 2015. - M. Radziunas, V.Z. Tronciu, E. Luvsandamdin, Ch. Kürbis, A. Wicht, H. Wenzel, ''Study of micro-integrated external-cavity diode lasers: simulations, analysis and experiments'', IEEE J. of Quantum Electronics,
**51**(2), art. no. 2000408, 2015. WIAS-Preprint, (1981). - S. Joshi, C. Calo, N. Chimot, M. Radziunas, R. Arkhipov, S. Barbet, A. Accard, A. Ramdane, F. Lelarge, ''Quantum dash based single section mode locked lasers for photonic integrated circuits'', Optics Express
**22**(9), pp. 11254-11266 , 2014. - M. Radziunas, A.G. Vladimirov, E.A. Viktorov, G. Fiol, H. Schmeckebier, D. Bimberg, ''Pulse broadening in quantum-dot mode-locked semiconductor lasers: simulation, analysis and experiments'', IEEE J. of Quantum Electronics
**47**(7), pp. 935-943, 2011. WIAS-Preprint, (1584). - M. Radziunas, K.-H. Hasler, B. Sumpf, Tran Quoc Tien, H. Wenzel, ''Mode transitions in DBR semiconductor lasers: Experiments, simulations and analysis'', J. Phys. B: At. Mol. Opt. Phys.
**44**, art. no. 105401, 2011. WIAS-Preprint, (1513). - O.V. Ushakov, N. Korneyev, M. Radziunas, H.-J. Wünsche, F. Henneberger, ''Excitability of chaotic transients in a semiconductor laser'', Europhysics Letters
**79**, 30004 (5pp), 2007. pdf file. - M. Radziunas, A. Glitzky, U. Bandelow, M. Wolfrum, U. Troppenz, J. Kreissl, W. Rehbein, ''Improving the modulation bandwidth in semiconductor lasers by passive feedback'', IEEE J. of Selected Topics in Quantum Electronics
**13**(1), pp. 136-142, 2007. WIAS-Preprint (1149). - U. Bandelow, M. Radziunas, A. Vladimirov, B. Hüttl, R. Kaiser, "Harmonic Mode-Locking in Monolithic Semiconductor Lasers: Theory, Simulations and Experiment", Optical and Quantum Electronics
**38**, pp. 495-512, 2006. WIAS-Preprint, (1039). - S. Bauer, O. Brox, J. Kreissl, B. Sartorius, M. Radziunas, J. Sieber, H.-J. Wünsche, F. Henneberger ''Nonlinear Dynamics of Semiconductor Lasers with Active Optical Feedback'', Phys. Rev. E
**69**, 016206, 2004. WIAS-Preprint, (866), 2003. - O. Brox, S. Bauer, M. Radziunas, M. Wolfrum, J. Sieber, J. Kreissl, B. Sartorius, H.-J. Wünsche, ''High-Frequency Pulsations in DFB-Lasers with Amplified Feedback'', IEEE J Quantum Elect.,
**39(11)**, pp. 1381-1387, 2003. WIAS-Preprint (849). - H.-J. Wünsche, O. Brox, M. Radziunas, F. Henneberger, "Excitability of a semiconductor laser by a two-mode homoclinic bifurcation", Phys. Rev. Lett.
**88(2)**, art. no. 023901, 2002. pdf file. - M. Radziunas, H.-J. Wünsche, O. Brox, F. Henneberger, ''Excitability of a DFB laser with short external cavity'', in
*SPIE Proceedings Series*,**(4646)**, pp. 420-428, 2002. WIAS-Preprint 712 . - M. Möhrle, B. Sartorius, C. Bornholdt, S. Bauer, O. Brox, A. Sigmund, R. Steingrüber, M. Radziunas, H.-J. Wünsche, "Detuned grating multisection-RW-DFB lasers for high-speed optical signal processing", IEEE J Selected Topics of Quantum Electron.
**7(2)**, pp. 217-223, 2001. pdf file. - M. Radziunas, H.-J. Wünsche, B. Sartorius, O. Brox, D. Hoffmann, K. Schneider, D. Marcenac, "Modeling self-pulsating DFB lasers with an integrated phase tuning section", IEEE J Quantum Elect.
**36(9)**, pp. 1026-1034, 2000. WIAS-Preprint 516 .

- M. Krüger, V.Z. Tronciu, A. Bawamia, Ch. Kürbis, M. Radziunas, H. Wenzel, A. Wicht, A. Peters, G. Tränkle, "Improving the spectral performance of extended cavity diode lasers using angled-facet laser diode chips," Appl. Phys. B

Page created and maintained by Mindaugas Radziunas. Last update on April 22, 2024.

## Contact

### Phone, E-mail

Tel.: 030 20372-441

**E-mail: ldsl@wias-berlin.de**

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