Laser Modeling and Simulation

The operation of lasers involves a complicated interplay of many effects that can affect a variety of important performance parameters. Some examples:

• Light propagates in a laser resonator with certain mode properties that affect beam quality, laser efficiency, and other performance parameters. The mode properties can be affected by thermal lensing and misalignment.
• The laser light interacts with the laser gain medium, resulting in gain amplification and gain saturation. Rate equation modeling is often used for such aspects.
• Laser dynamics arise especially in pulsed lasers, where the intracavity optical power evolves in time with the laser gain. In mode-locked lasers, pulse shaping can be achieved by a combination of frequency-dependent gain, nonlinear effects, chromatic dispersion, optical filters, saturable absorbers, etc.
• Nonlinear optical effects can have profound effects on light propagation, especially for light pulses in optical fibers.

Most aspects discussed in this article apply similarly to other types of devices, such as fiber amplifiers, optical parametric oscillators, and devices involving the propagation of ultrashort pulses through some optical elements.

Understanding Laser Operation Before Lab Experiments

For successful laser development, a good understanding of how various effects interact is essential. Furthermore, this is usually needed before the first prototype laser is built: otherwise, you are likely to encounter various problems that are difficult to diagnose and fix:

• If a laser does not work properly, e.g. does not deliver as much optical power as expected, or shows some instabilities, it is usually not clear what the reasons are. So you see a problem, but not its cause.

• There are various diagnostic tools for characterizing laser output – for example, power meters, beam profilers, and more sophisticated equipment for pulse characterization. However, these can usually only tell you what is coming out of the laser, not what is happening inside. Even the laser output can be difficult to fully characterize.
• When the cause is identified, its solution often requires replacing some of the parts used – for example, the laser crystal, some laser mirrors, or one needs to reconfigure the pump optics. You may have to order new components, wait for them to arrive, reassemble the prototype, and test it again. This is often very time consuming and expensive.

A powerful solution to this problem is to create a laser model that allows you to calculate and test certain aspects of how a laser works. The activity of constructing models is called modeling. This term can include the process of applying a model.

Essentially, a laser model then allows a simulation of the laser performance, or some particular aspects of it (and is sometimes called a laser simulator). Such a tool provides valuable insight not only into the expected laser output, but also into its internal workings – including many quantities that would be very difficult to access experimentally. When a simulation model reveals that a laser will not work as desired, it is comparatively easy, quick and inexpensive to identify the reasons and to fix problems with suitable changes of the laser design.

Splitting the Construction and Use of a Model

The person who constructs a computer model may be its only user. However, since the model development is very time-consuming, it is more efficient to have one person (or team) develop a laser simulator (which requires considerable expertise) that can then be applied by many others. Note that using a computer simulation model requires much less expertise than creating it:

• Although the user of a simulation model needs at least a basic understanding of the relevant physical processes (e.g. basic principles of light amplification by stimulated emission of radiation), one does not need a detailed knowledge of their quantitative features, e.g. of the differential equations describing them.
• Nor is it necessary to deal with numerical algorithms for solving complex systems of equations.
• Finally, there is no need to deal with the construction of a practical user interface.

Thus, a pure user of a laser simulator can fully concentrate on the difficulties of laser design (or problem analysis), while others can provide him or her with suitable software based on an appropriately chosen model (or set of models).

What a Model Is

It can be instructive to consider some basic aspects of modeling, as we do below.

A model is essentially an object that is constructed so that it resembles in some simplified way the properties of certain real objects. For example, an architectural model is usually a tiny version of a building, made of different materials and resembling the real house only to a certain extent. Physical models are generally mental representations of real objects that can be used as the basis for calculations and for software that performs computer simulations.

Some important aspects of models are:

• Abstraction: Although a model should resemble some part of reality, it is always much simpler than reality because many details have been omitted in the process of constructing the model. The level of detail in a model should be such that it is as simple as possible while still being useful for studying the phenomena of interest. The main reasons for using abstraction are:
  • It simplifies the construction of the model.
  • By omitting non-essential details, one also simplifies the use of the model and maximizes the chances of producing valuable insights with full clarity.
• Therefore, the most useful model is not necessarily the most detailed one, but the one that best fits the task: for example, allowing one to analyze certain aspects of laser operation while ignoring others. Modeling experience helps one decide on the most appropriate type of laser model.

• Ingredients: A model contains a number of variables and parameters related to physical quantities and properties, and a number of assumptions. These assumptions should be made explicit, as they are not always very well fulfilled in reality, and this can easily be overlooked. Examples of such assumptions in a laser model are that higher order resonator modes are not relevant and that propagation losses and thermal effects in the laser gain medium can be neglected.
• Laser models contain mathematical relations (e.g. differential equations) as essential components. Such relations relate different model variables (e.g. optical intensities and excitation densities of laser-active ions).
• Solving the equations: The resulting equations can sometimes be solved analytically, and the solutions can usually be used with some general-purpose software, e.g., to generate graphical plots. In many cases, however, it is necessary to use numerical methods that can be implemented in some computer software. Such models are often called computer models. (Strictly speaking, a model is a mental construction and the basis of some simulation software, but the term model is sometimes used for such programs that are based on models).

Specialized Models vs. Multi-physics Simulations

Different models can be used to study different aspects of a single laser. For example, one may first use modeling software for its laser resonator that calculates the mode properties (including beam radii at the output and in the laser crystal). The outputs of this can be used as another program (or software module) for the power dynamics of a Q-switched laser, or for the generation of ultrashort pulses in a mode-locked laser.

There are also multi-physics simulations that include multiple physical aspects of the laser. These may be needed when several physical effects interact in such a way that it is not sufficient to study them separately. For example, in a pulsed high-power laser system one may have time-dependent thermal lensing with significant effects on the mode composition of the laser light, and this can only be simulated based on a multi-physics model that includes laser amplification with dynamic heat generation and beam propagation. (In addition, one may want to include optical nonlinearities.) However, such multi-physics simulations are complex to construct and handle, and are only needed (and advantageous) where different physical effects interact in a significant way. In laser technology, it is often more practical to use a smaller model, or a combination of a few such models, for different aspects of interest.

Figure 1 illustrates how different models can handle the relationships between different aspects in a solid-state laser system.

Uses and Benefits of Laser Models

Typical benefits of laser modeling include the following:

• Laser development can become much more efficient, as explained above: one can identify problems very early (before even ordering parts) and avoid time-consuming and costly iterations of building, testing, analyzing, and improving laser designs.
• This is especially important when seeking maximum performance, which is difficult to achieve by trial and error alone.
• The same is true for laser research. For example, when trying to push the performance limits of laboratory prototypes, one faces roughly the same challenges as in industrial development.
• Scientific research can benefit from laser modeling in many other ways. For example, simplified models can be used to test whether certain effects are responsible for certain phenomena. Certain effects can be turned on and off in simulation software to see what difference it makes to the results.
• In the case of problems with existing lasers, a model can provide essential insight to analyze the cause and find a working solution. For example, a model can be used to test whether certain suspected effects (which can be turned on or off in the model) are responsible for observed misbehavior, and a simulation can show whether a proposed remedy would be sufficiently effective and what side effects it might have.
• In some cases, a model shows results which clearly deviate from what is seen on the real laser – simply because that one contains some yet unknown fault. Therefore, work with a model can reveal certain problems which are otherwise hard to find.

• Laser simulation models can be used to obtain huge educational benefits:
  • With laser simulations, a researcher can test his or her understanding by comparing simulated results with initial expectations. Often, one finds surprising results which help to identify overlooked aspects and wrong reasoning.
  • This process also stimulates more creativity. Ideas for new solutions of known problems or for exploiting observed phenomena in a creative way can often result from working with a simulation model.
  • Simulation, in contrast to reading textbook and research papers, is an active way of engaging with the subject, which thus produces a far deeper and longer-lasting understanding. This can be utilized in training courses for students or engineers. Even if they won't learn how exactly such a simulation is done (which may be too complicated for them), they will acquire a lot of expertise quickly – for example, by trying out what effects certain changes of parameters have, and by thinking about explanations for the observed (often unexpected) results.

Validation of Models

Results obtained with physical models may differ substantially from what is seen in reality – for various reasons:

• A model may neglect certain physical effects. If these are relevant for the result, it may produce wrong results.
• Wrong results can also simply be caused by errors in the equations used or in the numerical methods used to solve them.
• The properties of the parts used (e.g. a laser crystal or an active fiber) may differ from those specified by the manufacturer. This may reveal problems with the used parts (including defects) that would otherwise be difficult to find.

Model validation means applying methods of checking whether a simulation model provides correct results. Different such methods are available:

• Computer models often rely on sophisticated algorithms to produce results. Generally, they cannot be tested against analytical results as these often do not exist. However, analytical results are often available at least for certain simplified cases. Sufficiently flexible software allows such tests to be implemented for a wide range of cases, even in an automated fashion. Such tests are likely to reveal problems, and otherwise provide considerable confidence.
• Another method is comparison with experimental results. This is sometimes considered to be the most important, since agreement with reality is what ultimately matters. Note, however, that experimental results can be affected by a variety of errors, such as
  • deviation of material parameters from specified values
  • defects in some parts or the assembly
  • measurement inaccuracies
• On the other hand, agreement between simulated and measured results does not prove the correctness of a model, since one may have used incorrect parameter values to compensate for other deviations.
• Furthermore, experimental tests are often cumbersome and expensive. They can therefore often only cover a very limited range of cases.
• For these reasons, comparing model results with measurements should not be considered the gold standard for model validation.

Software for Laser Modeling

Laser modeling is usually done with software that can perform the necessary calculations and display or store the results. Software can also help with model construction, data organization (e.g., keeping parameter sets for different versions of a device), and convenient visualizations.

Researchers sometimes develop their own laser simulation software, if they are skilled in both laser physics and computer programming. This has the obvious advantages of being highly flexible and costing nothing (unless you count working time, which you should), plus a lot of learning. The disadvantages are the huge amount of time (including development, testing and validation) and the still missing perfection in usability. The latter may be acceptable to the creator of the program, but will make it difficult for others to work with it. That makes the time investment even less attractive. If commercial software with the required features and high quality is available, it will often be preferred. Note also that researchers need to find new results, not just produce tools which have been made by others in similar forms.

Kinds of Laser Simulation Software

As explained earlier, there are very different types of laser models that cover specific aspects such as

• the dynamics of Q-switched lasers,
• the formation of ultrashort pulses in mode-locked lasers, or
• the complex interaction of many effects in fiber lasers and amplifier systems.

Laser simulation software may focus on a particular type of such model, or it may offer features that allow several such models to be used, possibly in combination. In some cases, true multi-physics simulations are possible.

Numerical Efficiency and Reliability

Many cases in laser modeling do not cause a significant computational load and thus can be handled rather quickly even on an ordinary desktop computer. However, there are other cases where more computational time is required; then it becomes desirable to achieve high efficiency of the applied numerical algorithm and also high efficiency of its implementation with the available hardware. For example, the simulation of ultrashort pulse propagation in the context of supercontinuum generation can become quite slow unless an efficient solution is implemented using special CPU instructions for “numerical mass production”. (The CPUs of all modern desktop computers support such instructions, but most software does not use them.)

When computation times are long, it is possible to run long simulation runs overnight. In this case, it is highly desirable to have the flexibility to systematically scan a range of parameters and save the data in a convenient format for review the next day – perhaps including a large set of graphs.

Note that in many cases one can substantially cut down on computation time, e.g. by doing certain tests on simplified devices where the result can still be applied to other cases. It may also be worthwhile to optimize certain numerical parameters, as far as that is not done automatically by the software.

Numerical solutions often require iterative algorithms, where one cannot be sure that they converge in all cases. While it may be easy to find suitable numerical parameters (e.g. some step sizes) which work well in one given situation, it can be challenging to find a solution with which an algorithm works efficiently and reliably in a very wide range of cases.

User Interface

Laser simulation software should allow the user to concentrate on the task at hand. For this purpose, it must be convenient to work with separate sets of input data, to create appropriate diagrams, to configure them according to concrete needs, to save simulated data to files, etc. For self-made software to be used only by its creator, one will typically use some scripting approach with limited ease of use. For use by others, a more convenient solution is needed.

A fundamental challenge even for commercial software is to achieve a combination of high flexibility (to perform simulations on a wide range of cases) with good usability. Scripting provides flexibility, while forms are often preferred for ease of use; it is not easy (but possible) to combine these concepts for a combination of high flexibility and usability. Specifically for fiber lasers and amplifiers, RP Photonics has the RP Fiber Power software, which provides a powerful solution for this combination.

Problems of Modeling Exercises

Although laser modeling can provide extremely valuable benefits, modeling exercises can also fail to produce valid and useful results. Possible causes of such failure include the following:

• Oversimplification can effectively exclude relevant effects from a model, leading to incorrect results.
• Overly sophisticated approaches can be cumbersome to use or take too long to compute.
• Too many parameters make it difficult to interpret observed results correctly. Note that agreement between simulated and measured results does not prove the correctness of a model!
• In some cases, a lack of available input data is problematic. Note, however, that modeling software can help generate such data from experimental data.
• Errors in mathematical equations, programming logic, or physical assumptions may be difficult to detect and locate unless well developed validation procedures are available.

A Strategy for Successful Creation of a Simulation Model

It is highly recommended to use a systematic modeling strategy. This can be based on the following steps:

• It is important to first collect a set of specific questions, so that correct and comprehensive answers would be a real help, e.g. for a laser development project. Skipping this step can lead to unfocused, unsystematic, and therefore inefficient work.
• Based on this, it can be determined what kind of laser model is capable of providing the answers based on the available data and without excessive work.
• If the previous steps have been successful, the model can be constructed. This typically involves deciding which aspects and effects to include or exclude, assembling various equations, and often arranging for their evaluation with numerical methods on a computer. Powerful modeling software, such as simulation software, can be very helpful.
• Before a model can be used in practice, it must be carefully validated, i.e. checked for correctness. See above for how this works.

A Strategy for Successfully Using a Simulation Model

Using a simulation model in the form of existing high-quality simulation and design software is much easier than developing one. Some useful hints:

• Again, first formulate clear goals. (For example: design a prototype laser or develop an industrial laser product to generate ultrashort pulses according to certain specifications. Or: clarify whether a problem with an existing laser might be related to effect X, and how to solve it).
• One should carefully choose software that
  • provides appropriate physical models,
  • is trustworthy (e.g., based on sound scientific competence and carefully validated),
  • is flexible enough to be used not only for simple situations, but also for more complex scenarios,
  • is easy and convenient to use (not only in simple cases),
  • and where competent support is available.
• Then you can configure your first simulation models – perhaps starting with some simple generic models for situations where you can more or less predict what should happen. Then you can move on to more and more sophisticated cases.
• Simulation results can be compared with experimental data – taking into account various reasons for possible discrepancies, as explained above. One can try to explain discrepancies, e.g. with modified parameters of used components, but keep in mind that some “fudge factors” may not reflect reality.
• One should not forget to go back to the originally stated goals, e.g. to clarify certain questions of interest.

More to Learn

Tutorial on Modeling of Fiber Amplifiers and Lasers

Encyclopedia articles:

• lasers
• laser resonators
• rate equation modeling
• laser dynamics
• pulse propagation modeling

Blog articles:

• The Photonics Spotlight 2006-11-16: “Laser Models - not Always Useful”
• The Photonics Spotlight 2007-02-26: “No Laser, no Result?”
• The Photonics Spotlight 2008-11-08: “Validating Numerical Simulation Software”