Analytical modeling plays a crucial role in photonics engineering

Elevated stresses and deformations are the major contributor to malfunctions of, and failures in, microelectronics and photonics devices, packages, and systems.

by Ephraim Suhir

Be thankful for problems. If they were less difficult, someone with less ability might have your job.
—Unknown Engineer

Elevated stresses and deformations are the major contributor to malfunctions of, and failures in, microelectronics and photonics devices, packages, and systems. The most serious consequences of the elevated stresses in microelectronics are usually associated with structural failures. The situation is, to a great extent, different in photonics. In the majority of cases, the requirements for the mechanical behavior of the photonics materials and structures are based on their optical performance, rather than on mechanical reliability considerations. In other words, it is not the stresses, but the excessive linear or angular deformations (displacements) that are typically responsible for the short- and long-term reliability of a photonic product.

For instance, low-temperature microbending, which is apparently harmless in terms of the "static fatigue" (delayed fracture) of the silica material, is unacceptable from the standpoint of the elevated transmission losses. Small lateral misalignments (on the order of 0.5 μm) between lightguides might cause very low bending stresses, but are often detrimental from the coupling-efficiency point of view. Tiny displacements in a laser package can result in a complete optical failure of the device.

The ability to predict the magnitudes and the distributions of the stresses and displacements, and minimize them if necessary, is of obvious practical importance. It would not be an exaggeration to assert that successful prevention of induced failures in photonic devices and equipment can be achieved only if predictive modeling is widely used in addition and prior to experimental investigation and testing—whether carried out during the design stage, qualification and manufacturing of the product, or accelerated life testing.

Photonics engineering cannot do without different kinds of accelerated testing, and accelerated testing cannot do without a simple and meaningful theoretical model. Such a model enables a reliability engineer to decide which parameters in a system should be accelerated (such as temperature, stress level, or number of cycles), how to process or interpret the experimental data, and how to bridge the gap between what one "sees" during accelerated testing and what he or she will supposedly "get" in actual use conditions.

Mathematicians are like the French: when you talk to them, they translate your words into their language, and at once you cease to understand them.
—Johann Goethe, German poet

Modeling

Modeling is the basic approach of any science, whether "pure" or applied. Research and engineering models can be experimental or theoretical. While experimental models are typically of the same physical nature as the actual phenomenon or the object, theoretical models represent real phenomena and objects by using abstract notions. Such models usually employ more or less sophisticated mathematical methods of analysis, and can be either analytical (mathematical) or numerical (computational).

As a rule, today's numerical models are computer aided. The most widespread one is finite-element analysis (FEA). Since the mid-1950s, finite-element modeling has become the major research tool for theoretical evaluations in mechanical and structural engineering, including the area of photonics. This should be attributed to the availability of powerful and flexible computer programs, which enable one to obtain a solution to almost any stress-strain related problem within a reasonable time.

The ultimate goal of a theoretical model is to reveal relationships that are hidden in the initial information of the materials and structures. While experimental models can lead sometimes to unexpected results (there are numerous examples of experimentalists who discovered phenomena that were different from those they intended to address), no theoretical model can provide any "new" results—for example, results that are not contained in the input data, the assumptions, and the hypotheses.

Everything should be made as simple as possible, but not one bit simpler.

—Albert Einstein, German physicist

Analytical modeling

Pioneering work in modeling thermal stress in bodies consisting of dissimilar materials has been performed by Timoshenko, Völkersen, and Aleck.1, 2, 3 The first two authors based their treatment of the problem on a structural analysis (strength-of-materials) approach, while Aleck applied the theory-of-elasticity method. Both approaches were later extended in application to modeling of the mechanical behavior of assemblies and structures employed in various fields of engineering, including the area of microelectronics and photonics.

The structural analysis (engineering) approach enables one to determine, often with sufficient accuracy, the induced stresses and displacements. This approach is not straightforward and typically requires special assumptions for each particular problem, but if the model is adequate, it results in simple and easy-to-use formulas, and has been successfully used as a part of a physical design process for components and devices. It can be used to select materials, establish the dimensions of the structural elements, compare different designs from the standpoint of the stress-strain level, and so on.

The theory-of-elasticity method is based on rather general assumptions and equations of the elasticity theory and provides a rigorous mechanical treatment of the problem. When the theory-of-elasticity method is used, the simplifications are typically achieved, not by introducing special assumptions for the problem in question, but by narrowing the class of problems for which the given solution is applicable. It is not always easy to implement this method, or to make practical the solutions obtained by using it. The engineering and the theory-of-elasticity approaches should not be viewed as "competitors," but rather as different research tools that complement each other, play different roles, and have their merits and shortcomings and their different areas of application.

The practical value of mathematics is, in effect, a possibility to obtain, with its help, results simpler and faster.
—Andrey Kolmogorov, Russian mathematician

Potential advantages

Broad application of computers has not made analytical solutions unnecessary or even less important, whether exact, approximate, or asymptotic. Simple and easy-to-use analytical relationships have invaluable advantages, because of the clarity and "compactness" of the obtained information and clear indication of the role of various factors affecting the given phenomenon or the behavior of the given system. These advantages are especially significant when the parameter under investigation depends on more than one variable. The asymptotic techniques and formulas can be successful in those cases in which there are difficulties in the application of computational methods, such as in problems containing singularities.

Even when application of numerical methods encounters no difficulties, it is always advisable to investigate the problem analytically before carrying out computer-aided analyses. Such a preliminary investigation helps to reduce computer time and expense, to develop the most feasible and effective preprocessing model and, in many cases, to avoid fundamental errors. Those who have "hands-on" experience with FEA know very well that it is easy to obtain a solution using FEA, but it might not be easy to obtain the right solution. Many people who are not familiar with the field erroneously think that, although there are many unsolved problems in physics and chemistry, all mechanical problems had been solved many years ago, if not by Newton and Euler, then at least by Timoshenko and Roark. This reasoning continues further that all existing mechanical solutions have been incorporated in a "black box" with a keyboard, and as long as one pushes the right button, one gets the right answer. This is certainly not the case, and a lot of effort must be taken to make sure that an efficient and accurate solution is obtained.

This is particularly true in photonics applications when, in many cases, the existing FEA programs should be considerably modified to become accurate enough and suitable for the evaluation of the thermally induced displacements in a photonics package.

REFERENCES
1. S. Timoshenko, J. Opt. Soc. Am. 11 (1925).
2. O. Völkersen, Luftfahrtforschung 15 (1938; in German).
3. B. J. Aleck, ASME J. Appl. Mech. 16 (1949).

EPHRAIM SUHIR is vice president of reliability and packaging at Iolon Inc., 1870 Lundy Ave., San Jose, CA 95131; e-mail: esuhir@iolon.com.

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