Fiber Optics Systems and Reliability of Solder Materials
“Say not 7 have found the truth,' but rather, 7 have found a truth. ”
—Kahlil Gibran, Lebanese American artist, poet, and writer
In this chapter, it is shown how methods of analytical modeling in structural analyses of fiber optics systems can be effectively employed to evaluate stresses in, and provide recommendations for, the rational structural design of these systems. The emphasis is on stress analyses and reliability of solder materials—the reliability bottleneck in optical engineering. This undertaking actually forms a new direction— Fiber Optics Structural Analysis (FOSA) [1-8]. All the findings based on analytical modeling were confirmed, for the provided numerical examples, by finite element analysis (FEA) (see, for example, ). The developed analytical FOSA models can be or, actually, have been used to assess the reliability of the fiber optics materials and structures (see, for example, [10-13]). The Bell Labs researchers involved in reliability engineering of microelectronic and photonic systems used to say that “you could have the best electronic or a photonic chip in the world, but if you put it on a piece of junk that is called a substrate or a submount, you end up with a piece of junk.” Therefore, it was argued  that the selection and evaluation of the materials and structures in the emerging technologies in electronics and photonics engineering is critical: nothing happens to an electron or a photon per se, but it is the methods and approaches of materials science, structural analyses, and reliability engineering that deal with the “pieces of junk” that are needed to assure the failure-free performance of the electronic and photonic materials, structures, and systems of the future.
The contents of this chapter are mostly based on the author’s research conducted at Bell Labs, Physical Sciences and Engineering Research Division, Murray Hill, New Jersey, USA, during his approx. 20-year tenure with this company and, to a lesser extent, on his recent work in the field. In the past, we have addressed coated fibers [15-34]; low-temperature micro-bending of fibers intended for undersea communication systems [35-40]; mechanical behavior of optical fiber interconnects treated as flexible beams [41-57]; accelerated testing (“proof-testing”) of optical fibers [58-65]; free vibrations of fused bi-conical taper lightwave couplers (FBTLCs) [66-68]; strain- free planar optical waveguides ; fibers soldered into ferrules ; apparatus and method for thermostatic compensation of temperature sensitive, mostly optical and fiber optics-based, devices ; fiber “pigtails” bent on cylindrical surfaces [72, 73]; elastic stability of glass fibers in a micro-machined fiber-optic switch ; application of the probabilistic approach in thermal stress modeling  and in the interpretation of the accelerated testing results ; the attributes of aerospace optoelectronics [77, 78]; application of the Boltzmann-Arrhenius-Zhurkov (BAZ) model  and use of nanotechnologies to provide moisture resistant nanoparticle based material for an overclad in fiber optics [80-85] as well as elevated interfacial compliance in micro- and optoelectronic systems . In these publications, optical fibers subjected to thermal and/or mechanical loading (stresses) in bending, tension, compression, or to combinations of such loadings were addressed, and the considered structural elements included optical fibers of finite length (bare, jacketed and dual-coated fibers); fibers experiencing thermal loading; the roles of geometric and material nonlinearity; dynamic response of fiber systems to shocks and vibrations; and possible applications of nanomaterials in new generations of fiber optics coating and cladding systems. The objective of this chapter is to address, using the probabilistic-design-for-reliability (PDfR) concept and BAZ model, to quantify, on a probabilistic basis, the likelihood of operational failures of solder materials employed in fiber optics [87-95].
Fiber Optics Structural Analysis (FOSA) in Fiber Optics Engineering: Role and Attributes
The discipline of FOSA employs methods and approaches of reliability physics and structural analysis to evaluate stresses, strains, and displacements in fiber optics structures, to carry out physical (mechanical) design of these structures, and assess and assure short- and long-term reliability of fiber optics products. FOSA treats these products as structures, in which the materials interaction, their size and configuration, and the applied loads are as important as the properties and characteristics of the employed materials. As such, FOSA naturally complements fiber optics materials and optical fiber communication sciences.
The application of methods and approaches of FOSA enables one to design, fabricate, and operate viable and reliable fiber optics products. Like other branches of Structural Analysis (civil, aircraft, space, maritime, automotive), FOSA considers the specifics, associated with the properties of the materials used, typical structures employed, and the nature, magnitude and variability of the applied loads. Typical structures in fiber optics engineering are bare or composite (coated) rods and beams of various lengths and flexural rigidities. These rods and beams could be soldered into ferrules, adhesively bonded into capillaries, or embedded into various materials and media. Typical materials are, of course, silica glasses, but also polymers (coatings, adhesives, and even polymer light-guides) and solders, “hard” (e.g., gold- tin) or “soft” (e.g., silver-tin). Typical loads include internal thermal loads caused by dissimilar materials in the structure and/or by temperature gradients, and high- or low temperature environments (temperature extremes); external (“mechanical”) loads due to the inevitable or imposed deformations; possible dynamic loads due to shocks, vibrations and/or acoustic noise. High voltage, elevated electric current, ionizing radiation and/or extensive light output from a powerful laser source should also be considered, in accordance with the multiparametric BAZ equation, as loads (stressors, stimuli).
FOSA pursues, but is not limited to, the following major objectives:
- 1. Determine (and, to an extent possible, idealize, for the sake of the theoretical analysis) the most likely loading conditions;
- 2. Evaluate the stresses, strains, displacements, and, when methods and approaches of fracture mechanics are pursued, also fracture characteristics of the fiber optics materials and structures;
- 3. Assure, typically on a probabilistic basis, that the acceptable strength and reliability criteria will remain, during the lifetime of the optoelectronic product, within the limits that are allowable from the standpoint of its structural integrity, elastic stability, dependability, availability, and normal operation, both physical (structural) and functional (optical). While an optical engineer is and should be concerned, first of all, with the functional (optical) performance of an optoelectronic product, an adequate performance of such a product cannot be sustained and assured if the product’s ability to withstand elevated stresses (physical reliability) and to exhibit adequate environmental durability (ability to withstand degradation and aging) is not taken care of.
Fibers Soldered into Ferrules
Solder materials and joints are as important in photonics and, particularly, in fiber optics, as they are in microelectronics. There are, however, specific requirements for the solder materials and joints used in photonics: ability to achieve high alignment, requirement for a very low creep, and so on. Thermally induced stresses in optical fibers soldered into various ferrules were addressed in  (Figure 12.1). It has been shown that low' expansion enclosures with good thermal expansion (contraction) match w'ith the silica fiber is not always the right choice from the standpoint of the thermally induced stresses in the metalized optical fibers soldered into ferrules, as w'ell as, and first of all, from the standpoint of the stresses in the solder material itself. Indeed, the low' expansion enclosures result, at low' temperatures, in tensile radial stresses in the solder ring, and could lead to the delamination of the metallization from the fiber and/or to the excessive radial deformations in the solder material.
FIGURE 12.1 Optical glass fiber soldered into a ferrule.
On the other hand, high expansion (contraction) enclosures might result in high compressive stresses in the solder material, and, because of that, in low-cycle fatigue conditions during temperature cycling of the joint. The most feasible material of the enclosure can be found based on the developed model for the given thickness of the solder ring and solder material.
Thermal Stresses in a Cylindrical Soldered TriMaterial Body with Application to Optical Fibers
Ability to predict thermal stress failures in electronics and photonics is of obvious importance. Numerous predictive models for the evaluation of thermal stresses in cylindrical bi-and trimaterial bodies were developed during the last decade in application to dual-coated or jacketed optical fibers. Despite the success in understanding the physics of thermal stresses in various electronics and photonics assembles, there still exists an incentive for the development of practically useful and physically meaningful engineering predictive models for various particular applications, such as silicon photonics. This technology offers numerous novel solutions in different areas of optical communications, optical computer interconnects, sensing, bio-applications, and so on. It has been recognized, however, that the reliability of materials in silicon-photonics structures is not always satisfactory. Accordingly, the objective of the analysis that follows is to develop a simple engineering model for the evaluation of the thermal stresses in trimaterial cylindrical bodies of finite length, with application to silicon photonics and other optical fiber-based technologies.
The radial displacements in a circular ring subjected to the internal, p0, and to external, Pi, pressures applied to its inner and the outer boundaries of the radii r0 and rh can be evaluated, using plane strain approximation, by formula 
Here, r is the current radius, and E and v are the elastic constants of the material. This formula enables one to seek the interfacial radial displacements in a trimaterial body subjected to the change AT in temperature as follows:
Here, Ej,i = 0,1,2, are Young’s moduli of the materials, v(, i = 0,1,2, are their Poisson’s ratios, a,,i = 0,1,2, are the materials CTEs, rt,i = 0,1,2, are the outer radii of the
cylinders, y, = —= 0,1, are the radii ratios, u0 are the radial interfacial displace- П+1
ments of the zero (“core”) component, ui0 are the displacements of intermediate component#! (“bond”) at its interface with the zero component, un are the displacements of component #1 at its interface with outer component #2, u2 are the displacements of component #2 at its interface with component #1, Co is the stress at the zero interface (between the zero and component #1), and o, is the stress at the interface #1 (between components #1 and #2). The stresses o0 and Gi are considered positive if they are tensile, that is, if they are directed inward at the inner boundary of the intermediate material, and outward at its outer boundary. The conditions u0 = ui0, u2 = ul2 of the displacement compatibility result in equations:
From (12.6), we obtain
where D = 6И822 -6|282i is the determinant of equations (12.6). The radial, cr, and the tangential (circumferential), o,, normal stresses are equal to o0 throughout the zero component and are expressed as
for the intermediate component #1 (bond). At the outer surface of component #2, the radial normal stress is zero, and the tangential stress is
This stress fades away rapidly with an increase in the component’s thickness. When the ratio of the outer radius of outer component #2 to its inner radius is equal to 5, the tangential stress at the outer surface, where the radial stress is zero, is only 8.33% of the tangential stress at the component’s inner surface. The axial forces, Fj,i = 0,1,2, can be determined based on the condition that the thermally induced strain
is the same for all the components of the body. In this equation, X,- are the axial compliances of the components. These compliances can be evaluated by formulas
From condition (12.11), we have:
The equilibrium condition ^ F, =0 results in the following formula for the
induced strain: 8 = -а(,ДT, where
is the effective coefficient of thermal expansion (CTE) of the body. Formula (12.13) yields
The axial stresses can be found as
No appreciable longitudinal interfacial shearing stresses occur in the midportion of the body. These stresses might be significant, however, at its end portions. We use the concept of interfacial compliance (see Chapter 2) to address these stresses. In accordance with this concept, the interfacial longitudinal displacements of the body components can be sought as
Here, vv0(z) are the longitudinal interfacial displacements of the inner (zero) component at its interface with intermediate component #1; w10(z) are the displacements of
component #1 at its interface with the zero component; w12(z) are the displacements of component #1 at its interface with outer component #2; w2(z) are the displacements of outer component #2 at its interface with intermediate component #1; a0, a, and a2 are the CTE’s of the materials; X0, X| and X2 are the axial compliances of the components; AT is the change in temperature; F0(z), Fi(z) = -[F0(z) + E2(z)] and F2(z) are the axial forces; K0 is the longitudinal interfacial compliance of the zero component, кш is the compliance of component #1 at its interface with the zero component, кш is the compliance of component #1 at its interface with outer component #2, K2 is the compliance of component #2 at its interface with component #1; and T0(z) and x:(z) are the interfacial shearing stresses. They are related to the axial forces Fo(z) and F2(z) as follows:
Here, / is half the body’s length. The origin of the longitudinal z axis is at the body’s mid-cross section. The longitudinal interfacial compliances can be evaluated by formulas 
Here, G, =--— is the shear modulus of the intermediate material. The con-
ditions w0(z) = wt0(z), w2(z) = wt2(z) of the compatibility of the longitudinal interfacial displacements result in the following equations for the interfacial shearing stresses x0(z) and x: (z):
Differentiating these equations twice with respect to the coordinate z and considering relationships (12.19), we obtain the following equations for the shearing stress functions T0(z) and t[(z):
The shearing stresses are antisymmetric with respect to the mid-cross section of the body and could be sought as
Introducing these solutions into equations (22), we obtain the following two homogeneous equations:
The requirement that the determinant of these equations should be equal to zero for nonzero constants C0 and Q results in the following equation for the parameter к of the interfacial shearing stress:
Biquadratic equation (12.25) has the following solution:
We seek the axial forces at the body’s ends in the form
that satisfies the zero boundary conditions for the forces at the body’s ends. By differentiation, we obtain
Comparing these formulas with formulas (12.23), we find the following expressions for the constants C0 and Ct:
Then, formulas (12.23) yield
The maximum stresses occur at the body’s ends:
For long (large /) bodies with stiff (large к) interfaces, when the product kl is larger than, say, 3, the maximum shearing stresses become body length independent:
The distribution of the interfacial shearing stresses, in the case of long and/or stiff bodies, can be found from (12.31):
Thus, the interfacial stresses concentrate at the body’s ends.