Exact field solution to guided wave propagation in lossy thin films

The problem of light trapping in thin film solar cells is one of mode coupling into a lossy waveguide structure. This paper therefore derives the full-field solution for electromagnetic wave propagation in a three-layer dielectric system with arbitrary complex indices of refraction. The functional form of the eigenvalue equation is identical to that of the strictly lossless case, but with complex-valued propagation constants rather than real. Lossy mode propagation is then explored in the context of photovoltaics by modeling a thin film solar cell made of amorphous silicon with an aluminum back contact.


INTRODUCTION
The study of light trapping in thin-film solar cells is strongly related to the problem of mode coupling into a lossy waveguide structure. However, unlike the field of communications where loss is something to be minimized and neglected, the goal of photovoltaic structures is to absorb as much light as possible. It is therefore important for researchers in the field of light trapping to understand the fundamental physics of wave guidance in lossy dielectric films. Although waveguide theory certainly has a long and rich history with many classic references at the graduate level, 1-5 the problem of guided wave propagation in lossy media is not nearly as well-understood as the lossless case. Although significant progress was initially made in the late 1960s and early 1970s, 6-8 the scope of such analysis still remains fairly limited. Later research in the field of semiconductor lasers eventually prompted the study of wave guidance in amplifying media, 9 and such models are mathematically analogous to that of wave guidance in a lossy substrate. Eventually, the generalized problem of wave propagation in an n-layer guiding structure was solved in 1995, 10 though the generality of such a solution tends to obscure much of the underlying physics within the model. The goal of this paper is to derive a complete, full-field solution to the problem of wave guidance in an asymmetric three-layer slab. The symmetric case has been extensively analyzed in earlier work, 11 so this paper will build further on that analysis. The solution to the asymmetric problem serves as a useful model for thin-film photovoltaic devices and reveals many interesting insights into the nature of lossy wave guidance. It also serves as a useful benchmark from which to evaluate the reliability of low-loss approximations against their exact, analytical solutions.
The basic structure of interest for this work is the planar, three-layered dielectric structure depicted in Figure 1. Each region in the model is potentially either lossy or amplifying (we shall assume lossy) and may therefore be characterized by a complex index of refractionñ. The middle slab, also called the film region, has the complex indexñ f = n f + jκ f . The region defined by x > h is called the cladding region and has the complex indexñ c = n c + jκ c . The region where x < h is called the substrate and has the complex indexñ s = n s + jκ s . For the special case whenñ c =ñ s , the structure is said to be symmetric. Otherwise, the structure is asymmetric. The ultimate field solutions can likewise assume two possible polarizations called the transverse-electric (TE) and transverse-magnetic (TM) states. For the TE case, the electric field intensity E is polarized along theŷdirection. For the TM case, the magnetic field intensity H isŷ-polarized. We shall also adhere to the notational convention of a time-dependent phasor with the form of Re{e jkx e −jωt }, which is the standard in most optical literature.

TE POLARIZATION
Following a similar procedure outlined in earlier work, 11 we begin by assuming an asymmetric electric field profile with the functional form given by The constant E 0 determines the absolute magnitude of the field profile while A, B, and C represent the relative amplitudes of the wave functions in each region. The propagation constants k x , k z , γ s , and γ c all obey their respective dispersion relations given by where k 0 = 2π/λ 0 is the wavenumber associated with the free-space wavelength of excitation λ 0 . To solve for the magnetic field intensity, we apply Faraday's law ∇ × E = jωμ 0 H to produce Proc. of SPIE Vol. 8256 825606-2 The next step is to enforce continuity on the tangential field components at the planar boundaries. This leads us to the following system of equations: Combined with Equations (2)-(4), we now have a set of seven equations with seven unknowns and may therefore be solved uniquely. Using a little manipulation and substitution, it is possible to combine these expressions into a single equation in terms of k x . The TE asymmetric eigenvalue equation is therefore found to be where the propagation constants γ c and γ s are functions of k x that satisfy It is worth nothing that this eigenvalue equation is identical to that found for lossless systems, 4 but with complex propagation constants instead of purely real-valued. Values of k x which satisfy this expression can then be backsubstituted to solve for all other propagation constants and generate the total TE field solutions over all space.

TM POLARIZATION
Following an identical procedure to the TE case, the magnetic field intensity is first expressed as We can similarly derive the electric field intensity using Ampere's law ∇ × H = −jω 0ñ 2 E to arrive at By analogy with the TE case, we now we enforce continuity and manipulate the series of equations to produce the TM eigenvalue equation

MISFIT FUNCTION
Just like the case of the symmetric waveguide model, the asymmetric eigenvalue equations are nonlinear and can be solved only through iterative methods. We therefore define the residual functions using Note that these functions simply represent the complex error between the left and right sides of the eigenvalue equations. We may therefore define a function φ, called the misfit function, as the squared norm of the residual: Zeros in the misfit functions therefore represent solutions to the eigenvalue equations for k x and may be easily found using standard zero-finding algorithms. One simple algorithm that has performed well for our purposes is the steepest descent method with linear line search. An outline of this method is given in the appendix of our previous work. 11

BRANCH CUTS
Due to the presence of the square-root functions defining γ c and γ s (which each map to two unique values), the misfit φ maps to four unique solutions for any given value of k x . This produces branch cuts in the solution space that must be accounted for in order to properly generate a field profile. These branch cuts occur along the sets of points where the imaginary component to the radicals is equal to zero. Letting k x = β x + jα x , these branch cuts can be expressed as Another parameter of interest is the branch point associated with each branch cut. These occur when the entire argument of the radical itself is equal to zero. The cladding and substrate branch points therefore satisfy It is important to recognize that although many solutions exist where φ(k x ) = 0, not all of them are physically viable. If an incorrect sign is chosen for the propagation constants, mode profiles will exponentially grow as |x| → ∞ rather than decay to zero. It is therefore important to choose the proper branch cut in order to obtain bounded modes. For the example in Figure 2, it was necessary to choose the negative root for γ c and the positive root for γ s . However, the proper choice of roots will tend to vary as the model parameters are changed. It is therefore important to validate any mode solutions by testing for convergence.

APPLICATIONS TO THIN FILM PHOTOVOLTAICS
One simple way to model a photovoltaic cell is to consider a thin film of amorphous silicon (a-Si) backed by an aluminum (Al) contact. Neglecting any surface oxide layers or anti-reflective coatings, such a model is a reasonable representation of a typical thin film solar cell. Consider for example a thickness of 2h = 500 nm at an excitation wavelength of λ 0 = 600 nm. The dielectric properties for each layer may then be modeled using n c = 1, n f = 4.6 + j0.3, and n s = 1.26 + j7.2. The first four TE modes (M = 0, 1, 2, 3) for such a system are plotted in Figure 3 and the resultant propagation constants are summarized in Table 1. As expected, we see the typical mode profiles normally associated with wave guidance, but we also find a characteristic attenuation associated with the lossy nature of the structure.
Another parameter of special interest is that of longitudinal attenuation. This is important to the field of light trapping in thin films, as it represents the absorption length of a guided mode in the structure. We therefore define the longitudinal attenuation coefficient α z from the imaginary component to the wavenumber k z = β z + jα z . Figure 4 summarizes the computed values over 14 guided mode solutions for the solar cell model, including both TE and TM solutions. Modes 0-6 are all natural modes that would occur in a perfectly lossless system of the same general design. However, modes 7-13 are the so-called loss-guided modes that only occur in lossy systems. 11 These modes exhibit very strong longitudinal attenuation that may have useful applications for light trapping if normally-incident light can be effectively coupled into these modes.

SUMMARY AND CONCLUSIONS
This paper derives the full-field solution to wave guidance in a lossy, asymmetric structure. The solution is achieved by enforcing continuity at the planar boundaries and solving the resultant eigenvalue equation. Because of the transcendental nature of the expressions, iterative methods are necessary in order to reach a solution. We also find that the eigenvalue equations are identical in structure to their lossless counterparts, except the propagation constants take on complex values rather than purely real. Branch cuts present a unique obstacle with this problem due to the potential for divergent solutions in the mode profiles. Special care must therefore be taken to select only the physically viable modes. Finally, we demonstrate this method by modeling a simple thin-film photovoltaic device and solving for the resultant mode profiles.   Table 1. Propagation constants of the first four TE mode solutions to a 2h = 500 nm film of amorphous silicon backed by an aluminum contact. At an excitation wavelength λ0 = 600 nm, the indices of refraction are given byñc = 1, n f = 4.6 + j0.3, andñs = 1.26 + j7.2. Units are given in rad/μm.