## Exact field solution to guided wave propagation in lossy thin films |

Optics Express, Vol. 19, Issue 21, pp. 20159-20171 (2011)

http://dx.doi.org/10.1364/OE.19.020159

Acrobat PDF (1190 KB)

### Abstract

Wave guidance is an important aspect of light trapping in thin film photovoltaics making it important to properly model the effects of loss on the field profiles. This paper derives the full-field solution for electromagnetic wave propagation in a symmetric dielectric slab with finite absorption. The functional form of the eigenvalue equation is identical to the lossless case except the propagation constants take on complex values. Additional loss-guidance and anti-guidance modes appear in the lossy model which do not normally exist in the analogous lossless case. An approximate solution for the longitudinal attenuation coefficient *α _{z}* is derived from geometric optics and shows excellent agreement with the exact value. Lossy mode propagation is then explored in the context of photovoltaics by modeling a thin film solar cell made of amorphous silicon.

© 2011 OSA

## 1. Introduction

*μ*m or less, geometric optics no longer provides an accurate description of field propagation at optical wavelengths. It is therefore an important goal to better understand the problem of wave guidance in lossy dielectric films. The solution to this problem serves as a useful model for thin-film photovoltaic devices and reveals many interesting insights into the nature of lossy guidance. It also serves as a useful benchmark from which to evaluate the reliability of low-loss approximations against their exact, analytical solutions.

6. S. Miyanaga and H. Fujiwara, “Effects of absorption on the propagation constants of guided modes in an asymmetric slab optical waveguide,” Opt. Commun. **64**, 31–35 (1987). [CrossRef]

7. K. Fujii, “Dispersion relations of TE-mode in a 3-layer slab waveguide with imaginary-part of refractive index,” Opt. Commun. **171**, 245–251 (1999). [CrossRef]

8. T. D. Visser, H. Blok, and D. Lenstra, “Modal analysis of a planar waveguide with gain and losses,” IEEE J. Quantum Electron. **31**, 1803–1810 (1995). [CrossRef]

9. A. E. Siegman, “Propagating modes in gain-guided optical fibers,” J. Opt. Soc. Am. A **20**, 1617–1628 (2003). [CrossRef]

*h*placed between two half-space dielectrics. The dielectric slab, also called the film region, is lossy and may therefore be characterized by a complex index of refraction

*n*=

_{f}*n*+

*j*

*κ*. For simplicity, we shall assume that the cladding layers are both lossless and symmetric, such that they may both be defined by the purely real index of refraction

*n*. Further complications to the structure may include losses or asymmetries in the cladding layers but will not be considered in this work.

_{c}*e*

^{jkx}e^{−jωt}}. This is consistent with the majority of the optics literature but contrary to much of the RF and microwave literature. It also means that a positive value for the extinction coefficient

*κ*implies a lossy material rather than a gain material. When calculating the time derivative of a phasor, we may further make the substitution

*d*/

*dt*= −

*jω*. The ultimate field solutions can likewise assume two possible polarization states, which are the transverse-electric (TE) and transverse-magnetic (TM) polarizations. For the TE case, the electric field intensity

**E**is defined to be polarized along the

**ŷ**-direction. For the TM case, the magnetic flux density

**H**is defined to be

**ŷ**-polarized.

## 2. Eigenvalue equations

*k*=

*k*is a complex number that may be written as

_{f}*k*=

_{f}*k*

_{0}

*n*. The free-space wavenumber is then given by

_{f}*k*

_{0}= 2

*π*/

*λ*

_{0}, where

*λ*

_{0}is the free-space wavelength of excitation for the model. The functional expression for

**E**(

*x*,

*z*) can take on many forms, and the only condition for a “correct” choice is the constraint that the solution satisfy the boundary conditions imposed by the geometry of the model.

*x*| ≤

*h*), we shall assume that the electric field can be expressed in the following form: This expression represents two uniform plane waves propagating through a lossy medium. The constants

*β*and

_{x}*β*are the transverse and longitudinal phase constants, while

_{z}*α*and

_{x}*α*are the transverse and longitudinal attenuation coefficients, respectively. Both waves possess a forward component along the

_{z}**+ẑ**-direction and opposing components along

**x̂**. The ± sign is indicative of a choice between either even (+) or odd (−) symmetry along

*x. E*

_{0}is an arbitrary complex constant that determines the overall intensity and phase of the electric field.

*k*=

_{x}*β*+

_{x}*j*

*α*and

_{x}*k*=

_{z}*β*+

_{z}*j*

*α*. Plugging back into Equation (2) therefore leads to This expression is identical to the field profiles typically assumed for the lossless case, with the only change being the assumption of complex values for the wavenumbers. The procedure is therefore directly analogous to the lossless case found in most references, which we review in Appendix A. The resultant TE eigenvalue equations are therefore given as Similarly, for the TM case, the eigenvalue equations are Together, Equations (4) through (7) determine the allowed values for the transverse complex wavenumber of the even and odd modes under both polarizations.

_{z}## 3. Eigenvalue solutions

*f*(

*k*) as the error in the eigenvalue equation with respect to

_{x}*k*. For example, with the even TE case, this is simply while for the odd case we have Next, we define the misfit function

_{x}*ϕ*as the squared norm of the residual: Note that the asterisk {*} denotes the complex conjugate operation. Our goal is to solve for all complex values of

*k*such that

_{x}*ϕ*(

*k*) = 0. We shall also enforce the condition

_{x}*β*> 0. This task is accomplished using standard nonlinear optimization techniques [15

_{x}15. J. A. Snyman, *Practical Mathematical Optimization: An Introduction to Basic Optimization Theory and Classical and New Gradient-Based Algorithms (Applied Optimization)* (Springer, 2005). [PubMed]

*n*= 2.0 +

_{f}*j*0.5,

*n*= 1.5, and a normalized film thickness of

_{c}*h*/

*λ*

_{0}= 0.5. The logarithmic power of the misfit function, 10 log

_{10}

*ϕ*, is plotted in Figure 2(a) for the even modes. Mode solutions manifest as zeros in

*ϕ*(dark blue regions) with the

*M*= 0 and

*M*= 2 modes indicated. The initial trial solution is chosen by the

*M*= 2 mode for an equivalent lossless system (

*β*

_{x}λ_{0}= 7.27) and indicated by the “X” mark. After applying the steepest decent method, the exact lossy mode solution converges to a value of

*k*

_{x}λ_{0}= 7.89 +

*j*0.77, indicated by the “O” mark on the

*M*= 2 mode. Solving Equations (A.4) and (A.12) (see Appendix A) then produces the full set of propagation constants. Expressed in terms of electrical length, these are Figure 2(b) shows the electric fields along both

*x*and

*z*generated from Equation (A.7) and normalized to unit amplitude. The horizontal lines indicate the waveguide boundaries. We can also observe the exponential decay in field strength along

*z*, which we naturally expect from propagation through a lossy film.

*M*= 3 mode which actually does not exist as a viable solution for a lossless slab. This is a key difference between lossy and lossless waveguides, where lossy systems possess an equal or greater number of eigenmodes. These extra modes are referred to as “loss guided” modes [9

9. A. E. Siegman, “Propagating modes in gain-guided optical fibers,” J. Opt. Soc. Am. A **20**, 1617–1628 (2003). [CrossRef]

*k*/

_{z}*k*

_{0}= 1.14 +

*j*0.6, which is less than the real index of either the film or the cladding layers.

*M*= 2 mode as the extinction coefficient is incremented from

*κ*= 0.01 to

*κ*= 2. It is useful to note that as the film region grows more lossy, its field profile within the slab experiences relatively little perturbation with respect to the lossless case. The only significant change is that higher loss tends to result in greater field confinement within the slab.

*γ*possesses a real-valued component. Under lossless conditions,

*γ*would be purely imaginary, thereby producing strict exponential decay to the fields in the cladding region. When the film is lossy, the real component to

*γ*causes an additional phase oscillation with respect to

*x*on top of the decay fields. Using the diagram in Figure 4, we can understand this behavior by depicting the evanescent fields as rays that leave the waveguide at some point

*z*and then re-enter at some point

*z*

^{+}Δ

*z*(a common description for the Goos-Hanchen effect [10

10. A. W. Snyder and J. D. Love, “Goos-Hanchen shift,” Appl. Opt. **15**, 236–238 (1973). [CrossRef]

**R**

_{1}carries greater field intensity than the ray

**R**

_{2}. This means that, for |

*x*| >

*h*, the time-averaged Poynting’s vector

*x*-component that points toward the film region. Visually speaking, this manifests as a “skewing” of the fields in the cladding region as the constant phase fronts are pushed off-normal with respect to the waveguide. The effect is somewhat subtle for the case in Figure 2(b) but very dramatic for the case in Figure 6(b).

## 4. Branch cut solutions

*ϕ*due to the square-root function within the residual. The cut occurs when the imaginary component of the radical is set to zero, since this is the point where the square-root of a complex value switches phase during numerical computation. Carrying out this calculation therefore leads us to The branch point itself is found by setting the argument of the radical to zero, which occurs at

*ϕ*maps to two unique values along every point in the complex plane, only one at a time can be rendered on a single 2D graph. The default mapping is shown in Figure 5(a) using the positive value of the radical. An alternative mapping is found by switching the sign of the square-root function in Equations (4)–(7) and then re-deriving

*ϕ*accordingly. This function is shown in Figure 5(b) and reveals a potential set of new zeros in

*ϕ*. However, because of the negative sign on the radical, the imaginary component for

*γ*generally switches sign as well. Such solutions imply a field profile that diverges as |

*x*| → ∞ and are therefore not physically realizable as guided modes.

*κ*grows in value, the branch cut moves further away from the origin to expose zeros in

*ϕ*that otherwise would have been disallowed. Because of this trend, lossy waveguides will always possess an equal or greater number of viable mode solutions than their lossless counterparts. It also allows us to visualize the origins of anti-guidance, which violates the typical restrictions of total internal reflection (TIR) [9

9. A. E. Siegman, “Propagating modes in gain-guided optical fibers,” J. Opt. Soc. Am. A **20**, 1617–1628 (2003). [CrossRef]

*n*= 2.0 +

_{f}*j*0.1,

*n*= 2.25, and

_{c}*h*/

*λ*

_{0}= 0.5. Although the cladding index is greater than the real part of the film index, there still exists a mode solution at

*k*

_{x}λ_{0}= 2.8 +

*j*0.77. The anti-guided mode also has an effective index of

*k*/

_{z}*k*

_{0}= 1.95 +

*j*0.07, which is lower than either the film or cladding. The transition for this mode occurs at

*κ*≈ 0.03, below which the branch cut overlaps the zero in

*ϕ*and removes it as a physically viable solution. Low-loss and loss-less waveguides therefore do not possess guided mode solutions for

*n*>

_{c}*n*, which serves to reinforce the conventional TIR condition.

_{f}## 5. Longitudinal attenuation and low-loss approximation

*z*-direction increases with mode number. This behavior is demonstrated in Figure 7(a) using

*n*= 2.5+

_{f}*j*0.01,

*n*= 1.5,

_{c}*h*= 1.5

*μ*m and

*λ*

_{0}= 1.0

*μ*m. This prediction can be corroborated through numerical simulation with the finite-difference time-domain method [11

11. Lumerical Solutions Inc., http://www.lumerical.com/.

*z*. For the case of low-loss dielectrics (

*κ*<< 1), simple trigonometry suggests that where

*θ*= tan

^{−1}(

*β*/

_{x}*β*) is the elevation angle of the ray and

_{z}*α*

_{0}=

*k*

_{0}

*κ*is the intrinsic attenuation coefficient of the film. The coefficient Γ is a correction factor that accounts for the fraction of total power in the film region relative to the total power in the mode itself. Because the cladding layers are not lossy, the evanescent fields do not experience any Ohmic power loss and therefore need to be accounted for. A rigorous discussion of this problem is presented in [12

12. T. D. Visser, H. Blok, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. **33**, 1763–1766 (1997). [CrossRef]

*(*

_{m}*x*) is the normalized TE field profile along

*x*for the

*m*th mode. These mode profiles can be filled by the solution for an equivalent lossless slab, which is valid as long as the lossless field profiles generally match the lossy case. Using Figure 3 as a guide, we surmise that this will generally be true for

*κ*<< 1.

*(*

_{m}*x*) can be taken from the

*y*-component of the magnetic field for similar results [12

12. T. D. Visser, H. Blok, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. **33**, 1763–1766 (1997). [CrossRef]

*θ*dependence.

## 6. Applications to thin-film photovoltaics

*M*= 4 mode of an

*h*= 500 nm film of amorphous silicon at

*λ*

_{0}= 600 nm (

*n*= 4.6 +

_{f}*j*0.3,

*n*= 1) [13

_{c}13. G. K. M. Thutupalli and S. G. Tomlin, “The optical properties of amorphous and crystalline silicon,” J. Phys. C Solid State **10**, 467–477 (1977). [CrossRef]

*α*for all TE modes of the device. The solutions using the low-loss approximation are also indicated by the blue markers at each data point. Clearly, the low-loss approximation produces a very accurate measure of the mode attenuation. As expected, we can also see that high-order modes are generally more lossy than low-order modes, with the

_{z}*M*= 6 mode possessing an absorption coefficient that is roughly twice that of the fundamental

*M*= 0 mode. However, the lossy waveguide also possesses an extra five eigenmodes in addition to the seven that exist for the equivalent lossless model. These loss-guided modes cannot be derived from any low-loss approximation and are strictly unique to the lossy waveguide model. It is also apparent that such modes possess dramatically higher values for

*α*than do the classical modes.

_{z}*M*= 7 mode is illustrated in Figure 10(a). When viewed along the transverse axis, such modes generally behave in the familiar fashion one should expect from a guided mode. In particular, there are eight characteristic peaks and valleys, as well as a strong evanescent decay in the cladding region. Figure 10(b) shows the same mode when viewed along the longitudinal axis. From this perspective, the mode is almost completely attenuated within a single wavelength cycle. This behavior is characteristic for the additional modes in a lossy dielectric waveguide, which tend to have very high values for

*α*. Such results may have useful implications for light-trapping applications where the ultimate goal is to maximize light absorption in a finite film [14

_{z}14. Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. U.S.A. **107**, 17491–17496 (2010). [CrossRef] [PubMed]

## 7. Conclusion

## Appendix A: Full-field solutions

*E*

_{0}term during the substitution. If we plug either of these expressions back into the Helmholtz equation, we arrive at the dispersion relation

*x*| ≥

*h*). In this case, we shall assume that

**E**may be written as Note that because of the phase-matching condition between planar boundaries, the

*z*-component to the wavenumber is identical in both the slab and cladding regions. The constant coefficient

*C*will be determined later after enforcing continuity at the boundaries. The plus or minus sign is again determined by imposing even or odd symmetry. The complex propagation constant

*γ*=

*γ*+

_{r}*j*

*γ*then satisfies the dispersion relation where

_{i}*k*=

_{c}*k*

_{0}

*n*is the intrinsic wavenumber of the cladding region.

_{c}**H**(

*x*,

*z*) by applying Faraday’s law, ∇ ×

**E**=

*jωμ*

_{0}

**H**. For the case of even symmetry, this gives us with again a similar expression for the case of odd symmetry.

*x*= ±

*h*. Starting with the even case, continuity on the E-field implies Applying continuity on the

*z*-component to the H-field then gives us Now divide Equation (A.10) by Equation (A.9) to find To handle the

*γ*term, we combine Equations (A.4) and Equation (A.6) to arrive at The final result after substitution is Equation (4). Applying the same process to odd symmetry or the TM polarization produces Equations (5)–(7).

## Appendix B: Steepest descent method with linear line search

15. J. A. Snyman, *Practical Mathematical Optimization: An Introduction to Basic Optimization Theory and Classical and New Gradient-Based Algorithms (Applied Optimization)* (Springer, 2005). [PubMed]

*k*. For example, the even TE case produces with similar expressions for the odd case and TM polarization. Starting with a trial solution

_{x}*k*

_{x,0}, we then execute the following algorithm: This algorithm is repeated until

*ϕ*(

*k*

_{x,n}) falls below some given threshold of error tolerance (say, 10

^{−9}). For the case of relatively low-loss dielectrics, a good choice for

*k*

_{x,0}is the mode solution to an equivalent system without any loss. It is also important to bear in mind that the line-search algorithm applies a linear approximation to the derivative of the residual when determining the optimal step size

*u*. As a result,

_{n}*u*can often be over-estimated, thus leading to a divergence in the search algorithm. This can be alleviated by ensuring that each step of the algorithm enforces the descent condition, which is written as Whenever a new iteration fails to satisfy this condition, a very simple remedy is to reduce

_{n}*u*by a factor of 1/2. The iteration attempt is then repeated until Equation (B.8) finally holds, and the algorithm can then continue normally.

_{n}## References and links

1. | C. A. Balanis, |

2. | J. A. Kong, |

3. | G. H. Owyang, |

4. | C. R. Pollock and M. Lipson, |

5. | A. W. Snyder and J. D. Love, |

6. | S. Miyanaga and H. Fujiwara, “Effects of absorption on the propagation constants of guided modes in an asymmetric slab optical waveguide,” Opt. Commun. |

7. | K. Fujii, “Dispersion relations of TE-mode in a 3-layer slab waveguide with imaginary-part of refractive index,” Opt. Commun. |

8. | T. D. Visser, H. Blok, and D. Lenstra, “Modal analysis of a planar waveguide with gain and losses,” IEEE J. Quantum Electron. |

9. | A. E. Siegman, “Propagating modes in gain-guided optical fibers,” J. Opt. Soc. Am. A |

10. | A. W. Snyder and J. D. Love, “Goos-Hanchen shift,” Appl. Opt. |

11. | Lumerical Solutions Inc., http://www.lumerical.com/. |

12. | T. D. Visser, H. Blok, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. |

13. | G. K. M. Thutupalli and S. G. Tomlin, “The optical properties of amorphous and crystalline silicon,” J. Phys. C Solid State |

14. | Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. U.S.A. |

15. | J. A. Snyman, |

**OCIS Codes**

(310.2790) Thin films : Guided waves

(310.6805) Thin films : Theory and design

**ToC Category:**

Thin Films

**History**

Original Manuscript: July 29, 2011

Revised Manuscript: August 30, 2011

Manuscript Accepted: September 3, 2011

Published: September 29, 2011

**Citation**

James R. Nagel, Steve Blair, and Michael A. Scarpulla, "Exact field solution to guided wave propagation in lossy thin films," Opt. Express **19**, 20159-20171 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-21-20159

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### References

- C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, NY, 1989).
- J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, Cambridge, MA, 2000).
- G. H. Owyang, Foundations of Optical Waveguides (Elsevier, New York, NY, 1981).
- C. R. Pollock and M. Lipson, Integrated Photonics (Kluwer Academic Publishers, Boston, MA, 2003).
- A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, NY, 1983).
- S. Miyanaga and H. Fujiwara, “Effects of absorption on the propagation constants of guided modes in an asymmetric slab optical waveguide,” Opt. Commun.64, 31–35 (1987). [CrossRef]
- K. Fujii, “Dispersion relations of TE-mode in a 3-layer slab waveguide with imaginary-part of refractive index,” Opt. Commun.171, 245–251 (1999). [CrossRef]
- T. D. Visser, H. Blok, and D. Lenstra, “Modal analysis of a planar waveguide with gain and losses,” IEEE J. Quantum Electron.31, 1803–1810 (1995). [CrossRef]
- A. E. Siegman, “Propagating modes in gain-guided optical fibers,” J. Opt. Soc. Am. A20, 1617–1628 (2003). [CrossRef]
- A. W. Snyder and J. D. Love, “Goos-Hanchen shift,” Appl. Opt.15, 236–238 (1973). [CrossRef]
- Lumerical Solutions Inc., http://www.lumerical.com/ .
- T. D. Visser, H. Blok, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron.33, 1763–1766 (1997). [CrossRef]
- G. K. M. Thutupalli and S. G. Tomlin, “The optical properties of amorphous and crystalline silicon,” J. Phys. C Solid State10, 467–477 (1977). [CrossRef]
- Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. U.S.A.107, 17491–17496 (2010). [CrossRef] [PubMed]
- J. A. Snyman, Practical Mathematical Optimization: An Introduction to Basic Optimization Theory and Classical and New Gradient-Based Algorithms (Applied Optimization) (Springer, 2005). [PubMed]

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