## Propagation in planar waveguides and the effects of wall roughness

Optics Express, Vol. 9, Issue 9, pp. 461-475 (2001)

http://dx.doi.org/10.1364/OE.9.000461

Acrobat PDF (551 KB)

### Abstract

We consider the propagation of guided waves in a planar waveguide that has smooth walls except for a finite length segment that has random roughness. Maxwell’s equations are solved in the frequency domain for both TE and TM polarization in 2-D by using modal expansion methods. Obtaining numerical solutions is facilitated by using perfectly matched boundary layers and the **R**-matrix propagator.Varying lengths of roughness segments are considered and numerical results are obtained for guided wave propagation losses due to roughness induced scattering. The roughness on each waveguide boundary is numerically generated from an assumed Gaussian power spectrum. The guided waves are excited by a Gaussian beam incident on the waveguide aperture. Considerable numerical effort is given to determine the stability of the algorithm.

© Optical Society of America

## 1 Introduction

*z*=

*L*

_{t}to

*z*=

*L*

_{r}. At the end of the smooth segment begins a segment of roughness that has length extending from

*z*=

*L*

_{t}to

*z*=0. The length of the smooth portion is chosen so that at

*z*=

*L*

_{r}, the only remaining electromagnetic energy propagating downward is a guided wave mode. While transiting the roughness region, some of the guided wave energy is lost to scattering and after exiting the roughness region at

*z*=0, the remaining guided wave mode energy continues to propagate unimpeded in the semi-infinite smooth substrate region (

*z*<0) of the waveguide. The 1-D roughness is numerically generated with a Gaussian correlation function.

## 2 Theory

### 2.1 Maxwell’s Equations

*E*⃗ and magnetic

*B*⃗ fields in the frequency domain.

6. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys. **114**185–200 (1995). [CrossRef]

7. J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys. **127**363–379 (1996). [CrossRef]

*∊*(

*x*,

*z*), the magnetic conductivity σ

_{j}(

*x*,

*z*)* and the electric conductivity σ

_{j}(

*x*,

*z*). The PML conductivity terms, σ

_{j}(

*x*,

*z*) and σ

_{j}(

*x*,

*z*)

^{*}, control absorption for propagation of the electric and magnetic fields in the j-direction, respectively. The electric conductivity σ

_{j}(

*x*,

*z*) is not explicit in Eqs. (9) and (10) because the relationship σ

_{j}(

*x*,

*z*)=∊(

*x*,

*z*)

*x*,

*z*) is required for impedance matching. Ideally, impedance matching eliminates reflection when the PML absorbing layers are encountered. Outside of a PML absorption region, the σ

_{j}=

16. P. Lalanne and G. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. **13**779–784 (1996). [CrossRef]

17. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. **13**1870–1876 (1996). [CrossRef]

*L*

_{x}and discretize the x-coordinate. Second, we assume that the material parameters in Eq. (9) and (10) are z-invariant.

*N*points as

*x*

_{m}=

*m*Δ

*x*where integer

*m*=-

*N*/2 →

*N*/2 and Δ

*x*=

*L*

_{x}/

*N*. Likewise, we discretize the x-component of the wavevector

*k*

_{n}=

*n*Δ

*k*where

*n*=-

*N*/2 →

*N*/2 and Δ

*k*=2

*π*/

*L*

_{x}. Now that we have discretized and truncated the problem, we use matrix-vector notation to represent the Fourier transform and inverse Fourier transform:

**F**and

**F**

^{-1}. Such transform relations are written as

**F**is a

*N*×

*N*matrix and f is a

*N*-element column vector. Note that these are one dimensional transforms applied to the

*k*

_{x}-

*x*transform pair where

*k*

_{x}≡

*k*. Applying these transform operators to Eqs. (3)–(5) yields

**k**is interpreted as a diagonal matrix with elements

*k*

_{n}and the

**H**

_{y},

**E**

_{x}, and

**E**

_{z}are

*N*-element column vectors with each element corresponding to

*k*

_{n}. We have defined the

*N*×

*N*square matrices

*µ*

_{j}(

*x*),

*ε*

_{j}(

*x*),

*β*

_{j}(

*x*), and α

_{j}(

*x*) are interpreted as diagonal matrices with elements

*µ*

_{j}(

*x*

_{n}), etc.

**I**is the identity matrix. Since there are

*N*discrete

*k*values, Eq. (19) represents

*N*coupled differential equations. When

**M**is independent of z (this is the case when the material parameters are z-invariant), solution of this equation system is straightforward by diagonalization as

**S**

^{-1}

**MS**=

*ξ*

^{2}=

*ξ*·

*ξ*. The columns of the

*N*×

*N*matrix

**S**are the eigenvectors of

**M**and the diagonal matrix

*ξ*

^{2}contains the eigenvalues of

**M**. The solution to Eq. (19) is

*C*

_{±}are scalar constants and the

*e*

^{±ξz}represent diagonal matrices where the n-th diagonal element is an exponential term as exp(±

*ξ*

_{n}

*z*). The

*ξ*

_{n}is the n-th diagonal root-eigenvalue element of

*ξ*. In Eqs. (20) and (21) we now have solutions to Maxwell’s equations for an inhomogeneous solution space that has an x-dimension of length

*L*

_{x}and has material parameters that are z-invariant.

## 2.2 R-matrix Propagator

**R**-matrix algorithm which is but one of a class of several types of transfer matrices. First, for solutions within a single z-invariant layer bounded from

*z*→

*z*+

*δ*, a relationship is assumed to be given by

*δ*is the thickness of the layer. Inserting Eqs. (20) and (21) into Eq. (22) yields the

*N*×

*N*r-matrices as

*z*→

*z*+

*z*

_{t}, we use the

**R**-matrices which have a similar form as

*z*

_{t}is the cumulative thickness of all the z-invariant layers as described in Eq. (22). The

*N*×

*N*

**R**-matrices are calculated recursively by

**E**

_{x}and

**B**

_{y}across the boundaries separating z-invariant layers. It is seen from Eq. (22) and Eq. (25) that for the first z-invariant layer when

*z*

_{t}=

*δ*

_{1}, we initiate the recursion with R

_{ij}(

*δ*

_{1})=r

_{ij}(

*δ*

_{1}). If the next z-invariant layer has thickness

*δ*

_{2}, new r

_{ij}(

*δ*

_{2}) matrices are computed and the

**R**

_{ij}(

*δ*

_{1}) are used in Eqs. (26)–(29) to yield

**R**

_{ij}(

*δ*

_{1}+

*δ*

_{2}). These recursive relations are repeated until the desired thickness is achieved and when this occurs, we have a relation between the electric and magnetic fields at two planar boundaries that encompass the structure of interest.

## 2.3 Superstrate and Substrate

*z*≤

*L*

_{t}to solutions in the semi-infinite superstrate

*z*≥

*L*

_{t}and substrate

*z*≤0 and we can use Eq. (25) to provide such a relation. Setting

*z*=0 and

*z*

_{t}=

*L*

_{r}in Eq. (25), we can relate to the superstrate and substrate fields by invoking the continuity of the tangential components. If we have incident and reflected fields in the superstrate and a transmitted field in the substrate, then these continuity conditions are

*z*≥

*L*

_{t}is homogeneous. In the superstrate region, we let a Gaussian beam be incident on the waveguide aperture and our goal will be to calculate the resulting fields at the

*z*=0 and

*z*=

*L*

_{t}boundaries. This will allow us to further calculate the guided wave transmitted field after it has emerged from the roughness region and propagates into

*z*<0. Although we do not do so here, we can also calculate the reflected field for

*z*>

*L*

_{t}.

## 2.3.1 Transmitted field z≤0

*k*,

*z*) and

*k*,

*z*) transmitted fields. Solutions of the form given by Eqs. (20) and (21) are still valid for the transmitted region by setting C-=0 and this yields

*z*≤0 has only downward propagating waves, the eigenvalues associated with Eqs. (32) and (33) must have ℜ

*ξ*≥0 and

*ℑξ*≤0. The latter equations in Eqs. (32) and (33) give the z-dependence after the guided wave exits the roughness region. These equations can be examined numerically after we have obtained the solution for

*k*, 0). Equations Eqs. (32) and (33) may be combined to give the relationship between the electric and magnetic transmitted fields in the inhomogeneous region

*z*≤0 which is

## 2.3.2 Incident and Reflected Fields: *z*≥*L*_{t}

18. A. A. Maradudin, T. Michel, A. R. McGurn, and E. Mendez, “Enhanced backscattering of light from a random grating,” Annals of Physics **203**225–307 (1990). [CrossRef]

*ωσ*/c and the angle of incidence centered about

*θ*

^{i}. We write the x and y-components of the incident field as

*k*,

*z*) and

*k*,

*z*) where the n-th element is given by

*k*

_{n}<(

*ω*/c) and zero when

*k*

_{n}>(

*ω*/c). The wavevector

*k*

_{n}=2πn/

*L*

_{x}and sin

*θ*

_{n}=

*k*

_{n}/(

*θ*/c).

*k*,

*z*)=(

*q*c/

*ω*)

*k*,

*z*). With this and consistent with the notation of Eq. (31), we relate the column vectors associated with the reflected field as

**Z**(k) is

## 2.3.3 Surface Fields: *z*=0 and *z*=*L*_{t}

## 3 Numerical Results

## 3.1 General Comments

*L*

_{t}-

*L*

_{r}=3000λ and nominal waveguide channel width

*ω*=1λ. This length

*L*

_{t}-

*L*

_{r}is sufficient to allow the remaining transmitted energy that reaches the

*z*=

*L*

_{r}plane to only be one or more guided wave modes. With this, only a guided wave mode is incident on the roughness region. The length

*L*

_{r}of the roughness region is a parameter of study in this work that varies over many values. In the numerical algorithm, the length

*L*

_{r}=

*N*

_{r}

*δ*is the cumulative thickness of

*N*

_{r}z-invariant layers each having thickness

*δ*. This is illustrated in Fig. 2. Within each layer of thickness

*δ*, the waveguide channel width is dependent on the roughness at each waveguide boundary. We have let

*δ*=0.1λ for all calculations presented here. The waveguide permittivity values are ∊

_{1}=(2.25, 0.) and ∊

_{2}=(2.50, 0.). In most cases, we assume that the Gaussian beam is normally incident (

*θ*

^{i}=0) and the ratio

*σ*/λ=5 (see Eq. (35)). In Figs. 3–5, the extent of the computational region in the x-dimension is

*L*

_{x}=16.3λ and this length is divided into

*N*=299 segments of length Δ

*x*=

*L*

_{x}/

*N*=0.055λ.

*x*| in the vicinity of

*L*

_{x}/2eac h region has a total thickness denoted by

*δ*

_{PML}. If

*η*is the magnitude of the penetration depth into the PML region, where

*η*ranges over 0 →

*δ*

_{PML}, we assume that the PML absorption increases quadratically with penetration depth as

*N*

_{PML}layers each with thickness Δ

*x*and the total thickness may then be written as

*δ*

_{PML}=

*N*

_{PML}Δ

*x*. In all cases except the data shown in Fig. 9, we have set

*N*

_{PML}=24 and the value of

*A*

_{PML}=8. Since the x coordinate is discretized, we calculate the

*N*

_{PML }values of Eq. (42) with

*η*set to the middle value of the appropriate layer in a PML region.

## 3.2 Transmission Loss

*P*

_{t}/

*P*

_{i}where

*L*

_{r}increases. These three plots consider both TE and TM polarization and each plot includes four roughness realizations. The difference between the three plots is the rms roughness used to generate the realizations where the rms roughness values are 0.05λ, 0.10λ, and 0.20λ, respectively. The transmitted power is only plotted for every 10

*δ*=1λ increase in

*L*

_{r}. Thus, each curve in Figs. 3–5 contain 160 data points. However, since the numerical algorithm actually calculates in increments of

*δ*=0.1λ, the total number of sublayers over the roughness region ranges from 0 to 1600 over the range of

*L*

_{r}. To include the smooth region

*L*

_{r}≤

*z*≤

*L*

_{t}, we use four sublayers with

*δ*=750λ for a total thickness

*L*

_{t}-

*L*

_{r}=3000λ. As would be expected, it is seen in Figs. 3–5 that the rate of scattered power lost as

*L*

_{r}increases becomes greater as the rms roughness increases. These curves are summarized later in Section 4.

## 3.3 Convergence

*N*is varied with all other parameters held constant. Figure 6 is similar to Fig. 3 through Fig. 5 where the residual transmitted power versus

*L*

_{r}is shown, but only for realization 1. The main purpose of Fig. 6 is to show convergence as the number of digitization points

*N*increases for constant

*L*

_{x}=16.3λ. The data in Figs. 3–5 were generated for

*N*=299 whereas in Fig. 6, the seven TM curves cover

*N*=199 → 449 and the five TE cover

*N*=199 → 399. These data indicate numerical convergence as

*N*increases, or equivalently, as Δ

*x*decreases.

*x*is held fixed. We choose

*L*

_{x}and

*N*such that

*L*

_{x}/

*N*=Δ

*x*≈0.055λ. Values of

*L*

_{r}range from 14.3λ to 24.3λ and it is seen that as

*L*

_{r}increases, the transmission at

*z*=-10000λ is noticeably greater for the larger values of

*N*and

*L*

_{x}. The short answer for this discrepancy is that for larger

*L*

_{x}, the plane wave solutions are absorbed by the PML layers at a lower rate as

*z*→ -∞. As an example, when

*L*

_{x}=16.3λ one of the 299 calculated complex eigenvalues is

*ξ*=(

*ω*/

*c*)(0.9340×10

^{-4}-1.5i). When

*L*

_{x}=2 4.3λ there are 441 calculated eigenvalues and the corresponding eigenvalue is now

*ξ*=(

*ω*/

*c*)(0.1217×10

^{-4},-1.5i). In the transmission region, the solutions are proportional to exp(

*ξz*) (see Eqs. (32) and (33)) and it is clear that these particular eigenvalues pertain to the downward propagating plane wave solution in the

*∊*=2.25 bounding medium of the waveguide structure. The absorptive real part of ξ is generated by the presence of the PML layers. Note that at

*z*=-10000λ and the two

*ξ*values above, the products ℜ(

*ξ*)

*z*are -0.9340(

*ω*/

*c*) and -0.1217(

*ω*/

*c*). For these values it is clear that exp(

*ξz*) is still contributing a non-negligible solution value to the transmitted field. In hindsight, this discrepancy could have been avoided simply by choosing

*z*<<-10000λ such that ℜ(

*ξ*)

*z*<<0 and all plane wave solutions would have been sufficiently decayed to be a negligible contribution to the transmitted field. We conclude that the discrepancy noted in Fig. 7 is not a convergence problem.

*N*

_{PML}and the absorption coefficient

*A*

_{PML}are varied. For curves labeled

*N*

_{PML}=1 through 24, we set

*A*

_{PML}=8 and vary

*N*

_{PML}as indicated. For the curve labeled

*N*

_{PML}=0, we have for numerical reasons actually set

*N*

_{PML}=1 with

*A*

_{PML}=0.1. This latter case does not completely eliminate PML absorption, but the effect is relatively minimal. We see that the

*N*

_{PML}=0 curve shows little absorption at

*z*=-10000λ and examination of the plane wave eigenvalues as was done in Fig. 7 shows a very low rate of absorption coefficient. As the PML absorption is increased, the curves quickly converge to a common result.

*L*

_{x}of the computational domain. Unlike the previous figures, there is no initial smooth region followed by a roughness region. Here, all z-values are measured relative to the input aperture to the waveguide. It is seen that the PML layers perform quite well. The onset of PML layers is shown by the vertical lines near the x-limits of the computational region. In this example the total thickness of a PML region is 1.47λ. The intensity curve labeled

*z*=-205λ will eventually become a single mode guided wave (as is the case in Figs. 3–5) after the plane wave and evanescent modes have vanished. This guided wave mode has eigenvalue

*ξ*=(

*ω*/

*c*)(0,-1.552i).

*y*(

*x*)=

*m*

_{0}10

^{m1x}where

*y*is the normalized transmission and

*x*is

*L*

_{r}/λ. The

*m*

_{0}and

*m*

_{1}are fitting parameters. In Figs. 3–5, the fitting parameters are

*m*

_{0}=0.293, 0.286, and 0.277, and

*m*

_{1}=-0.00045, -0.00168, and -0.00542, respectively. These analytical curves are plotted in Fig. 10 and the loss can be expressed in dB as -10Log

_{10}[

*y*(

*x*)/

*y*(0)] which yields 0.0045

*L*

_{r}/λ, 0.0168

*L*

_{r}/λ, and 0.0542

*L*

_{r}/λ, respectively.

## 4 Concluding Remarks

**R**-matrix algorithm allows very large computational domains to be considered with numerical stability.

## References and links

1. | J. M. Elson and P. Tran, “R-matrix propagator with perfectly matched layers for the study of integrated optical components,” J. Opt. Soc. Am. A |

2. | J. M. Elson and P. Tran, “Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal,” Phys. Rev. B |

3. | J. M. Elson and P. Tran, “Band structure and transmission of photonic media: a real-space finite-difference calculation with the R-matrix propagator,” |

4. | L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A |

5. | L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. |

6. | J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys. |

7. | J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys. |

8. | D. Marcuse, “Mode conversion caused by surface imperfections in a dielectric slab waveguide,” Bell Sys. Tech. J. |

9. | D. Marcuse, “Radiation losses of dielectric waveguides in terms of the power spectrum of the wall distortion function,” Bell Sys. Tech. J. |

10. | F. P. Payne and J. P. R. Lacey, “A theoretical analysis of scattering loss from planar optical waveguides,” Opt. and Quantum Elec. |

11. | J. P. R. Lacey and F. P. Payne, “Radiation loss from planar waveguides with random wall imperfections,” IEE Proc. J. |

12. | K. K. Lee, D. R. Lim, H. Luan, A. Agarwal, J. Foresi, and L. Kimerling, “Effect of size and roughness on light transmission on a Si/SiO |

13. | F. Ladouceur, J. D. Love, and T. J. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc.-Optoelectron. |

14. | F. Ladouceur, J. D. Love, and T. J. Senden, “Measurement of surface roughness in buried channel waveguides,” Electron. Lett. |

15. | J. Rodríguez, R. D. Crespo, S. Fernández, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibáñez, and J. Virgós, “Radiation losses on discontinuities in integrated optical waveguides,” Opt. Engr. |

16. | P. Lalanne and G. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. |

17. | L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. |

18. | A. A. Maradudin, T. Michel, A. R. McGurn, and E. Mendez, “Enhanced backscattering of light from a random grating,” Annals of Physics |

**OCIS Codes**

(230.7390) Optical devices : Waveguides, planar

(290.5880) Scattering : Scattering, rough surfaces

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 28, 2001

Published: October 22, 2001

**Citation**

J. Merle Elson, "Propagation in planar waveguides and the effects of wall roughness," Opt. Express **9**, 461-475 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-9-461

Sort: Journal | Reset

### References

- J. M. Elson and P. Tran, "R-matrix propagator with perfectly matched layers for the study of integrated optical components," J. Opt. Soc. Am. A 16, 2983-2989 (1999). [CrossRef]
- J. M. Elson and P. Tran, "Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal," Phys. Rev. B 54, 1711-1715 (1996). [CrossRef]
- J. M. Elson and P. Tran, "Band structure and transmission of photonic media: a real-space finite-difference calculation with the R-matrix propagator," NATO ASI Series E: Applied Sciences on Photonic Band Gap Materials Vol. 315 341-354 Crete, Greece June 15-29 (1995).
- L. Li, "Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity," J. Opt. Soc. Am. A 11, 2816-2828 (1993). [CrossRef]
- L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. 13, 1024-1035 (1996). [CrossRef]
- J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comp. Phys. 114, 185-200 (1995). [CrossRef]
- J. P. Berenger, "Three-dimensional perfectly matched layer for the absorption of electromagnetic waves," J. Comp. Phys. 127, 363-379 (1996). [CrossRef]
- D. Marcuse, "Mode conversion caused by surface imperfections in a dielectric slab waveguide," Bell Sys. Tech. J. 48, 3187-3215 (1969).
- D. Marcuse, "Radiation losses of dielectric waveguides in terms of the power spectrum of the wall distortion function," Bell Sys. Tech. J. 48, 3233-3242 (1969).
- F. P. Payne and J. P. R. Lacey, "A theoretical analysis of scattering loss from planar optical waveguides," Opt. and Quantum Elec. 26, 977-986 (1994). [CrossRef]
- J. P. R. Lacey and F. P. Payne, "Radiation loss from planar waveguides with random wall imperfections," IEEE Proc. J. 137, 282-288 (1990).
- K. K. Lee, D. R. Lim, H. Luan, A. Agarwal, J. Foresi and L. Kimerling, "Effect of size and roughness on light transmission on a Si/SiO2 waveguide: Experiment and model," Appl. Phys. Lett. 77, 1617-1619 (2000). [CrossRef]
- F. Ladouceur, J. D. Love and T. J. Senden, "Effect of side wall roughness in buried channel waveguides," IEEE Proc.-Optoelectron. 141, 242-248 (1994). [CrossRef]
- F. Ladouceur, J. D. Love and T. J. Senden, "Measurement of surface roughness in buried channel waveguides," Electron. Lett. 28, 1321-1322 (1992). [CrossRef]
- J. Rodrguez, R. D. Crespo, S. Fernandez, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibanez, J. Virgos, "Radiation losses on discontinuities in integrated optical waveguides," Opt. Engr. 38, 1896-1906 (1999). [CrossRef]
- P. Lalanne and G. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am. 13, 779-784 (1996). [CrossRef]
- L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. 13, 1870-1876 (1996). [CrossRef]
- A. A. Maradudin, T. Michel, A. R. McGurn, and E. Mendez, "Enhanced backscattering of light from a random grating," Annals of Physics 203, 225-307 (1990). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

OSA is a member of CrossRef.