## Solving dielectric and plasmonic waveguide dispersion relations on a pocket calculator

Optics Express, Vol. 17, Issue 26, pp. 24112-24129 (2009)

http://dx.doi.org/10.1364/OE.17.024112

Acrobat PDF (854 KB)

### Abstract

We present a robust iterative technique for solving complex transcendental dispersion equations routinely encountered in integrated optics. Our method especially befits the multilayer dielectric and plasmonic waveguides forming the basis structures for a host of contemporary nanophotonic devices. The solution algorithm ports seamlessly from the real to the complex domain—i.e., no extra complexity results when dealing with leaky structures or those with material/metal loss. Unlike several existing numerical approaches, our algorithm exhibits markedly-reduced sensitivity to the initial guess and allows for straightforward implementation on a pocket calculator.

© 2009 Optical Society of America

## 1. Introduction

2. R. Zia, J. A. Schuller, and M. L. Brongersma, “Plasmonics: The next chip-scale
technology,” Materials Today **9**, 20–27 (2006). [CrossRef]

3. R. A. Pala, J. S. White, E. S. Barnard, J. Liu, and M. L. Brongersma, “Design of plasmonic thin-film solar cells with
broadband absorption enhancements,” Adv. Mater. **21**, 1–6 (2009). [CrossRef]

*transcendental*and except in a few simple cases, their solutions cannot be expressed in terms of elementary mathematical functions.

6. J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional guiding of a
one-dimensional optical beam with nanometer diameter,”
Opt. Lett. **22**, 475–477
(1997). [CrossRef] [PubMed]

7. J.-C. Weeber, Y. Lacroute, and A. Dereux, “Optical near-field distributions of surface plasmon
waveguide modes,” Phys. Rev. B **68**, 115401 (2003). [CrossRef]

8. R. Zia, A. Chandran, and M. L. Brongersma, “Dielectric waveguide model for guided surface
polaritons,” Opt. Lett. **30**, 1473–1475
(2005). [CrossRef] [PubMed]

9. R. Zia, J. A. Schuller, and M. L. Brongersma, “Near-field characterization of guided polariton
propagation and cutoff in surface plasmon waveguides,”
Phys. Rev. B **74**, 165415 (2006). [CrossRef]

10. R. Zia, M. D. Selker, and M. L. Brongersma, “Leaky and bound modes of surface plasmon
waveguides,” Phys. Rev. B **71**, 165431 (2005). [CrossRef]

11. G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal
subwavelength plasmonic waveguides,” Appl. Phys.
Lett. **87**, 131102 (2005). [CrossRef]

12. G. Veronis and S. Fan, “Guided subwavelength plasmonic mode supported by a
slot in a thin metal film,” Opt. Lett. **30**, 3359–3361
(2005). [CrossRef]

13. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components
including interferometers and ring resonators,”
Nature **440**, 508–511
(2006). [CrossRef] [PubMed]

14. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless
and lossy planar multilayer optical waveguides: reflection pole method and
wavevector density method,” J. Lightwave Technol. **17**, 929–941
(1999). [CrossRef]

15. R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface
plasmon modes,” J. Opt. Soc. Am. A **21**, 2442–2446
(2004). [CrossRef]

16. S. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal
waveguides,” Phys. Rev. B **79**, 035120 (2009). [CrossRef]

17. J. P. McKelvey, “Simple iterative procedures for solving
transcendental equations with the electronic slide rule,”
Am. J. Phys. **43**, 331–334
(1975). [CrossRef]

## 2. Iterative technique: A simple example

*F*(

*x*) is a combination of elementary mathematical functions. We reformulate this as an iterative problem by converting Eq. (1) to:

*f*is obtained by manipulating the parent function

*F*. Next, we start with an initial guess

*x*

_{1}and obtain a sequence according to:

*x*⋍0.5. To find the root more accurately by the iterative method, we recast the equation in a form similar to Eq. (2) by choosing

*f*(

*x*)=1-sin

*x*, setting up an iteration scheme following Eq. (3):

*x*=0.5109734293885691. We compare the 16-decimal agreement between this value and the solution

*x*=0.5109734293885691 computed by Mathematica

^{™}’s FindRoot function that implements an optimized version of the Newton’s method [4] commonly used for solving transcendental equations. Figure 2(c) shows the LHS and RHS of the iteration scheme in Eq. (5) and Fig. 2(a) shows how the procedure converges to the intersection point of the two curves, starting from the initial guess. Making an initial guess corresponds to choosing a point on the curve

*y*=

*x*shown by the red curve in Fig. 2(a). This point is indicated in Fig. 2(a) by the red dot. The vertical lines in the spiral represent computation of

*f*(

*x*) for the chosen

*x*and the horizontal lines represent re-substitution of

*x*by

*f*(

*x*) for the next iteration.

*F*(

*x*), there are usually multiple ways to choose

*f*(

*x*) which differ in their convergence behavior. For example, an equally-legitimate way of setting up an iteration scheme for Eq. (4) would be to choose

*f*(

*x*)=sin

^{-1}(1-

*x*) and obtain the sequence {

*x*} according to:

_{n}*x*} of iterations according to:

_{n}*f*(

*x*) is steeper than x (i.e., if |

*f*′(

*x*)|>1), the successive computations and re-substitutions spiral away from the intersection and the solution diverges. On the other hand, if

*f*(

*x*) rises gently compared to

*x*(i.e., if |

*f*′(

*x*)|<1), then the scheme spirals toward the intersection and the solution converges. For convergent functions, the rate of convergence is governed by the magnitude of |

*f*′(

*x*)|. The RHS of Eq. (8) is much flatter (|

*f*′(

*x*)|≪1) near the solution compared to the RHS of Eq. (5), which results in the latter’s considerably slower convergence. While the preceding justification is not a rigorous proof, it provides a basis for choosing the manipulations required to obtain convergent forms of the practically-useful equations we consider in the forthcoming sections [18].

## 3. Dispersion equation of a general asymmetric three-layer structure

*ε*. The bottom and top cladding layers are called the substrate (with permittivity

_{f}*ε*) and cover (with permittivity

_{s}*ε*), respectively.

_{c}*ε*>

_{f}*ε*,

_{s}*ε*. The structures in Figs. 4(c) and (d) are two basic configurations of plasmonic waveguides. The first, shown in Fig. 4(c), consists of an dielectric sandwiched between two (possibly different) metals and is commonly known as a metal-dielectric-metal (MDM) waveguide. The second structure, shown in Fig. 4(d), consists of a metal layer between two (possibly different) dielectrics; it is commonly known as an dielectric-metal-dielectric (DMD) waveguide.

_{c}20. J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy
metal films,” Phys. Rev. B **33**, 5186–5201
(1986). [CrossRef]

*k*(i.e.,

*k*→

*iκ*,

*κ*∊ ℝ). Having solved for

*k*or

*κ*, the mode’s complex effective index

*n*

_{eff}may be calculated using relations in Table 1.

## 4. Modes of dielectric waveguides

### 4.1. Strong-confinement dielectric waveguides

*M*), and the polarization (

*p*and

*q*) of the desired mode. For TE polarization (

*p*=

*q*=1) and even parity (+ sign), the effective indices of successive even TE modes may be simply calculated by setting

*M*=0,1, …

*m*and iterating according to Eq. (12); this yields the TE

_{0}, TE

_{2}, … TE

*modes. For TE polarization with odd parity (- sign), setting*

_{2m}*M*=1,2, …,

*m*yields the TE

_{1}, TE

_{3}, … TE

_{2m-1}modes. Note that for even modes, the mode order begins with

*M*=0, whereas for odd modes, it begins with

*M*=1. The procedure for obtaining TM mode indices is identical, with proper input of (

*p,q*) as shown in Table 1.

*µ*m-thick silicon-on-insulator (SOI) slab waveguide operating at 1550 nm as a test structure. The refractive indices of the silicon film, oxide substrate, and air cover are assumed to be √

*ε*=3.50,√

_{f}*ε*=1.45, and √

_{s}*ε*=1.00 respectively. Using the TE

_{c}_{0}mode as an example, Fig. 6 depicts how Eq. (12) converges to the solution. Fig. 6(b) and (c) show the real and imaginary values of the first twenty iterates of Eq. (12). Although the iteration is carried out in terms of

*k*, we have chosen to depict the convergence in terms of the effective index

*n*

_{eff}=

*β*/

*k*

_{0}since this is a more familiar quantity in waveguide analysis. Figure 6(a) shows the LHS and RHS of Eq. (12) and how the iterative scheme converges to the solution. Notice that the slope condition mentioned in section 1 is satisfied for this particular case.

*k*

_{1}=

*K*. Moreover, the convergence of the iterative technique is completely insensitive to the exact value of this initial guess: we obtained the same effective

_{s}*K*to 10

_{s}^{5}

*K*(including complex values). While a precise mathematical characterization of the convergence behavior is outside the scope of our paper, this exercise does suggest the robustness of the iterative technique with respect to the accuracy of the initial guess.

_{s}#### 4.1.1. Symmetric strong-confinement waveguides

*ε*and

_{s}*ε*:

_{c}*M*assumes values starting from 0 for even modes and 1 for odd modes.

#### 4.2. Weak-confinement dielectric waveguides

*µ*m-thick air-clad GaAs waveguide on an Al

_{0.1}Ga

_{0.9}As substrate operating at 1550 nm. The refractive indices of GaAs and Al

_{0.1}Ga

_{0.9}As are assumed to be √

*ε*=3.300 and √

_{f}*ε*=3.256 respectively. We attempt to find the fundamental TE mode iteratively by inputting the corresponding parameters to Eq. (12) (

_{s}*M*=0,

*p*=

*q*=1, and + sign). Figures 7(b) and (c) show the behavior of the real and imaginary parts of the calculated effective index for first 50 iterates. The real part of the effective index initially diverges to reach a maximum at iteration number 30 and begins to converge thereafter. Starting from the thirtieth iteration, the imaginary part oscillates between ±0.01. These oscillations do not damp out even if we increase the number of iterations to 10

^{4}. A graphical solution using Mathematica

^{™}, however, indicates that this structure supports TE

_{0}and TM

_{0}modes with indices of approximately 3.266 and 3.263, respectively. We conclude that the iterative scheme in Eq. (12) fails to converge for this example.

*f*′

_{1}(

*κ*)|>1. As a result, the iterates begin to diverge away from the intersection point. In fact, |

*f*′

_{1}(

*κ*)| continues to increase away from the intersection, causing a rapidly-increasing divergence. However, at

*κ*⋍0.537

*k*

_{0}, |

*f*′

_{1}(

*κ*)| abruptly becomes <1, causing the real part of the iteration to converge. This convergence of the real part is misleading, however since the imaginary part shows undamped oscillations. The behavior of the strong confinement formula in this case is similar to the example problem in section 1 illustrated in Fig. 2(d–f).

^{2}

*z*=1/cos

^{2}

*z*-1, rewriting Eq. (9) as:

*k*contained in the

*γ*term on the right hand side of Eq. (14):

_{c}γ_{s}_{0.1}Ga

_{0.9}As waveguide using Eq. (15) yields the effective indices of the TE

_{0}and TM

_{0}modes in excellent agreement with the Newton’s method, as seen from Table 3. The convergence behavior of the effective index is plotted in Fig. 7(d–f). The convergence of Eq. (15) improves as the core–cladding index contrast decreases; similarly, the convergence of Eq. (12) improves for

*increasing*core–cladding index contrast.

#### 4.2.1. Symmetric weak-confinement dielectric waveguides

#### 4.3. Extension to photonic wire waveguides

*w*and thickness

*h*. In the first step, we disregard the confinement in the

*x*-direction and solve for the effective index

*n*′. Depending on the polarization of the desired mode, we will need to solve either the TE or TM equation. In the second step, we consider a silica-clad slab waveguide with a core index

*n*′ confined in the

*x*-direction. Note that the orientation of the effective waveguide is orthogonal to that in the first step and hence the equation in this step is for the polarization orthogonal to that in the previous step. That is, if the first step uses the TE dispersion equation, then the second step uses the TM equation(and vice versa). The effective index

*n*″ determined in this second step is our final answer for the effective mode index of the original PW waveguide.

*E*-polarized mode of a sub-micron PW waveguide shown in Fig. 9 (a) with h=450nm and

_{y}*w*=300nm. In the first step, we solve for the TE mode equation for the structure shown in Fig. 8(b). Since SOI waveguide constitutes a strong-confinement system, the first step is readily accomplished by iteration of Eq. (12) for the TE condition (namely,

*M*=0,

*p*=

*q*=1). This yields the first effective index as

*n*′=3.073930677459340. In the second step, we consider an

*x*-confined slab with core-index

*n*′ and iterate Eq. (12) with the fundamental TM mode condition. This yields the effective index of the PW waveguide as

*n*″=2.652766507502340.

*n*″=2.612594. It is evident that even for sub-mircron dimensions, the relative error defined as |

*n*″

_{iter}-

*n*″

_{FEM}|/

*n*″

_{FEM}is only 1.5%. Figure 9(b) shows the variation of the relative error with waveguide dimensions for a fixed width-to-height aspect ratio of 1.5, from which we note the exponential drop in the relative error with increasing waveguide size. The reason for the decrease of accuracy with decreasing size lies in the increased interaction of the electro-magnetic fields with the corner regions of the waveguide. By decomposing a two-dimensional mode-solving problem into two one-dimensional problems, we effectively chose to neglect the interaction of the fields with the waveguide corners. This assumption becomes decreasingly accurate with decreasing waveguide size. Figure 9(c) and (d) show the y-component of the electric field, calculated using FEM and the iterative method, for the sub-micron waveguide considered above. Note the strong electric field at the waveguide corners in Fig. 9(c) calculated using FEM, and its absence in Fig. 9(d) calculated using the iterative method. In addition to highlighting this difference for small waveguides, this exercise illustrates the well-known fact that while EIM successfully calculates the effective indices, it does not guarantee satisfaction of the boundary conditions. It should therefore not be used to calculate fields for wavelength-sized structures.

## 5. Modes of plasmonic waveguides

### 5.1. Metal-dielectric-metal waveguides

16. S. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal
waveguides,” Phys. Rev. B **79**, 035120 (2009). [CrossRef]

#### 5.1.1. Gap-plasmon modes

*µ*m allows the structure to support gap-plasmon modes of both symmetries. We calculate the effective indices iteratively using Eq. (18) with + and - signs for odd and even gap-plasmon modes respectively. Once again, the iterative method agrees with the Newton’s method (Table 4).

#### 5.1.2. TM-like waveguide modes

*κ*is purely imaginary and plasmon Eq. (10) transforms to dielectric Eq. (9). The iterative form for obtaining the indices of these modes is identical to Eq. (12), with only a few slight differences as regards its implementation.

*M*assumed integer values starting from 0 for even modes and 1 for odd modes. For MDM waveguides, this is reversed:

*M*assumes integer values starting from 1 for even modes and 0 for odd modes. This is a consequence of the signs of

*p*and

*q*being reversed for MDM waveguides due to the negative permittivity of the metal “claddings.” Additionally, because of the large index difference between metals and dielectrics, the TM-like modes of MDM waveguides are almost always calculated using the strong-confinement formula in Eq. (12).

#### 5.1.3. Symmetric MDM waveguides

#### 5.2. Dielectric-metal-dielectric waveguides

*h*<100nm) metal films.

*A*and

*B*are defined in Table 1. Considering this as a quadratic equation in

*A*and

*B*leads us to the desired iterative forms. We start by identifying the lower-index dielectric as the substrate and using initial guesses of

*A*

_{0}=

*k*

_{0},

*B*

_{0}=0, and

*κ*

_{0}=

*k*

_{0}. We then iterate to obtain five different quantities successively, using the following equations in the order shown:

*ε*=1). We start by specifying

_{c}*A*

_{0},

*B*

_{0}, and

*κ*

_{0}, and obtain their successive values according to Eq. (21). The convergence of

*A*, and

_{n},B_{n}*κ*is shown in Fig. 11; Table 5 compares the solution obtained using the iterative scheme with that computed using the Newton’s method. In our computations, the Newton’s method was unable to return the complex mode index unless we gave a very precise

_{n}*and*the imaginary parts. On the other hand, the iterative method computed the complex mode index regardless of the initial guess.

#### 5.2.1. Symmetric DMD waveguides

## 6. Extension to multilayer structures

14. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless
and lossy planar multilayer optical waveguides: reflection pole method and
wavevector density method,” J. Lightwave Technol. **17**, 929–941
(1999). [CrossRef]

## 7. Conclusion

## Acknowledgements

## References and links

1. | M. L. Brongersma and P. G. Kik, eds., |

2. | R. Zia, J. A. Schuller, and M. L. Brongersma, “Plasmonics: The next chip-scale
technology,” Materials Today |

3. | R. A. Pala, J. S. White, E. S. Barnard, J. Liu, and M. L. Brongersma, “Design of plasmonic thin-film solar cells with
broadband absorption enhancements,” Adv. Mater. |

4. | W. H. Press, S. A. Teukolsky, W. J. Vetterling, and B. P. Flannery, |

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

6. | J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional guiding of a
one-dimensional optical beam with nanometer diameter,”
Opt. Lett. |

7. | J.-C. Weeber, Y. Lacroute, and A. Dereux, “Optical near-field distributions of surface plasmon
waveguide modes,” Phys. Rev. B |

8. | R. Zia, A. Chandran, and M. L. Brongersma, “Dielectric waveguide model for guided surface
polaritons,” Opt. Lett. |

9. | R. Zia, J. A. Schuller, and M. L. Brongersma, “Near-field characterization of guided polariton
propagation and cutoff in surface plasmon waveguides,”
Phys. Rev. B |

10. | R. Zia, M. D. Selker, and M. L. Brongersma, “Leaky and bound modes of surface plasmon
waveguides,” Phys. Rev. B |

11. | G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal
subwavelength plasmonic waveguides,” Appl. Phys.
Lett. |

12. | G. Veronis and S. Fan, “Guided subwavelength plasmonic mode supported by a
slot in a thin metal film,” Opt. Lett. |

13. | S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components
including interferometers and ring resonators,”
Nature |

14. | E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless
and lossy planar multilayer optical waveguides: reflection pole method and
wavevector density method,” J. Lightwave Technol. |

15. | R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface
plasmon modes,” J. Opt. Soc. Am. A |

16. | S. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal
waveguides,” Phys. Rev. B |

17. | J. P. McKelvey, “Simple iterative procedures for solving
transcendental equations with the electronic slide rule,”
Am. J. Phys. |

18. | J. Dugundji and A. Granas, |

19. | C. R. Pollock, |

20. | J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy
metal films,” Phys. Rev. B |

21. | H. Raether, “Surface plasmons on smooth and rough surfaces and on
gratings,” Springer Tracts Mod. Phys. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(240.6680) Optics at surfaces : Surface plasmons

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: October 13, 2009

Revised Manuscript: December 11, 2009

Manuscript Accepted: December 14, 2009

Published: December 17, 2009

**Citation**

Rohan D. Kekatpure, Aaron C. Hryciw, Edward S. Barnard, and Mark L. Brongersma, "Solving dielectric and plasmonic waveguide dispersion relations on a pocket calculator," Opt. Express **17**, 24112-24129 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-26-24112

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### References

- M. L. Brongersma and P. G. Kik, eds., Surface Plasmon Nanophotonics, Springer series in Optical Sciences (Springer, 2007) Vol. 131.
- R. Zia, J. A. Schuller, and M. L. Brongersma, "Plasmonics: The next chip-scale technology," Maters. Today 9, 20-27 (2006). [CrossRef]
- R. A. Pala, J. S. White, E. S. Barnard, J. Liu, and M. L. Brongersma, "Design of plasmonic thin-film solar cells with broadband absorption enhancements," Adv. Mater. 21, 1-6 (2009). [CrossRef]
- W. H. Press, S. A. Teukolsky,W. J. Vetterling, and B. P. Flannery, Numerical recipes in C++, The art of scientific computing (Cambridge University Press, 2002), 2nd ed.
- A. W. Snyder and J. Love, Optical Waveguide Theory (Science Paperbacks, 1983).
- J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, "Guiding of a one-dimensional guiding of a one-dimensional optical beam with nanometer diameter," Opt. Lett. 22, 475-477 (1997). [CrossRef] [PubMed]
- J.-C. Weeber, Y. Lacroute, and A. Dereux, "Optical near-field distributions of surface plasmon waveguide modes," Phys. Rev. B 68, 115401 (2003). [CrossRef]
- R. Zia, A. Chandran, and M. L. Brongersma, "Dielectric waveguide model for guided surface polaritons," Opt. Lett. 30, 1473-1475 (2005). [CrossRef] [PubMed]
- R. Zia, J. A. Schuller, and M. L. Brongersma, "Near-field characterization of guided polariton propagation and cutoff in surface plasmon waveguides," Phys. Rev. B 74, 165415 (2006). [CrossRef]
- R. Zia, M. D. Selker, and M. L. Brongersma, "Leaky and bound modes of surface plasmon waveguides," Phys. Rev. B 71, 165431 (2005). [CrossRef]
- G. Veronis and S. Fan, "Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides," Appl. Phys. Lett. 87, 131102 (2005). [CrossRef]
- G. Veronis and S. Fan, "Guided subwavelength plasmonic mode supported by a slot in a thin metal film," Opt. Lett. 30, 3359-3361 (2005). [CrossRef]
- S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440, 508-511 (2006). [CrossRef] [PubMed]
- E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, "Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method," J. Lightwave Technol. 17, 929-941 (1999). [CrossRef]
- R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, "Geometries and materials for subwavelength surface plasmon modes," J. Opt. Soc. Am. A 21, 2442-2446 (2004). [CrossRef]
- S. E. Kocabas¸, G. Veronis, D. A. B. Miller, and S. Fan, "Modal analysis and coupling in metal-insulator-metal waveguides," Phys. Rev. B 79, 035120 (2009). [CrossRef]
- J. P. McKelvey, "Simple iterative procedures for solving transcendental equations with the electronic slide rule," Am. J. Phys. 43, 331-334 (1975). [CrossRef]
- J. Dugundji and A. Granas, Fixed Point Theory (Springer-Verlag, 2003).
- C. R. Pollock, Fundamentals of Optoelectronics (McGraw-Hill Professional Publishing, 2003).
- J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186-5201 (1986). [CrossRef]
- Q1Q2. H. Raether, "Surface plasmons on smooth and rough surfaces and on gratings," Springer Tracts Mod. Phys. 111, 1-133 (1988).

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