## Excitation of s-polarized surface electromagnetic waves in inhomogeneous dielectric media

Optics Express, Vol. 16, Issue 17, pp. 13354-13363 (2008)

http://dx.doi.org/10.1364/OE.16.013354

Acrobat PDF (326 KB)

### Abstract

We consider a model of an inhomogeneous dielectric slab first studied by Shvartzburg, Petite and Auby [J. Opt. Soc. Am. B 16, 966 (1999)] and several variations of that model and study the excitation of *s*-polarized surface electromagnetic waves on the surface of inhomogeneous dielectric media. Using the invariant imbedding theory of wave propagation in stratified media, we calculate the reflectance and the absorptance of an *s* wave incident obliquely on a dielectric slab in the Otto configuration, as a function of incident angle and frequency. We also calculate the spatial distribution of the electric field intensity in the inhomogeneous region. We find that in all cases we have considered, *s*-polarized surface waves are excited at certain incident angles and frequencies. We discuss the physical mechanism of the surface wave generation and the possibility of experimental observations of these effects.

© 2008 Optical Society of America

## 1. Introduction

*p*-polarized. In addition, because of the large difference in momentum between external electromagnetic waves and surface plasmons, there has to be a way to provide the momentum difference. For this purpose, one uses the Kretschmann or Otto configuration, or breaks the translational symmetry by etching gratings or drilling holes into metal [1, 8

8. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature **391**, 667–669 (1998). [CrossRef]

*s*-polarized incident waves can also excite surface electromagnetic waves [9–15

9. G. I. Stegeman, J. D. Valera, C. T. Seaton, J. Sipe, and A. A. Maradudin, “Nonlinear s-polarized surface plasmon polaritons,” Solid State Commun. **52**, 293–297 (1984). [CrossRef]

*s*-polarized surface electromagnetic waves which can be excited on the surface of inhomogeneous semiconductors [14

14. A. Shvartzburg, G. Petite, and N. Auby, “S-polarized surface electromagnetic waves in inhomogeneous media: exactly solvable models,” J. Opt. Soc. Am. B **16**, 966–970 (1999). [CrossRef]

14. A. Shvartzburg, G. Petite, and N. Auby, “S-polarized surface electromagnetic waves in inhomogeneous media: exactly solvable models,” J. Opt. Soc. Am. B **16**, 966–970 (1999). [CrossRef]

*s*-polarized surface electromagnetic waves are excited in all cases we have studied. This suggests that they can exist in a fairly broad range of inhomogeneous dielectrics.

*s*-polarized surface waves in this model. In Section 3, we describe the dispersion relation in the special case considered in [14

14. A. Shvartzburg, G. Petite, and N. Auby, “S-polarized surface electromagnetic waves in inhomogeneous media: exactly solvable models,” J. Opt. Soc. Am. B **16**, 966–970 (1999). [CrossRef]

## 2. Model

*s*wave of vacuum wave number

*k*

_{0}=

*ω*/

*c*incident on a stratified medium where the dielectric permittivity

*ε*varies only in the

*z*direction. We assume that the medium lies in 0≤

*z*≤

*L*and the wave propagates in the

*xz*plane. Then the complex amplitude of the electric filed,

*E*=

*E*(

*z*), satisfies

*q*is the

*x*component of the wave vector. We assume that the wave is incident from the region where

*z*>

*L*and

*ε*=

*ε*

_{1}and is transmitted to the region where

*z*<0 and

*ε*=

*ε*

_{1}. When

*θ*is defined as the angle of incidence,

*q*is equal to

*h̄*

^{2}

*k*

_{2}/2

*m*) incident on a potential

*V*(

*z*) in one dimension, if we identify

*V*(

*z*)/𝓔 with [1-

*ε*(

*z*)/

*ε*

_{1}]/cos

^{2}

*θ*and

*k*with

24. K. Kim, F. Rotermund, and H. Lim, “Disorder-enhanced transmission of a quantum mechanical particle through a disordered tunneling barrier in one dimension: Exact calculation based on the invariant imbedding method,” Phys. Rev. B **77**, 024203 (2008). [CrossRef]

*ε*(

*z*)/

*ε*

_{1}]/cos

^{2}

*θ*plays the role of a potential in quantum mechanics. This leads us to claim that in order to excite

*s*-polarized surface electromagnetic waves in linear dielectric media, it is necessary to have strong inhomogeneity in the

*z*dependence of

*ε*so that it can provide a confining potential for surface waves, similarly to the cases where quantum mechanical particles are confined in a potential well.

**16**, 966–970 (1999). [CrossRef]

*D*, which lies in 0≤

*z*≤

*D*, is given by

*F*(

*z*) describes the spatial dependence of

*ε*near the surface at

*z*=

*D*and

*a*is the length scale representing the thickness of the inhomogeneous region. The dimensionless parameter

*b*controls the strength of inhomogeneity. The spatial variation of

*ε*(

*z*) is supposed to describe approximately the carrier depletion layer formed on the surface of a degenerate semiconductor [26–28

26. I. P. Ipatova, A. Yu. Maslov, L. V. Udod, G. Benedek, and G. Panzarini, “The enhancement factor of hyper-Raman scattering from an inhomogeneous semiconductor surface,” Surf. Sci. **377–379**, 436–439 (1997). [CrossRef]

*ε*is the frequency-independent part of

_{L}*ε*and

*γ*is the damping parameter. The plasma frequency

*ω*is expressed in terms of

_{p}*N*

_{0}, which is the density of charge carriers at

*z*=

*D*, and the carrier effective mass

*m*

_{eff}. The dispersion relation for surface waves when

*α*=1 was derived and analyzed in detail in [14

**16**, 966–970 (1999). [CrossRef]

*α*=1 case. We consider the cases where

*α*=0.5, 0.75, 1, 1.25 and 1.5 and calculate the reflectance, the absorptance and the electric field distribution inside the inhomogeneous region.

*ε*(

*z*) considered here. We use the parameter values

*N*

_{0}=3×10

^{16}cm

^{-3},

*ε*=12,

_{L}*m*

_{eff}=0.01

*m*,

_{e}*γ*=0 and

*b*=-0.53, where

*m*is the electron mass. The value of the plasma frequency

_{e}*ω*is about 2.82×10

_{p}^{13}/sec. We choose

*ω*=5×10

^{13}/sec and

*α*=0.5,1,1.5 and plot (1-

*ε*), which plays the role of an effective potential when

*θ*=0, as a function of the normalized distance from the surface of a semiconductor in contact with the air. We note that there is a deep potential well near the surface. Since the effective potential is inversely proportional to cos

^{2}

*θ*, the potential well becomes infinitely deeper as

*θ*→

*π*/2.

*p*waves in layered metallic structures, it is necessary to use the Kretschmann or Otto configuration to compensate the large difference in momentum between the incident electromagnetic waves and the surface waves. For this purpose, we place a uniform dielectric slab with

*ε*=

*ε*(<

_{d}*ε*

_{1}) in

*D*<

*z*<

*L*.

## 3. Dispersion relation in a special case

*α*is equal to 1 in Eq. (2), it is possible to transform Eq. (1) into the Bessel equation

*u*increases to infinity, we need the condition that

*s*

^{2}should be always positive. Then the solution of Eq. (3) is given by the modified Bessel function of the second kind,

*K*

_{ρ}(

*su*). In [14

**16**, 966–970 (1999). [CrossRef]

^{2}should also be nonnegative in order to have a decaying solution. This latter condition, however, is unnecessary, since

*K*

_{ρ}(

*x*) with a pure imaginary ρ decays exponentially as

*x*→∞, too [29

29. T. M. Dunster, “Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter,” SIAM J. Math. Anal. **21**, 995–1018 (1990). [CrossRef]

*E*and the transverse component of the magnetic field,

*H*, in the medium in terms of the Bessel function

_{x}*K*

_{ρ}(

*su*) and using the continuity of these fields across the surface, we obtain the dispersion relation for surface waves propagating in the

*x*direction along the surface:

*q*and

*ω*in an implicit manner.

## 4. Invariant imbedding method

*Ẽ*(

*x*,

*z*)=

*E*(

*z*)exp(

*iqx*)=exp[

*ip*(

*L*-

*z*)+

*iqx*], where

*z*>

*L*. The quantities of main interest are the reflection and transmission coefficients,

*r*=

*r*(

*L*) and

*t*=

*t*(

*L*), defined by the wave functions outside the medium:

*r*and

*t*:

*=*ε ˜

*ε*/

*ε*

_{1}[19]. These are supplemented with the initial conditions

*r*(0)=0 and

*t*(0)=1. For given values of

*k*

_{0}and

*θ*and for an arbitrary function

*ε*(

*l*), we integrate Eq. (7) from

*l*=0 to

*l*=

*L*and obtain the reflection and transmission coefficients

*r*(

*L*) and

*t*(

*L*).

*E*is considered to be a function of

*z*and

*l*. For a given

*z*(0<

*z*<

*L*),

*E*(

*z*,

*L*) is obtained by integrating this equation, together with the equation for

*r*(

*l*) in Eq. (7), from

*l*=

*z*to

*l*=

*L*using the initial condition

*E*(

*z*,

*z*)=1+

*r*(

*z*).

## 5. Results

*α*is equal to 1. In Fig. 4, we show the reflectance

*R*(=|

*r*|

^{2}) and the absorptance

*A*(=1-|

*r*|

^{2}-|

*t*|

^{2}) of an

*s*wave of frequency

*ω*=5.2×10

^{13}/sec as a function of incident angle. In all results shown in this section, we use the parameter values

*N*

_{0}=3×10

^{16}cm

^{-3},

*ε*=12,

_{L}*m*

_{eff}=0.01

*m*,

_{e}*a*=10

^{-4}cm,

*ε*

_{1}=2.25,

*ε*=1,

_{d}*D*=100

*a*and

*b*=-0.53. In Fig. 4, the thickness of the gap between the inhomogeneous dielectric slab and the prism is 15

*a*and the damping parameter

*γ*is 0.001 in (a) and (b) and 0.01 in (c) and (d). We observe that there appears a very sharp absorption peak at

*θ*=64.88°, which is due to the excitation of a surface wave. This angle agrees perfectly with the value obtained from the dispersion relation shown in Fig. 3 and is well above the critical angle

*θ*. We find that as

_{c}*γ*increases, the absorption peak gets broadened and the maximum value of

*A*decreases.

*θ*=65° versus frequency, obtained for the same parameters as in Fig. 4. When

*γ*=0.001, we find an extremely sharp and narrow absorption peak at

*ω*=5.2×10

^{13}/sec, which is again due to the excitation of a surface wave. Similarly to the behavior shown in Fig. 4, as

*γ*increases, the absorption peak gets broadened and the maximum value of

*A*decreases.

*s*wave of frequency

*ω*=5.2×10

^{13}/sec is incident at

*θ*=64.88° and 64°, obtained for the same parameters as in Figs. 4 and 5. The vertical dashed line indicates the boundary between the inhomogeneous medium located in 0≤

*z*≤100

*a*and the air gap located in 100

*a*≤

*z*≤115

*a*. The wave is assumed to be incident from the right and transmitted to the left. We find that the electric field is very strong near the boundary in the

*θ*=64.88° case, which clearly demonstrates the excitation of a surface wave. The fact that the electric field becomes much weaker when the incident angle is slightly different from 64.88° shows that the surface wave excitation is a resonant phenomenon.

*α*is equal to 1. We have done similar calculations for several other values of the exponent

*α*and found that the surface waves are excited in those cases, too. In Figs. 7(a–d), we show the absorptance

*A*of an

*s*wave incident obliquely on an inhomogeneous dielectric slab, the permittivity of which is given by Eq. (2) with

*α*=0.5, 0.75, 1.25 and 1.5, in the Otto configuration as a function of incident angle. The thickness of the air gap is 20

*a*and the damping parameter

*γ*is 0.001. The frequency of the incident wave is 5×10

^{13}/sec in (a) and (b) and 5.05×10

^{13}/sec in (c) and (d). All other parameters are the same as in Fig. 4. We find sharp absorption peaks in all cases, at

*θ*=68.82°, 53.01°, 48.12° and 47°, when

*α*is 0.5, 0.75, 1.25 and 1.5 respectively. In Figs. 7(e–h), we show the spatial distributions of the electric field intensity corresponding to the absorption peaks listed above. We find that the surface waves are indeed excited near the boundary in all of these cases. It appears that they are more strongly and narrowly confined to the surface when

*α*is smaller.

## 6. Discussion and conclusion

*s*-polarized surface waves we have considered so far are similar in nature to the waveguide modes propagating in an inhomogeneous dielectric slab. They are also similar to the bound states of a quantum mechanical particle in a potential well. In our case, the effective potential well has a sharp minimum at the surface so that the bound states are strongly localized near there. The fact that we found

*s*-polarized surface waves for all cases we considered suggests that they exist for a fairly broad range of functional forms of the dielectric permittivity. As long as

*ε*decreases sufficiently rapidly near the surface of a dielectric and a deep effective potential well is formed there, there is a possibility to observe

*s*-polarized surface waves. Since this condition is expected to be satisfied in heavily doped degenerate semiconductors due to the formation of a carrier depletion layer [26–28

26. I. P. Ipatova, A. Yu. Maslov, L. V. Udod, G. Benedek, and G. Panzarini, “The enhancement factor of hyper-Raman scattering from an inhomogeneous semiconductor surface,” Surf. Sci. **377–379**, 436–439 (1997). [CrossRef]

15. A. B. Shvartsburg, V. Kuzmiak, and G. Petite, “Optics of subwavelength gradient nanofilms,” Phys. Rep. **452**, 33–88 (2007). [CrossRef]

30. L. A. A. Pettersson, L. Hultman, and H. Arwin, “Porosity depth profiling of thin porous silicon layers by use of variable-range spectroscopic ellipsometry: a porosity graded-layer model,” Appl. Opt. **37**, 4130–4136 (1998). [CrossRef]

*s*-polarized surface electromagnetic waves for several models of an inhomogeneous dielectric slab. We have calculated the reflectance, the absorptance and the electric field distribution associated with an

*s*wave incident obliquely on an inhomogeneous dielectric slab in the Otto configuration. In all models we have considered,

*s*-polarized surface waves are found to be excited at certain angles and frequencies. We have discussed the physical mechanism of the surface wave generation and argued that these effects can be observed experimentally in degenerate semiconductors and artificially fabricated materials.

## Acknowledgments

## References and links

1. | H. Raether, |

2. | W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature |

3. | E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science |

4. | A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. |

5. | S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. |

6. | K. A. Willets and R. P. Van Duyne, “Localized surface plasmon resonance spectroscopy and sensing,” Annu. Rev. Phys. Chem. |

7. | X. D. Hoa, A. G. Kirk, and M. Tabrizian, “Towards integrated and sensitive surface plasmon resonance biosensors: A review of recent progress,” Biosens. Bioelectron. |

8. | T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature |

9. | G. I. Stegeman, J. D. Valera, C. T. Seaton, J. Sipe, and A. A. Maradudin, “Nonlinear s-polarized surface plasmon polaritons,” Solid State Commun. |

10. | W. Chen and A. A. Maradudin, “S-polarized guided and surface electromagnetic waves supported by a nonlinear dielectric film,” J. Opt. Soc. Am. B |

11. | H. W. Schürmann, V. S. Serov, and Yu. V. Shestopalov, “TE-polarized waves guided by a lossless nonlinear three-layer structure,” Phys. Rev. E |

12. | W. Youfa, W. Qi, and B. Jiashan, “Nonlinear TE surface waves on an antiferromagnetic crystal,” J. Appl. Phys. |

13. | V. E. Kravtsov, E. I. Firsov, V. A. Yakovlev, and G. N. Zhizhin, “TE-surface polaritons at the interface of the isotropic media,” Solid State Commun. |

14. | A. Shvartzburg, G. Petite, and N. Auby, “S-polarized surface electromagnetic waves in inhomogeneous media: exactly solvable models,” J. Opt. Soc. Am. B |

15. | A. B. Shvartsburg, V. Kuzmiak, and G. Petite, “Optics of subwavelength gradient nanofilms,” Phys. Rep. |

16. | R. Bellman and G. M. Wing, |

17. | V. I. Klyatskin, “The imbedding method in statistical boundary-value wave problems,” Prog. Opt. |

18. | R. Rammal and B. Doucot, “Invariant-imbedding approach to localization. I. General framework and basic equations,” J. Phys. (Paris) |

19. | K. Kim, H. Lim, and D.-H. Lee, “Invariant imbedding equations for electromagnetic waves in stratified magnetic media: Applications to one-dimensional photonic crystals,” J. Korean Phys. Soc. |

20. | K. Kim, D.-H. Lee, and H. Lim, “Theory of the propagation of coupled waves in arbitrarily inhomogeneous stratified media,” Europhys. Lett. |

21. | K. Kim and D.-H. Lee, “Invariant imbedding theory of mode conversion in inhomogeneous plasmas. I. Exact calculation of the mode conversion coefficient in cold, unmagnetized plasmas,” Phys. Plasmas |

22. | K. Kim and D.-H. Lee, “Invariant imbedding theory of mode conversion in inhomogeneous plasmas. II. Mode conversion in cold, magnetized plasmas with perpendicular inhomogeneity,” Phys. Plasmas |

23. | K. Kim, D. K. Phung, F. Rotermund, and H. Lim, “Propagation of electromagnetic waves in stratified media with nonlinearity in both dielectric and magnetic responses,” Opt. Express |

24. | K. Kim, F. Rotermund, and H. Lim, “Disorder-enhanced transmission of a quantum mechanical particle through a disordered tunneling barrier in one dimension: Exact calculation based on the invariant imbedding method,” Phys. Rev. B |

25. | D. J. Yu and K. Kim, “Influence of bottom topography on the propagation of linear shallow water waves: an exact approach based on the invariant imbedding method,” Waves Random Complex Media |

26. | I. P. Ipatova, A. Yu. Maslov, L. V. Udod, G. Benedek, and G. Panzarini, “The enhancement factor of hyper-Raman scattering from an inhomogeneous semiconductor surface,” Surf. Sci. |

27. | T. Inaoka, “Elementary excitations at doped polar semiconductor surfaces with carrier-depeltion layers,” Appl. Surf. Sci. |

28. | J. W. L. Yim, R. E. Jones, K. M. Yu, J. W. Ager III, W. Walukiewicz, W. J. Schaff, and J. Wu, “Effects of surface states on electrical characteristics of InN and In |

29. | T. M. Dunster, “Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter,” SIAM J. Math. Anal. |

30. | L. A. A. Pettersson, L. Hultman, and H. Arwin, “Porosity depth profiling of thin porous silicon layers by use of variable-range spectroscopic ellipsometry: a porosity graded-layer model,” Appl. Opt. |

31. | X. Wang, H. Masumoto, Y. Someno, and T. Hirai, “Design and experimental approach of optical reflection filters with graded refractive index profiles,” J. Vac. Sci. Technol. A |

32. | H. Ren and S.-T. Wu, “Inhomogeneous nanoscale polymer-dispersed liquid crystals with gradient refractive index,” Appl. Phys. Lett. |

**OCIS Codes**

(240.6690) Optics at surfaces : Surface waves

(260.2710) Physical optics : Inhomogeneous optical media

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: July 9, 2008

Manuscript Accepted: August 11, 2008

Published: August 14, 2008

**Citation**

Kihong Kim, "Excitation of s-polarized surface electromagnetic waves in inhomogeneous dielectric media," Opt. Express **16**, 13354-13363 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13354

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### References

- H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).
- W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003). [CrossRef] [PubMed]
- E. Ozbay, "Plasmonics: Merging photonics and electronics at nanoscale dimensions," Science 311, 189-193 (2006). [CrossRef] [PubMed]
- A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, "Nano-optics of surface plasmon polaritons," Phys. Rep. 408, 131-314 (2005). [CrossRef]
- S. A. Maier and H. A. Atwater, "Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures," J. Appl. Phys. 98, 011101 (2005). [CrossRef]
- K. A. Willets and R. P. Van Duyne, "Localized surface plasmon resonance spectroscopy and sensing," Annu. Rev. Phys. Chem. 58, 267-297 (2007). [CrossRef]
- X. D. Hoa, A. G. Kirk, and M. Tabrizian, "Towards integrated and sensitive surface plasmon resonance biosensors: A review of recent progress," Biosens. Bioelectron. 23, 151-160 (2007). [CrossRef] [PubMed]
- T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998). [CrossRef]
- G. I. Stegeman, J. D. Valera, C. T. Seaton, J. Sipe, and A. A. Maradudin, "Nonlinear s-polarized surface plasmon polaritons," Solid State Commun. 52, 293-297 (1984). [CrossRef]
- W. Chen and A. A. Maradudin, "S-polarized guided and surface electromagnetic waves supported by a nonlinear dielectric film," J. Opt. Soc. Am. B 5, 529-538 (1988). [CrossRef]
- H. W. Sch¨urmann, V. S. Serov, and Yu. V. Shestopalov, "TE-polarized waves guided by a lossless nonlinear three-layer structure," Phys. Rev. E 58, 1040-1050 (1998). [CrossRef]
- W. Youfa, W. Qi, and B. Jiashan, "Nonlinear TE surface waves on an antiferromagnetic crystal," J. Appl. Phys. 84, 6233-6238 (1998). [CrossRef]
- V. E. Kravtsov, E. I. Firsov, V. A. Yakovlev, and G. N. Zhizhin, "TE-surface polaritons at the interface of the isotropic media," Solid State Commun. 50, 741-743 (1984). [CrossRef]
- A. Shvartzburg, G. Petite, and N. Auby, "S-polarized surface electromagnetic waves in inhomogeneous media: exactly solvable models," J. Opt. Soc. Am. B 16, 966-970 (1999). [CrossRef]
- A. B. Shvartsburg, V. Kuzmiak, and G. Petite, "Optics of subwavelength gradient nanofilms," Phys. Rep. 452, 33-88 (2007). [CrossRef]
- R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding (Wiley, 1976).
- V. I. Klyatskin, "The imbedding method in statistical boundary-value wave problems," Prog. Opt. 33, 1-127 (1994). [CrossRef]
- R. Rammal and B. Doucot, "Invariant-imbedding approach to localization. I. General framework and basic equations," J. Phys. (Paris) 48, 509-526 (1987). [CrossRef]
- K. Kim, H. Lim, and D.-H. Lee, "Invariant imbedding equations for electromagnetic waves in stratified magnetic media: Applications to one-dimensional photonic crystals," J. Korean Phys. Soc. 39, L956-L960 (2001).
- K. Kim, D.-H. Lee, and H. Lim, "Theory of the propagation of coupled waves in arbitrarily inhomogeneous stratified media," Europhys. Lett. 69, 207-213 (2005). [CrossRef]
- K. Kim and D.-H. Lee, "Invariant imbedding theory of mode conversion in inhomogeneous plasmas. I. Exact calculation of the mode conversion coefficient in cold, unmagnetized plasmas," Phys. Plasmas 12, 062101 (2005). [CrossRef]
- K. Kim and D.-H. Lee, "Invariant imbedding theory of mode conversion in inhomogeneous plasmas. II. Mode conversion in cold, magnetized plasmas with perpendicular inhomogeneity," Phys. Plasmas 13, 042103 (2006). [CrossRef]
- K. Kim, D. K. Phung, F. Rotermund, and H. Lim, "Propagation of electromagnetic waves in stratified media with nonlinearity in both dielectric and magnetic responses," Opt. Express 16, 1150-1164 (2008). [CrossRef] [PubMed]
- K. Kim, F. Rotermund, and H. Lim, "Disorder-enhanced transmission of a quantum mechanical particle through a disordered tunneling barrier in one dimension: Exact calculation based on the invariant imbedding method," Phys. Rev. B 77, 024203 (2008). [CrossRef]
- D. J. Yu and K. Kim, "Influence of bottom topography on the propagation of linear shallow water waves: an exact approach based on the invariant imbedding method," Waves Random Complex Media 18, 325-341 (2008). [CrossRef]
- I. P. Ipatova, A. Yu. Maslov, L. V. Udod, G. Benedek, and G. Panzarini, "The enhancement factor of hyper-Raman scattering from an inhomogeneous semiconductor surface," Surf. Sci. 377-379, 436-439 (1997). [CrossRef]
- T. Inaoka, "Elementary excitations at doped polar semiconductor surfaces with carrier-depeltion layers," Appl. Surf. Sci. 169-170, 51-56 (2001). [CrossRef]
- J.W. L. Yim, R. E. Jones, K. M. Yu, J.W. AgerIII, W. Walukiewicz, W. J. Schaff, and J. Wu, "Effects of surface states on electrical characteristics of InN and In1-xGaxN," Phys. Rev. B 76, 041303(R) (2007). [CrossRef]
- T. M. Dunster, "Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter," SIAM J. Math. Anal. 21, 995-1018 (1990). [CrossRef]
- L. A. A. Pettersson, L. Hultman, and H. Arwin, "Porosity depth profiling of thin porous silicon layers by use of variable-range spectroscopic ellipsometry: a porosity graded-layer model," Appl. Opt. 37, 4130-4136 (1998). [CrossRef]
- X. Wang, H. Masumoto, Y. Someno, and T. Hirai, "Design and experimental approach of optical reflection filters with graded refractive index profiles," J. Vac. Sci. Technol. A 17, 206-211 (1999). [CrossRef]
- H. Ren and S.-T. Wu, "Inhomogeneous nanoscale polymer-dispersed liquid crystals with gradient refractive index," Appl. Phys. Lett. 81, 3537-3539 (2002). [CrossRef]

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