## Optical sensor based on resonant porous silicon structures

Optics Express, Vol. 13, Issue 10, pp. 3754-3764 (2005)

http://dx.doi.org/10.1364/OPEX.13.003754

Acrobat PDF (197 KB)

### Abstract

We propose a new design for an optical sensor based on porous silicon structures. We present an analysis based on a pole expansion, which allows for the easy identification of the parameters important for the operation of the sensor, and the phenomenological inclusion of scattering losses. The predicted sensitivity of the sensor is much greater than detectors utilizing surface plasmon resonance.

© 2005 Optical Society of America

## 1. Introduction

3. E. Kretschmann, “Decay of non radiative surface plasmons into light on rough silver films. Comparison of experimental and theoretical results.,” Opt. Comm. **6**, 185–187 (1972). [CrossRef]

4. I. Pockrand, “Surface plasma oscillations at silver surfaces with thin transparent and absorbing coatings,” Surf. Sci. **72**, 577–588 (1978). [CrossRef]

5. J. D. Swalen, “Optical wave spectroscopy of molecules at surfaces,” J. Phys. Chem. **83**, 1438–1445 (1979). [CrossRef]

6. H. Kano and S. Kawata, “Surface-plasmon sensor for absorption-sensitivity enhancement,” Appl. Opt. **33**, 5166–5170 (1994). [CrossRef] [PubMed]

7. J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, “Simulation on wavelength-dependent complex refractive index of liquids obtained by phase retrieval from reflectance dip due to surface plasmon resonance,” Appl. Spectrosc. **57**, 288–292 (2003). [CrossRef] [PubMed]

8. R. J. Green, R. A. Frazier, K. M. Skakesheff, M. C. Davies, C. J. Roberts, and S. J. B. Tendler, “Surface plasmon resonance analysis of dynamic biological interactions with biomaterials,” Biomaterials **21**, 1823–1835 (2000). [CrossRef] [PubMed]

11. S. M. Weiss and P. M. Fauchet, “Electrically tunable porous silicon active mirrors,” Phys. Stat. Sol. A **197**, 556–560 (2003). [CrossRef]

12. J. E. Lugo, J. A. del Rio, and J. Tagüeña-Martínez, “Influence of surface coverage on the effective optical properties of porous silicon modeled as a Si-wire array,” J. Appl. Phys. **81**, 1923–1928 (1997). [CrossRef]

## 2. Pole expansion analysis of the reflectance

13. P. E. Schmid, “Optical absorption in heavily doped silicon,” Phys. Rev. B. **23**, 5531–5536 (1981). [CrossRef]

14. J. von Behren, L. Tsybeskov, and P. M. Fauchet
, “Preparation, properties and applications of free-standing porous silicon films,” in *Microcrystalline and nanocrystalline semiconductors*, Vol. 358,
R. W. Collins, C. C. Tsai, M. Hirose, F. Koch, and L. Brus, eds. (Mat. Res. Proc., 1995), pp. 333–338.

15. J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B **4**, 481–489 (1987). [CrossRef]

**κ**, assumed to be real. Hence, the wave number perpendicular to the surface

*w*

_{i}is given by

*ω̃*=

*ω/c*and εi is the dielectric function of the ith medium. The square root is defined such that Im{

*z*

^{1/2}}≥0 and Re{

*z*

^{1/2}}≥0, if Im{

*z*

^{1/2}}=0. We analyze the reflectance by means of the Fresnel reflection and transmission coefficients, which in this notation are [15

15. J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B **4**, 481–489 (1987). [CrossRef]

*i*and

*j*from medium

*i*. The expressions (2)–(3) are valid for absorbing medium as well, and these coefficients satisfy the Fresnel coefficient identities

16. J. E. Sipe, “Surface plasmon-enhanced absorption of light by adsorbed molecules,” Solid State Commun. **33**, 7–9 (1980). [CrossRef]

17. J. E. Sipe and J. Becher, “Surface energy transfer enhanced by optical cavity excitation: a pole analysis,” J. Opt. Soc. Am. **72**, 288–295 (1982). [CrossRef]

### 2.1. Pole expansion of the reflectance R51 from the SPR sensor

*r*

_{31}, given by Eq. (3) for p-polarized light, diverges [16

16. J. E. Sipe, “Surface plasmon-enhanced absorption of light by adsorbed molecules,” Solid State Commun. **33**, 7–9 (1980). [CrossRef]

*r*

_{31}has a pole at a particular wave number, and this pole signals the dispersion relation for a surface plasmon,

*κ*

_{SPR}is the complex wave number of the surface plasmon, with

*ε*

_{1}and

*ε*

_{3}denoting the dielectric function of the sample and the metal, respectively. We restrict ourselves to a single wavelength, allowing the dispersion of the constituent materials to be omitted. Of course, the analysis here can be performed for any wavelength, and by superposition could thus be extended to treat the response of the device to pulsed irradiation.

*r*

_{31}in the neighborhood of the pole (5), and we express the reflection coefficient in the form

*κ*

_{s}is found to be

*ε*

_{1}) as well. We demonstrate the validity of the pole expansion by comparing calculated reflectance to the exact one in a case, where the SPR sensor is in vacuum. This choice is made only for the sake of simplicity, for with it the pole strength parameter can be approximated as real for typical metals, such as Ag and Au, whose real part of the dielectric function is much larger than the imaginary part

*i.e*. |Re{

*ε*

_{3}}|≫|Im{

*ε*

_{3}}|. Hence, the pole strength is given approximately by [16

16. J. E. Sipe, “Surface plasmon-enhanced absorption of light by adsorbed molecules,” Solid State Commun. **33**, 7–9 (1980). [CrossRef]

*R*

_{51}we consider the squared modulus of complex reflectivity

*r̃*

_{51}. By a tilde we denote an effective Fresnel coefficient connecting the two indicated media, where there may be any number of layers of other media in between. For example, referring to Fig. 1a,

*r*

_{31}.

*r̃*

_{51}are easily obtained and physically interpreted; we will see another advantage in Section 2.2. We write reflection coefficient

*r*

_{53}=-

*r*

_{35}=

*A*exp(-i

*ϕ*), where both

*A*and

*ϕ*are real. Thus we are in the total-internal-reflection regime, and for highly reflecting metals

*A*≈1 and |

*r*

_{53}|=|

*r*

_{35}|=1. The phase of this coefficient in this approximation is obtained from the expression

*w*

_{3}and

*w*

_{5}calculated at the pole with

*κ*=

*κ*

_{SPR}. It is convenient to separate the real and imaginary parts of the complex pole

*κ*

_{SPR}and express

*r*

_{31}as follows:

*R*

_{51}=|

*r̃*

_{51}|

^{2}is found, inserting these expressions into Eq. (9), to be given by

*λ*=1.532

*µ*m to have a reasonable comparison between the conventional SPR sensor and our suggested PS sensor.

*n*

_{2}=0.462+9.20i [18], thickness

*d*=40 nm) on top of the rutile prism (refractive index

*n*

_{3}=2.55). The comparison between the exact SPR reflectance (solid line) and the SPR reflectance obtained from the pole expansion (dashed line) is presented in the Fig. 2. The exact curve is calculated from Eq. (9) and the pole expansion curve from Eq. (12). We observe a good fit close to the pole, near the minimum of the reflectance. The discrepancy between the exact calculation and the pole analysis arises largely, because

*β*varies over the angular spread of the surface plasmon dip, while in the pole analysis it is treated as a constant.

*κ*

_{m}

*, κ*

_{s}

*, β*, and

*ϕ*on which the performance of the device depends. It also leads to a natural comparison of the standard SPR sensors with the PS sensor to which we now turn.

### 2.2. Pole expansion of the reflectance R51 from the PS sensor

*ε*

_{xx}=

*ε*

_{yy}=

*ε*‖ and

*ε*

_{zz}=

*ε*⊥, where

*z*is the direction normal to the interfaces. Here we assume the air pores to be small compared to the wavelength of light, as they are for many fabrication protocols [11

11. S. M. Weiss and P. M. Fauchet, “Electrically tunable porous silicon active mirrors,” Phys. Stat. Sol. A **197**, 556–560 (2003). [CrossRef]

*ε*‖; the analysis of p-polarized light, which involves both

*ε*‖ and

*ε*⊥, is left to a future communication.

*n*

_{2}

*n*

_{1}

*ω̃*,

*n*

_{3}

*r*

_{31}are associated with the usual dispersion relation for the waveguide modes [19]

*w*

_{1}=

*iq*,

*w*

_{2}=

*h*, and

*w*

_{3}=i

*p*with

*q,h, p*∈ℝ, and where

*d*is the thickness of the waveguide layer. Equation (13) is an implicit equation for

*κ*as a function of the resonator layer thickness

*d*; the waveguide modes, labelled by

*κ*

_{m}, are the solutions of this equation. These wave numbers are real, as we assume silicon to be non-absorbing at the frequency of interest.

*r̃*

_{31}from the waveguide layer in Fig. 1b is given by Eq. (9) with the replacements 3→2 and 5→3. The pole expansion takes the form

*κ*

_{s}is found, after some tedious algebra, to be [17

17. J. E. Sipe and J. Becher, “Surface energy transfer enhanced by optical cavity excitation: a pole analysis,” J. Opt. Soc. Am. **72**, 288–295 (1982). [CrossRef]

*κ*=

*κ*

_{m}. As an example, we consider the same incident wavelength of

*λ*=1.532

*µ*m as in SPR calculations; the refractive index of doped silicon is practically the same as for undoped silicon (

*n*

_{Si}=3.4784 [18]), and absorption is not taken into account in the thin porous silicon layers considered here. The porosities of the resonator and coupling layers are chosen here to be 50% and 75%, respectively. From a Maxwell Garnett (MG) effective medium theory for cylinders [12] we then obtain the following indices for the resonator and coupling layers:

*n*‖

_{Res}=2.213,

*n*⊥

_{Res}=2.559 and

*n*‖

_{Cpl}=1.642, and

*n*⊥

_{Cpl}=1.943. In Fig. 3 we present the numerical solution of the dispersion relation for

*κ*

_{m}and pole strength parameter

*κ*

_{s}as a function of the waveguide thickness

*d*. We observe a cutoff value in the thickness of the waveguide that can support waveguide modes in the asymmetric waveguide, as expected [19]. The pole strength shown in Fig. 3 displays a maximum at

*d*=287.2 nm.

*r̃*

_{31}has been considered as purely real, and all the incident light in the geometry of Fig. 1b would be reflected, since no energy is dissipated in the structure and in the evanescent region of interest none could propagate into medium 1. However, there exists scattering losses from the pores in the PS layers. As a first approximation we neglect the scattering losses in the coupling layer, since at resonance the electric field is concentrated in the waveguide layer. We introduce a phenomenological scattering parameter

*γ*to take into account losses in the resonator layer, and adjust the pole expansion Eq. (14),

*γ*is to be determined experimentally. Thus the pole κ

_{m}is now shifted to a complex wave number κ

_{m}+

*iγ*in analogy to the pole κ

_{SPR}in the SPR pole expansion. This shows another strength of the pole analysis, as the approach gives a simple phenomenological treatment of scattering. Of course, one could phenomenologically include scattering losses using the transfer-matrix approach by adding an imaginary part to the dielectric function of the waveguide layer. However, the pole expansion gives us analytical expressions for the optimization of the structure. As well,

*γ*can be set immediately from the experimentally determined waveguide loss; for porous silicon structures such as those we consider here that loss can range from 1–90 dB/cm [20

20. G. Amato, L. Boarino, S. Borini, and A. M. Rossi, “Hybrid approach to porous silicon integrated waveguides,” Phys. Stat. Sol. (a) **182**, 425–430 (2000). [CrossRef]

*R*

_{51}from the PS sensor can now be calculated from the usual combinations of Fresnel coefficients that appear in thin film optics. For the ideal structure shown in Fig. 1b, but taking into account the scattering described above, we would have

*R*

_{51}=|

*r̃*

_{51}|

^{2}, where

*w4s*) in (17). Hence we can set the denominator in the second term of (17) equal to unity and neglect the interference between the two terms. This yields

*R*

_{41}=|

*r̃*

_{41}|

^{2}. The absorption in the silicon substrate (thickness

*s*=500

*µ*m) is estimated from the doping level, which is less than 10

^{18}B atoms per cm

^{3}. Using Fig. 2 from Schmid [13

13. P. E. Schmid, “Optical absorption in heavily doped silicon,” Phys. Rev. B. **23**, 5531–5536 (1981). [CrossRef]

*α*=1 cm

^{-1}for the absorption, which corresponds to a value 1.22×10

^{-5}for the extinction coefficient in the doped silicon substrate.

*r*

_{43}=-

*r*

_{34}=exp(-i

*ϕ*), and we approximate

*ϕ*by its value at the waveguide resonance κ

_{m},

*w*

_{3}by its value at the resonance,

*w*

_{3}=i

*p*

_{m}, and put exp(i2

*w*

_{3}

*D*)≈

*β*=exp(-2

*p*

_{m}

*D*). Then

*R*

_{41}is exactly the same as the first one of Eqs. (12), except that we have

*r̃*′

_{31}instead of

*r*

_{31}. Using (19), we then find

_{m}-βκ

_{s}cos

*ϕ*, which corresponds to the critical angle of incidence

*θ*

_{crit}. In silicon substrate (κ=

*n*

_{4}

*ω̃*sin

*θ*) this angle is given by

*γ*=

*βκ*

_{s}sin

*ϕ*. This condition corresponds to a situation where all the incident light is coupled into a waveguide mode and absorbed by the scattering losses. We obtain the optimal coupler thickness

*D*

_{opt}, using the definition of

*β*=exp(-2

*p*

_{m}

*D*), in the form

*D*. The thickness of the resonator layer can be chosen arbitrarily, but the best sensitivity of the sensor is obtained where the pole strength is largest. The minimum value and the width of the reflectance dip is larger for smaller coupler thicknesses as the light is coupled too efficiently back to the silicon substrate. On the other hand, the excitation of the waveguide mode is reduced if the coupling layer is thicker than the optimal value, leading to an increase in the minimum value of the reflectance.

## 3. Numerical comparison between PS and SPR sensors

*n*=2.55) placed below the silicon substrate to facilitate optical coupling into the waveguide resonance. We use the pole expansion, developed in the previous section, which is limited to angles of incidence for which fields are evanescent in both the coupler and air but not in the resonator; we include these scattering losses by adding the appropriate imaginary part

*γ*to the resonance wave number associated with the waveguide mode. We take

*γ*=115.13 m

^{-1}, corresponding to a waveguide loss of 10 dB/cm. This is well within the range of losses observed for structures such as these [20

20. G. Amato, L. Boarino, S. Borini, and A. M. Rossi, “Hybrid approach to porous silicon integrated waveguides,” Phys. Stat. Sol. (a) **182**, 425–430 (2000). [CrossRef]

*d*=287.2 nm, which corresponds to the maximum value of the pole strength curve presented in Fig. 3. The optimal thickness of the coupling layer is found to be

*D*=1235.8 nm from Eq. (25). From an experimental point of view the etching of 290 nm low-porosity layer on top of 1.24 micron high porosity layer on silicon substrate is straightforward [11

11. S. M. Weiss and P. M. Fauchet, “Electrically tunable porous silicon active mirrors,” Phys. Stat. Sol. A **197**, 556–560 (2003). [CrossRef]

*D*(thin solid lines) the system is undercoupled and we obtain smaller reflectivity dips, while for smaller

*D*(dashed lines) the dips become broader as the light is coupled too efficiently back to the reflected field.

*n*=1.59, and compare the reflectance with that of the structure before filling. Assuming that the nanoparticles fill 1% of the volume of the pores, we model the new refractive index of these pores with the Maxwell Garnett theory for spheres in air. For the resonator layer, the MG effective medium theory for cylinders [12

12. J. E. Lugo, J. A. del Rio, and J. Tagüeña-Martínez, “Influence of surface coverage on the effective optical properties of porous silicon modeled as a Si-wire array,” J. Appl. Phys. **81**, 1923–1928 (1997). [CrossRef]

*n*‖

_{Res}=2.215. The predicted reflectances before and after filling are shown in Figs. 4a and 4b, respectively. For the air pores we obtain a narrow reflectance dip at an angle of incidence of 45.872°, while for pores partially filled with nanoparticles the corresponding resonance angle is 45.919°, yielding a shift of Δ

*θ*=0.047°. This shift is well in the detection range as the half-width of the dip, which is approximately 0.004° with optimized coupler thickness.

*n*

_{Ag}=0.462+9.20i at

*λ*=1.532

*µ*m [18], on top of a rutile prism. We observe the best sensitivity of the device operation by taking a metal thickness of

*d*=40 nm, as shown in Fig. 5a. For larger thicknesses the reflection minimum increases as the excitation of surface plasmon is reduced due to absorption of the metal film; for smaller thicknesses the surface plasmon is coupled too strongly back to the reflected field and the reflectance dips become broader as seen from Figs. 5a and b.

*A*of the detector was (0.5)(287nm)

*A*, of which 1% was filled with nanoparticles with

*n*=1.59. The same amount of material per area

*A*here, spread uniformly over the surface of the SPR sensor, would result in an overlayer of thickness (0.01)(0.5)(287nm)=1.44nm. We calculate the optical response here using the transfer matrices, without applying a pole approximation. But we note that a calculation with the pole approximation would lead to essentially the same shifts and widths we find below [21

21. For an overlayer thickness *l* with an index *n*_{l} the effective Fresnel coefficient *r̃*_{51} from the prism in Fig. 1a is exactly given by Eq. (9) but with *r*_{31} replaced by *r̂*_{31}=(*r*_{3l}+*r*_{l1} exp(2i*w*_{l}*l*))/(1-*r*_{l3}*r*_{l1} exp(2i*w*_{l}*l*)) in an obvious notation. Using Fresnel coefficient identities, that new equation can be written as *r̂*_{31}=(*r*_{31}+*r̂*_{1l})/(1+*r*_{31}*r̂*_{1l}), where *r̂*_{1l}=(*r*_{1l}+*r*_{l1} exp(2i*w*_{l}*l*))/(1-*w*_{l}*l*)). Using the pole approximation (11) for *r*_{31} in this new expression for *r̃*_{51}, we predict a shift of the resonance dip due to the overlayer which deviates from an exact calculation of that shift by only 0.002°.

*θ*=0.011° with a half-width of the SPR resonance dip approximately 0.06° for the optimized metal thickness. Furthermore, this shift is smaller than that predicted shift for the PS sensor, and can be associated with the fact that the target material is exposed only to evanescent fields in the SPR sensor; in the PS sensor, on the other hand, the target material exists within the volume of the waveguide and is exposed to the full field of the waveguide mode. Perhaps more importantly, the dips are significantly broader in the SPR sensor than predicted for the PS sensor. This arises because the scattering loss of the PS sensor waveguide mode, even at 10 dB/cm, is less than the absorption loss of typical surface plasmons; the

*γ*for the SPR sensor that follows from (11) corresponds to 214 dB/cm. Note that any surface roughness or defects near the metal surface would only increase the SPR resonance width.

## 4. Conclusions

## Acknowledgments

## References and links

1. | J. Räty, K.-E. Peiponen, and T. Asakura, |

2. | H. Räther, |

3. | E. Kretschmann, “Decay of non radiative surface plasmons into light on rough silver films. Comparison of experimental and theoretical results.,” Opt. Comm. |

4. | I. Pockrand, “Surface plasma oscillations at silver surfaces with thin transparent and absorbing coatings,” Surf. Sci. |

5. | J. D. Swalen, “Optical wave spectroscopy of molecules at surfaces,” J. Phys. Chem. |

6. | H. Kano and S. Kawata, “Surface-plasmon sensor for absorption-sensitivity enhancement,” Appl. Opt. |

7. | J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, “Simulation on wavelength-dependent complex refractive index of liquids obtained by phase retrieval from reflectance dip due to surface plasmon resonance,” Appl. Spectrosc. |

8. | R. J. Green, R. A. Frazier, K. M. Skakesheff, M. C. Davies, C. J. Roberts, and S. J. B. Tendler, “Surface plasmon resonance analysis of dynamic biological interactions with biomaterials,” Biomaterials |

9. | B. J. Sedlak, “Next-generation microarray technologies - Focus is on higher sensitivity, drug discovery, and lipid cell signaling,” Genetic Engineering News |

10. | P. M. Fauchet
, “Silicon: Porous,” in |

11. | S. M. Weiss and P. M. Fauchet, “Electrically tunable porous silicon active mirrors,” Phys. Stat. Sol. A |

12. | J. E. Lugo, J. A. del Rio, and J. Tagüeña-Martínez, “Influence of surface coverage on the effective optical properties of porous silicon modeled as a Si-wire array,” J. Appl. Phys. |

13. | P. E. Schmid, “Optical absorption in heavily doped silicon,” Phys. Rev. B. |

14. | J. von Behren, L. Tsybeskov, and P. M. Fauchet
, “Preparation, properties and applications of free-standing porous silicon films,” in |

15. | J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B |

16. | J. E. Sipe, “Surface plasmon-enhanced absorption of light by adsorbed molecules,” Solid State Commun. |

17. | J. E. Sipe and J. Becher, “Surface energy transfer enhanced by optical cavity excitation: a pole analysis,” J. Opt. Soc. Am. |

18. | E. D. Palik, |

19. | A. Yariv, |

20. | G. Amato, L. Boarino, S. Borini, and A. M. Rossi, “Hybrid approach to porous silicon integrated waveguides,” Phys. Stat. Sol. (a) |

21. | For an overlayer thickness |

**OCIS Codes**

(230.3990) Optical devices : Micro-optical devices

(230.5750) Optical devices : Resonators

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 3, 2005

Revised Manuscript: May 3, 2005

Published: May 16, 2005

**Citation**

Jarkko Saarinen, Sharon Weiss, Philippe Fauchet, and J. E. Sipe, "Optical sensor based on resonant porous silicon structures," Opt. Express **13**, 3754-3764 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-10-3754

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### References

- J. Räty, K.-E. Peiponen, and T. Asakura, UV-visible reflection spectroscopy of liquids (Springer, Heidelberg, 2004).
- H. Räther, Surface plasmons on smooth and rough surfaces and on gratings (Springer, Berlin, 1988).
- E. Kretschmann, �??Decay of non radiative surface plasmons into light on rough silver films. Comparison of experimental and theoretical results.,�?? Opt. Comm. 6, 185�??187 (1972). [CrossRef]
- I. Pockrand, �??Surface plasma oscillations at silver surfaces with thin transparent and absorbing coatings,�?? Surf. Sci. 72, 577�??588 (1978). [CrossRef]
- J. D. Swalen, �??Optical wave spectroscopy of molecules at surfaces,�?? J. Phys. Chem. 83, 1438�??1445 (1979). [CrossRef]
- H. Kano and S. Kawata, �??Surface-plasmon sensor for absorption-sensitivity enhancement,�?? Appl. Opt. 33, 5166�??5170 (1994). [CrossRef] [PubMed]
- J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, �??Simulation on wavelength-dependent complex refractive index of liquids obtained by phase retrieval from reflectance dip due to surface plasmon resonance,�?? Appl. Spectrosc. 57, 288�??292 (2003). [CrossRef] [PubMed]
- R. J. Green, R. A. Frazier, K. M. Skakesheff, M. C. Davies, C. J. Roberts, and S. J. B. Tendler, �??Surface plasmon resonance analysis of dynamic biological interactions with biomaterials,�?? Biomaterials 21, 1823�??1835 (2000). [CrossRef] [PubMed]
- B. J. Sedlak, �??Next-generation microarray technologies - Focus is on higher sensitivity, drug discovery, and lipid cell signaling,�?? Genetic Engineering News 23, 20�??20 (2003).
- P. M. Fauchet, �??Silicon: Porous,�?? in Encyclopedia of applied physics, update 2, G. L. Trigg, ed. (Wiley-VCH Verlag, New York, 1999), pp. 249�??272.
- S. M. Weiss and P. M. Fauchet, �??Electrically tunable porous silicon active mirrors,�?? Phys. Stat. Sol. A 197, 556�??560 (2003). [CrossRef]
- J. E. Lugo, J. A. del Rio, and J. Tagüeña-Martínez, �??Influence of surface coverage on the effective optical properties of porous silicon modeled as a Si-wire array,�?? J. Appl. Phys. 81, 1923�??1928 (1997). [CrossRef]
- P. E. Schmid, �??Optical absorption in heavily doped silicon,�?? Phys. Rev. B. 23, 5531�??5536 (1981). [CrossRef]
- J. von Behren, L. Tsybeskov, and P. M. Fauchet, �??Preparation, properties and applications of free-standing porous silicon films,�?? in Microcrystalline and nanocrystalline semiconductors, Vol. 358, R. W. Collins, C. C. Tsai, M. Hirose, F. Koch, and L. Brus, eds. (Mat. Res. Proc., 1995), pp. 333�??338.
- J. E. Sipe, �??New Green-function formalism for surface optics,�?? J. Opt. Soc. Am. B 4, 481�??489 (1987). [CrossRef]
- J. E. Sipe, �??Surface plasmon-enhanced absorption of light by adsorbed molecules,�?? Solid State Commun. 33, 7�??9 (1980). [CrossRef]
- J. E. Sipe and J. Becher, �??Surface energy transfer enhanced by optical cavity excitation: a pole analysis,�?? J. Opt. Soc. Am. 72, 288�??295 (1982). [CrossRef]
- E. D. Palik, Handbook of optical constants of solids (Academic Press, New York, 1985).
- A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1988).
- G. Amato, L. Boarino, S. Borini, and A. M. Rossi, �??Hybrid approach to porous silicon integrated waveguides,�?? Phys. Stat. Sol. (a) 182, 425�??430 (2000). [CrossRef]
- For an overlayer thickness l with an index nl the effective Fresnel coefficient �?r51 from the prism in Fig. 1a is exactly given by Eq. (9) but with r31 replaced by �?r31 = (r3l+rl1 exp(2iwll))/(1-rl3rl1 exp(2iwll)) in an obvious notation. Using Fresnel coefficient identities, that new equation can be written as �?r31 = (r31 + �?r1l)/(1+r31 �?r1l), where �?r1l = (r1l +rl1 exp(2iwll))/(1-r2 l1 exp(2iwll)). Using the pole approximation (11) for r31 in this new expression for �?r51, we predict a shift of the resonance dip due to the overlayer which deviates from an exact calculation of that shift by only 0.002.

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