## Modeling and optimization of complex photonic resonant cavity circuits

Optics Express, Vol. 11, Issue 4, pp. 381-391 (2003)

http://dx.doi.org/10.1364/OE.11.000381

Acrobat PDF (204 KB)

### Abstract

The simple method for modeling of circuits of weakly coupled lossy resonant cavities, previously developed in quantum mechanics, is generalized to enable calculation of the transmission and reflection amplitudes and group delay of light. Our result is the generalized Breit-Wigner formula, which has a clear physical meaning and is convenient for fast modeling and optimization of complex resonant cavity circuits and, in particular, superstructure gratings in a way similar to modeling and optimization of electric circuits. As examples, we find the conditions when a finite linear chain of cavities and a linear chain with adjacent cavities act as bandpass and double bandpass filters, and the condition for a Y-shaped structure to act as a bandpass 50/50 light splitter. The group delay dependencies of the considered structures are also investigated.

© 2003 Optical Society of America

## 1. Introduction

1. N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B **57**, 12127 (1998). [CrossRef]

12. R. Slavic and S. LaRochelle, “Large-band periodic filters for DWDM using multiple-superimposed fiber Bragg gratings,” IEEE Photon. Technol. Lett. **14**, 1704 (2002). [CrossRef]

1. N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B **57**, 12127 (1998). [CrossRef]

10. S. Nishikawa, S. Lan, N. Ikeda, Y. Sugimoto, H. Ishikawa, and K. Asakawa, “Optical characterization of photonic crystal delay lines based on one-dimensional coupled defects,” Opt. Lett. **27**, 2079 (2002). [CrossRef]

11. C. M. de Sterke, “Superstructure gratings in the tight-binding approximation,” Phys. Rev. E **57**, 3502 (1998). [CrossRef]

12. R. Slavic and S. LaRochelle, “Large-band periodic filters for DWDM using multiple-superimposed fiber Bragg gratings,” IEEE Photon. Technol. Lett. **14**, 1704 (2002). [CrossRef]

## 2. Generalized Breit-Wigner formula

*i, j, k*, …) and ports (

*l, m*, …). It is essential to consider a photonic device not as an isolated structure but as a structure coupling to the rest of the optical system through ports [15

15. Y. Xu, R. K. Lee, and A. Yariv, “Adiabatic coupling between conventional dielectric waveguides and waveguides with discrete translational symmetry,” Opt. Lett. **25**, 755 (2000). [CrossRef]

18. P. Sanchis, J. Marti, J. Blasco, A. Martinez, and A. Garcia, “Mode matching technique for highly efficient coupling between dielectric waveguides and planar photonic crystal circuits,” Opt. Express **10**, 1391 (2002). [CrossRef] [PubMed]

_{n}+ ½

*i*γ

_{n}, where the lossy component γ

_{n}defines the decay time of the corresponding eigenstate at resonant wavelength λ

_{n}, τ =

*c*γ

_{n}), and

*c*is the speed of light. Generalizing this concept, if

*N*cavities and

*M*ports are weakly coupled to each other then the whole structure can be described by introducing λ

_{n}, γ

_{n}, and also the coupling coefficient between cavities

*i*and

*k*, δ

_{ik}, and the coupling coefficient between cavity

*j*and the port

*m*, γ

_{jm}, (see Fig.1). The resonant condition assumes that the eigenwavelengths λ

_{n}are close to each other and the wavelength λ is close to λ

_{n}. Below we also assume that the coupling between cavities and ports is relatively weak, so that the characteristic bandwidths are relatively small: γ

_{i}~ δ

_{kl}~ |λ

_{i}– λ

_{j}| ≪λ

_{i}. The single particle theory of resonant propagation of electrons based on this approach was developed in [20

20. M. Sumetskii, “Modeling of complicated nanometer resonant tunneling devices with quantum dots”, J.Phys.: Condens. Matter **3**, 2651 (1991). [CrossRef]

22. M. Sumetskii, “Narrow current dip for the double quantum dot resonant tunneling structure with three leads: Sensitive nanometer Y-branch switch,” Appl. Phys. Lett. **63**, 3185 (1993). [CrossRef]

20. M. Sumetskii, “Modeling of complicated nanometer resonant tunneling devices with quantum dots”, J.Phys.: Condens. Matter **3**, 2651 (1991). [CrossRef]

22. M. Sumetskii, “Narrow current dip for the double quantum dot resonant tunneling structure with three leads: Sensitive nanometer Y-branch switch,” Appl. Phys. Lett. **63**, 3185 (1993). [CrossRef]

20. M. Sumetskii, “Modeling of complicated nanometer resonant tunneling devices with quantum dots”, J.Phys.: Condens. Matter **3**, 2651 (1991). [CrossRef]

*j*is close to the port

*m*. In the absence of port

*m*, the cavity eigenvalue is λ

_{j}. If the port

*m*is weakly coupling to this cavity then the corresponding eigenstate is no longer stationary and decays in time. Formal continuation of the eigenstate calculated in the absence of the port into the port region will generate there an outgoing wave as well as an incoming wave. Because there are no external sources, the incoming wave should be absent. Often it can be eliminated by introducing the complex quasi-eigenwavelength λ

_{j}+ Δ

_{jm}+ ½

*i*γ

_{jm}and choosing it so that the coefficient in front of the incoming wave becomes zero. In our calculations we include the real shift Δλ

_{jm}into λ

_{j}writing down the quasi-eigenwavelength in the form λ

_{j}+ ½

*i*γ

_{j}where γ

_{j}summarizes the contribution of all ports coupled to the cavity and also other possible losses.

*u*

_{n}(

**r**) corresponding to quasi-eigenwavelengths λ

_{j}+ ½

*i*γ

_{j}and the incident wave

**r**) in port

*m*:

*u*

_{n}(

**r**), can be found by direct continuation of

*u*

_{n}(

**r**) into the region of the corresponding ports. The principal difference of our approach compared to the conventional tight-binding approximation [13,11

11. C. M. de Sterke, “Superstructure gratings in the tight-binding approximation,” Phys. Rev. E **57**, 3502 (1998). [CrossRef]

*u*(

**r**) satisfy the equation:

*H*is the Hamiltonian of the considered system. For example, for the scalar wave equation

*H*=

*n*(

**r**)

^{-2}Δr where

*n*(

**r**) is the refractive index. Similar as it is done in derivation of the equations of the tight-binding approximation, we chose

*u*

_{n}(

**r**) and

**r**) to satisfy the equations:

*m*and in cavity

*n*with adjacent ports, respectively, in the absence of the rest of the system. We substitute Eq. (1) into Eq. (2), use Eqs. (3) for each term of Eq. (1), and then multiply both sides of Eq.(2) by

*u*

_{k}(

**r**) and integrate over space. As the result, we arrive at the following equation for

*C*

_{n}:

_{j}, χ

_{jm}and δ

_{ij}are given in the Appendix. In this paper we consider systems of coupling cavities with very close resonant eigenvalues, which obey the scalar wave equation. Then, in the tight-binding approximation, matrix Λ is symmetric, δ

_{ij}=δ

_{ji}, and has real non-diagonal coefficients δ

_{ij}. Physically, the coefficient χ

_{jm}determines coupling between cavity

*j*and port

*m*, and coefficient δ

_{ij}determines coupling between two cavities,

*i*and

*j*. For convenience, we introduce the coupling coefficient between a cavity and a port having the same units as λ

_{j}, γ

_{j}, and δ

_{jk}:

*l*is coupled only to cavity

*i*and the port

*m*is coupled only to cavity

*j*as shown in Fig. 1. Assume also, that the ports are the single mode waveguides, for which the incoming wave (like the one introduced in Eq. (1)) and the outgoing wave into the same port are conjugate to each other within a constant factor. Then, the transmission amplitude from port

*l*to port

*m*,

*A*

_{lm}(λ), and the reflection amplitude to port

*l*,

*R*

_{ll}(λ), are defined by equations:

## 3. A single-cavity structure

*A*

_{12}(λ) = 1, if there is no internal loss, γ

_{1}=γ

_{11}+ γ

_{12}, if it is symmetric, γ

_{11}= γ

_{12}, and if the wavelength of incident light is resonant, λ = λ

_{1}. This condition corresponds to the maximum delay time τ =

*c*γ

_{1}). In a glass with refractive index

*n*= 1.5 the speed of light

*c*= 2·10

^{8}m/s and for λ

_{1}= 1500 nm and γ

_{1}= 2 nm, the delay time τ = 1.2 ps. Figure 2 shows the corresponding transmission spectrum (b) and group delay (c) for γ

_{11}= γ

_{12}= 1 nm. By decreasing γ

_{1}we increase the cavity Q-factor,

*Q*= λλ

_{1}/γ

_{1}, and, proportionally, increase the delay time which, physically, is the light dwell time in the cavity. If the second port is absent, this structure works as an all-reflecting filter with the group delay characteristic defined by Eq. (10).

## 4. A few cavity bandpass filters and a Y-splitter

_{ij}and γ

_{im}, and minimize it in several iterations. The examples of this section do not pretend to determine the smallest possible transmission ripple with minimum reflection in a certain bandwidth but rather demonstrate some designs having very flat passbands.

### 4.1. Three cavity all-pass filter

_{0}|<δ

_{0}the transmission coefficient |

*A*

_{12}|

^{2}> 0.99.

_{0}|<δ

_{0}, the group delay is varying. The characteristic value of the group delay has the same order of magnitude as the group delay defined by Eqs. (10). For example, for the characteristic bandwidth δ

_{0}= 1nm, the group delay is of order 1 ps, while for δ

_{0}= 0.1nm it is of order 10 ps. Note that similar procedure of optimization of the linear chain of resonant cavities was performed in Ref. [5

5. S. Lan, S. Nishikawa, and H. Ishikawa, “Design of impurity band-based photonic crystal waveguides and delay lines for ultrashort optical pulses,” J. Appl. Phys. **90**, 4321 (2001). [CrossRef]

5. S. Lan, S. Nishikawa, and H. Ishikawa, “Design of impurity band-based photonic crystal waveguides and delay lines for ultrashort optical pulses,” J. Appl. Phys. **90**, 4321 (2001). [CrossRef]

### 4.2. Long linear chain of resonant cavities

*A*

_{12}|

^{2}> 0.97 in the interval | λ – λ

_{0}|<1.5δ

_{0}. The transmission (curve a) and group delay spectrums shown in Fig. 4 correspond to the linear chain consisting of 50 cavities. The port-chain matching condition determined here is analogous to the apodization condition commonly used in fiber Bragg grating fabrication in order to suppress the Fabry-Perot-like oscillations in transmission and reflection spectrum [24]. However, while in the case of weak Bragg gratings the apodization region includes very large number of grating periods, we were able to accurately solve the problem by variation of 3 coupling coefficients only. Note, that the resonant cavity chain experimentally investigated in Ref. [9

9. S. Oliver, C. Smith, M. Rattier, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, and U. Oesterlé, “Miniband transmission in a photonic crystal coupled-resonator optical waveguide,” Opt. Lett. **26**, 1019 (2002). [CrossRef]

_{int}= 0.01δ

_{0}(b). While the decrease in transmission is significant and grows with the number of cavities, the effect of this loss on the group delay spectrum is negligible.

### 4.3. Double-channel bandpass filter

_{1}is turning on then the all-pass band in Fig. 3b is splitting into two all-pass bands as shown in Fig. 5. Interestingly, the distance between the centers of the created bands is approximately equal to the coupling coefficient δ

_{1}. For the transmission and group delay spectrums shown in Fig. 5, we put δ

_{1}= 2δ

_{0}.

### 4.4. Bandpass 50/50 Y-splitter

*A*

_{12}|

^{2}> 0.9994 in the interval | λ – λ

_{0}|< 0.75δ

_{0}by variation of only a single coupling coefficient to the center cavity as shown in Fig. 6(a). The behavior of the transmission and the group delay spectrum is shown in Fig. 6(b). The fine behavior of transmission spectrum in the all-pass band is shown in the insert of the first plot of Fig. 6(b).

## 5. Discussion

23. G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,”, IEEE J. Quant. Electron. **27**, 525 (2001). [CrossRef]

25. G. Lenz and C. K. Madsen, “General optical all-pass filter structures for dispersion control in WDM systems,”, J. Lightwave Technol. **17**, 1248 (1999). [CrossRef]

11. C. M. de Sterke, “Superstructure gratings in the tight-binding approximation,” Phys. Rev. E **57**, 3502 (1998). [CrossRef]

12. R. Slavic and S. LaRochelle, “Large-band periodic filters for DWDM using multiple-superimposed fiber Bragg gratings,” IEEE Photon. Technol. Lett. **14**, 1704 (2002). [CrossRef]

**14**, 1704 (2002). [CrossRef]

_{k}+

*i*γ

_{k}and to model coupling between cavities using the single coupling parameter δ

_{jk}independent of λ. However, the formalism developed in this paper can be applied to this general case too. Actually, Fig. 7b illustrates how to make a simple model of a device consisting of cavities having internal hexagonal symmetry. The complexstructured cavities (large circles) can be assembled of elementary cavities (small circles). The photonic device is then built of these complex cavities as shown in Fig. 7b. The basic concept of such modeling originates from similar concept in quantum mechanics [26

26. Yu. N. Demkov and V. N. Ostrovskii, *Zero-range potentials and their applications in atomic physics*, (Plenum Press, 1988). [CrossRef]

**3**, 2651 (1991). [CrossRef]

27. Y. Meir and N. S. Wingreen, “Landauer Formula for the current through an interacting electron region,” Phys. Rev. Lett. **68**, 2512 (1992). [CrossRef] [PubMed]

8. M. Soljačić, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic- crystal slowlight enhancement of non-linear phase sensitivity,” J. Opt. Soc. Am. B **19**, 2052 (2002). [CrossRef]

29. M. Sumetskii, “Forming of wave packets by one-dimensional tunneling structures having a time-dependent potential,” Phys. Rev. B **46**, 4702 (1992). [CrossRef]

## 6. Summary

## Acknowledgment

## References

1. | N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B |

2. | A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. |

3. | M. Bayindir, B. Temelkuran, and E. Ozbay, “Tight-binding description of the coupled defect modes in threedimensional photonic crystals,” Phys. Rev. Lett. |

4. | E. Ozbay, M. Bayindir, I. Bulu, and E. Cubukcu, “Investigation of localized coupled-cavity modes in twodimensional photonic bandgap structures,” IEEE J. Quant. Electron. |

5. | S. Lan, S. Nishikawa, and H. Ishikawa, “Design of impurity band-based photonic crystal waveguides and delay lines for ultrashort optical pulses,” J. Appl. Phys. |

6. | S. Mookherjea and A. Yariv, “Coupled resonator optical waveguides,” IEEE J. Sel. Topics in Quant. Electron. |

7. | K. Hosomi and T. Katsuyama, “A dispersion compenstor using coupled defects in a photonic crystal,”, IEEE J. Quant. Electron. |

8. | M. Soljačić, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic- crystal slowlight enhancement of non-linear phase sensitivity,” J. Opt. Soc. Am. B |

9. | S. Oliver, C. Smith, M. Rattier, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, and U. Oesterlé, “Miniband transmission in a photonic crystal coupled-resonator optical waveguide,” Opt. Lett. |

10. | S. Nishikawa, S. Lan, N. Ikeda, Y. Sugimoto, H. Ishikawa, and K. Asakawa, “Optical characterization of photonic crystal delay lines based on one-dimensional coupled defects,” Opt. Lett. |

11. | C. M. de Sterke, “Superstructure gratings in the tight-binding approximation,” Phys. Rev. E |

12. | R. Slavic and S. LaRochelle, “Large-band periodic filters for DWDM using multiple-superimposed fiber Bragg gratings,” IEEE Photon. Technol. Lett. |

13. | N. W. Ashcroft and N. D. Mermin, |

14. | L. P. Kouwenhoven, C. M. Marcus, P. L. McEuen, S. Tarucha, R. M. Westervelt, and N. S. Wingreen, “Electron Transport in Quantum Dots,” Proceedings of the NATO Advanced Study Institute on Mesoscopic Electron Transport, edited by L. L. Sohn, L. P. Kouwenhoven, and G. Schön, 1997, pp.105–214. |

15. | Y. Xu, R. K. Lee, and A. Yariv, “Adiabatic coupling between conventional dielectric waveguides and waveguides with discrete translational symmetry,” Opt. Lett. |

16. | A. Mekis and J. D. Joannopoulos, “Tapered couplers for efficient interfacing between dielectric and photonic crystal waveguides,” J. Lightwave Technol. |

17. | D. W. Prather, J. Murakowski, S. Shi, S. Venkataraman, A. Sharkawy, C. Chen, and D. Pustai, “Highlyefficiency coupling structure for a single-line-defect photonic-crystal waveguide,” Opt. Lett. |

18. | P. Sanchis, J. Marti, J. Blasco, A. Martinez, and A. Garcia, “Mode matching technique for highly efficient coupling between dielectric waveguides and planar photonic crystal circuits,” Opt. Express |

19. | L. D. Landau and E. M. Lifshitz, |

20. | M. Sumetskii, “Modeling of complicated nanometer resonant tunneling devices with quantum dots”, J.Phys.: Condens. Matter |

21. | M. Sumetskii, “Resistance resonances for resonant-tunneling structures of quantum dots, Phys. Rev. B |

22. | M. Sumetskii, “Narrow current dip for the double quantum dot resonant tunneling structure with three leads: Sensitive nanometer Y-branch switch,” Appl. Phys. Lett. |

23. | G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,”, IEEE J. Quant. Electron. |

24. | R. Kashyap, |

25. | G. Lenz and C. K. Madsen, “General optical all-pass filter structures for dispersion control in WDM systems,”, J. Lightwave Technol. |

26. | Yu. N. Demkov and V. N. Ostrovskii, |

27. | Y. Meir and N. S. Wingreen, “Landauer Formula for the current through an interacting electron region,” Phys. Rev. Lett. |

28. | S. Datta, “ |

29. | M. Sumetskii, “Forming of wave packets by one-dimensional tunneling structures having a time-dependent potential,” Phys. Rev. B |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(230.7370) Optical devices : Waveguides

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 9, 2003

Revised Manuscript: February 19, 2003

Published: February 24, 2003

**Citation**

Michael Sumetsky and Benjamin Eggleton, "Modeling and optimization of complex photonic resonant cavity circuits," Opt. Express **11**, 381-391 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-4-381

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### References

- N. Stefanou and A. Modinos, �??Impurity bands in photonic insulators,�?? Phys. Rev. B 57, 12127 (1998). [CrossRef]
- A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, �??Coupled-resonator optical waveguide: a proposal and analysis,�?? Opt. Lett. 24, 711 (1999). [CrossRef]
- M. Bayindir, B. Temelkuran, and E. Ozbay, �??Tight-binding description of the coupled defect modes in threedimensional photonic crystals,�?? Phys. Rev. Lett. 84, 2140 (2000). [CrossRef] [PubMed]
- E. Ozbay, M. Bayindir, I. Bulu, and E. Cubukcu, �??Investigation of localized coupled-cavity modes in twodimensional photonic bandgap structures,�?? IEEE J. Quant. Electron. 38, 837 (2002). [CrossRef]
- S. Lan, S. Nishikawa, and H. Ishikawa, �??Design of impurity band-based photonic crystal waveguides and delay lines for ultrashort optical pulses,�?? J. Appl. Phys. 90, 4321 (2001). [CrossRef]
- S. Mookherjea and A. Yariv, �??Coupled resonator optical waveguides,�?? IEEE J. Sel. Topics in Quant. Electron. 3, 448 (2002). [CrossRef]
- K. Hosomi and T. Katsuyama, �??A dispersion compenstor using coupled defects in a photonic crystal,�?? IEEE J. Quant. Electron. 38, 825 (2002). [CrossRef]
- M. Soljacic, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, �??Photonic- crystal slowlight enhancement of non-linear phase sensitivity,�?? J. Opt. Soc. Am. B 19, 2052 (2002). [CrossRef]
- S. Oliver, C. Smith, M. Rattier, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, and U.Oesterlé, �??Miniband transmission in a photonic crystal coupled-resonator optical waveguide,�?? Opt. Lett. 26, 1019 (2002). [CrossRef]
- S. Nishikawa, S. Lan, N. Ikeda, Y. Sugimoto, H. Ishikawa, and K. Asakawa, �??Optical characterization of photonic crystal delay lines based on one-dimensional coupled defects,�?? Opt. Lett. 27, 2079 (2002). [CrossRef]
- C. M. de Sterke, �??Superstructure gratings in the tight-binding approximation,�?? Phys. Rev. E 57, 3502 (1998). [CrossRef]
- R. Slavic and S. LaRochelle, �??Large-band periodic filters for DWDM using multiple-superimposed fiber Bragg gratings,�?? IEEE Photon. Technol. Lett. 14, 1704 (2002). [CrossRef]
- N. W. Ashcroft and N. D. Mermin, Solid state physics, (Saunders College, Philadelphia, 1976).
- . L. P. Kouwenhoven, C. M. Marcus, P. L. McEuen, S. Tarucha, R. M. Westervelt, and N. S. Wingreen, �??Electron Transport in Quantum Dots,�?? Proceedings of the NATO Advanced Study Institute on Mesoscopic Electron Transport, edited by L. L. Sohn, L. P. Kouwenhoven, and G. Schön, 1997, pp.105-214.
- Y. Xu, R. K. Lee, and A. Yariv, �??Adiabatic coupling between conventional dielectric waveguides and waveguides with discrete translational symmetry,�?? Opt. Lett. 25, 755 (2000). [CrossRef]
- A. Mekis and J. D. Joannopoulos, �??Tapered couplers for efficient interfacing between dielectric and photonic crystal waveguides,�?? J. Lightwave Technol. 19, 861 (2001). [CrossRef]
- D. W. Prather, J. Murakowski, S. Shi, S. Venkataraman, A. Sharkawy, C. Chen, and D. Pustai, �??Highlyefficiency coupling structure for a single-line-defect photonic-crystal waveguide,�?? Opt. Lett. 27, 1601 (2002). [CrossRef]
- P. Sanchis, J. Marti, J. Blasco, A. Martinez, and A. Garcia, �??Mode matching technique for highly efficient coupling between dielectric waveguides and planar photonic crystal circuits,�?? Opt. Express 10, 1391 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1391">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1391</a> [CrossRef] [PubMed]
- L. D. Landau and E. M. Lifshitz, Quantum mechanics, (Pergamon Press, 1958, pp. 440-449).
- M. Sumetskii, �??Modeling of complicated nanometer resonant tunneling devices with quantum dots,�?? J. Phys: Condens. Matter 3, 2651 (1991). [CrossRef]
- M. Sumetskii, �??Resistance resonances for resonant-tunneling structures of quantum dots," Phys. Rev. B 48, 4586 (1993). [CrossRef]
- M. Sumetskii, �??Narrow current dip for the double quantum dot resonant tunneling structure with three leads: Sensitive nanometer Y-branch switch,�?? Appl. Phys. Lett. 63, 3185 (1993). [CrossRef]
- G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, �??Optical delay lines based on optical filters,�?? IEEE J. Quant. Electron. 27, 525 (2001). [CrossRef]
- R. Kashyap, Fiber Bragg Gratings, (Academic Press, 1999).
- G. Lenz and C. K. Madsen, �??General optical all-pass filter structures for dispersion control in WDM systems,�?? J. Lightwave Technol. 17, 1248 (1999). [CrossRef]
- Yu. N. Demkov and V. N. Ostrovskii, Zero-range potentials and their applications in atomic physics, (Plenum Press, 1988). [CrossRef]
- Y. Meir and N. S. Wingreen, �??Landauer Formula for the current through an interacting electron region,�?? Phys. Rev. Lett. 68, 2512 (1992). [CrossRef] [PubMed]
- S. Datta, Electronic transport in mesoscopic systems, (Cambridge University Press, 1995).
- M. Sumetskii, �??Forming of wave packets by one-dimensional tunneling structures having a time-dependent potential,�?? Phys. Rev. B 46, 4702 (1992). [CrossRef]

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