## Theory of SNAP devices: basic equations and comparison with the experiment |

Optics Express, Vol. 20, Issue 20, pp. 22537-22554 (2012)

http://dx.doi.org/10.1364/OE.20.022537

Acrobat PDF (4272 KB)

### Abstract

A SNAP (Surface Nanoscale Axial Photonics) device consists of an optical fiber with introduced nanoscale effective radius variation, which is coupled to transverse input/output waveguides. The input waveguides excite whispering gallery modes circulating near the fiber surface and slowly propagating along the fiber axis. In this paper, the theory of SNAP devices is developed and applied to the analysis of transmission amplitudes of simplest SNAP models exhibiting a variety of asymmetric Fano resonances and also to the experimental characterization of a SNAP bottle microresonator and to a chain of 10 coupled microresonators. Excellent agreement between the theory and the experiment is demonstrated.

© 2012 OSA

## 1. Introduction

6. M. Sumetsky, D. J. DiGiovanni, Y. Dulashko, X. Liu, E. M. Monberg, and T. F. Taunay, “Photo-induced SNAP: fabrication, trimming, and tuning of microresonator chains,” Opt. Express **20**(10), 10684–10691 (2012). [CrossRef] [PubMed]

6. M. Sumetsky, D. J. DiGiovanni, Y. Dulashko, X. Liu, E. M. Monberg, and T. F. Taunay, “Photo-induced SNAP: fabrication, trimming, and tuning of microresonator chains,” Opt. Express **20**(10), 10684–10691 (2012). [CrossRef] [PubMed]

7. F. Luan, E. Magi, T. Gong, I. Kabakova, and B. J. Eggleton, “Photoinduced whispering gallery mode microcavity resonator in a chalcogenide microfiber,” Opt. Lett. **36**(24), 4761–4763 (2011). [CrossRef] [PubMed]

3. M. Sumetsky, D. J. DiGiovanni, Y. Dulashko, J. M. Fini, X. Liu, E. M. Monberg, and T. F. Taunay, “Surface nanoscale axial photonics: robust fabrication of high-quality-factor microresonators,” Opt. Lett. **36**(24), 4824–4826 (2011). [CrossRef] [PubMed]

6. M. Sumetsky, D. J. DiGiovanni, Y. Dulashko, X. Liu, E. M. Monberg, and T. F. Taunay, “Photo-induced SNAP: fabrication, trimming, and tuning of microresonator chains,” Opt. Express **20**(10), 10684–10691 (2012). [CrossRef] [PubMed]

8. H. G. Limberger, P.-Y. Fonjallaz, R. P. Salathé, and F. Cochet, “Compaction‐ and photoelastic‐induced index changes in fiber Bragg gratings,” Appl. Phys. Lett. **68**(22), 3069–3071 (1996). [CrossRef]

9. A. D. Yablon, M. F. Yan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. **84**(1), 19–21 (2004). [CrossRef]

2. M. Sumetsky and J. M. Fini, “Surface nanoscale axial photonics,” Opt. Express **19**(27), 26470–26485 (2011). [CrossRef] [PubMed]

2. M. Sumetsky and J. M. Fini, “Surface nanoscale axial photonics,” Opt. Express **19**(27), 26470–26485 (2011). [CrossRef] [PubMed]

2. M. Sumetsky and J. M. Fini, “Surface nanoscale axial photonics,” Opt. Express **19**(27), 26470–26485 (2011). [CrossRef] [PubMed]

*N*waveguides, one of which is the input/output waveguide and others are the output waveguides, can be expressed through the solution of the one-dimensional Schrödinger equation of Refs [1,2

**19**(27), 26470–26485 (2011). [CrossRef] [PubMed]

*N*complex constants, which determine the lossy SF/waveguide coupling. In the case of lossless coupling, the number of constants is reduced to

*N*. In Section 4, the developed theory is applied to the investigation of transmission amplitudes through a localized state and also through a uniform SF coupled to one waveguide, which exhibits a variety of asymmetric Fano resonances. In Section 5, it is shown that sharp transmission resonances can characterize a lossy SNAP device coupled to two waveguides. In Section 6, the experimental characterization of the SNAP bottle microresonator and of a chain of 10 coupled microresonators is demonstrated to be in the excellent agreement with the developed theory. It is shown that the effective radius variation of a SNAP fiber and all of the transmission amplitude parameters can be accurately determined from the experiment. The results of this paper are discussed and summarized in Section 7.

## 2. WGMs in a SNAP fiber in the absence of input/output waveguides

*adiabatically*propagating along the axis

*z*of an SF with

*nanoscale*smooth radius variation (and/or equivalent refractive index variation)

*in the absence of input/output waveguides*is defined aswhere

*m*is the discrete azimuthal quantum number,

*p*is the discrete radial quantum number, and

*q*is the discrete or continuous axial quantum number. Function

**19**(27), 26470–26485 (2011). [CrossRef] [PubMed]

*m*and

*p*. These numbers are omitted below for brevity.

## 3. Theory of a SNAP device

10. A. D. Stone and A. Szafer, “What is measured when you measure a resistance?-The Landauer formula revisited,” IBM J. Res. Develop. **32**(3), 384–413 (1988). [CrossRef]

16. M. Sumetsky and B. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express **11**(4), 381–391 (2003). [CrossRef] [PubMed]

*renormalized Green’s function*of the cavity was introduced. This function modifies the Green’s function of the isolated cavity (i.e., the cavity uncoupled from the waveguides) called the

*bare Green’s function*, and takes into account the losses due to coupling to the waveguides. It was found that the transmission amplitudes through the waveguides can be expressed through the overlap integrals between the renormalized Green’s function and the travelling waves propagating along the waveguides.

10. A. D. Stone and A. Szafer, “What is measured when you measure a resistance?-The Landauer formula revisited,” IBM J. Res. Develop. **32**(3), 384–413 (1988). [CrossRef]

15. Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **62**(5), 7389–7404 (2000). [CrossRef] [PubMed]

11. H. U. Baranger, R. A. Jalabert, and A. D. Stone, “Quantum-chaotic scattering effects in semiconductor microstructures,” Chaos **3**(4), 665–682 (1993). [CrossRef] [PubMed]

16. M. Sumetsky and B. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express **11**(4), 381–391 (2003). [CrossRef] [PubMed]

*N*transverse waveguides WG

_{1}, WG

_{2}, …, WG

*. Assume that WG*

_{N}_{1}serves as the input and output waveguide while all other waveguides are the output waveguides only (in Fig. 1,

*while its imaginary part takes into account the radiation loss through WG*

_{n}*, In Eq. (4), coupling to waveguides WG*

_{n}*is modeled with*

_{n}*zero-range potentials*

_{1}is simply found as (Appendix 1)and the transmission amplitude from WG

_{1}to WG

*iswhere constants*

_{n}*coupling parameters and*

_{n}*N*complex constants

## 4. Transmission amplitude of a SNAP device coupled to a single waveguide

_{1}coupled to an SF (

19. M. Sumetsky, “Whispering-gallery-bottle microcavities: the three-dimensional etalon,” Opt. Lett. **29**(1), 8–10 (2004). [CrossRef] [PubMed]

*ideal coupling*which is confirmed experimentally for weak coupling and a single mode waveguide [20

20. S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. **91**(4), 043902 (2003). [CrossRef] [PubMed]

*N*waveguides,

### 4.1. Transmission through localized states of an SF

19. M. Sumetsky, “Whispering-gallery-bottle microcavities: the three-dimensional etalon,” Opt. Lett. **29**(1), 8–10 (2004). [CrossRef] [PubMed]

21. A. E. Miroshnichenko, S. Flach, and Yu. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. **82**(3), 2257–2298 (2010). [CrossRef]

_{1}replaces the resonance width

*, which is proportional to*

_{n}_{1}[2

**19**(27), 26470–26485 (2011). [CrossRef] [PubMed]

21. A. E. Miroshnichenko, S. Flach, and Yu. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. **82**(3), 2257–2298 (2010). [CrossRef]

### 4.2. Transmission through a uniform SF

_{1}does not cause a phase shift. In the presence of a phase shift, the behavior of the transmission amplitude can be more complex. This is illustrated in Fig. 3(b) for

## 5. Transmission amplitudes of a SNAP device coupled to two waveguides

_{1}, WG

_{2}, …, WG

*, i.e., the transmission amplitudes*

_{N}_{1},

_{1}and exiting WG

_{2},

### 5.1. Transmission through the localized states of an SF

_{2}. Thus, the presence of the second waveguide modifies the widths and shifts of the individual resonances (e.g., those in a bottle microresonator); however, it does introduce qualitative changes in the spectral behavior.

### 5.2. Transmission through a uniform SF

_{2}is negligible then the renormalized Green’s function defined by Eqs. (18) and (19) coincides with that of a single waveguide device defined by Eq. (7). Here we consider the opposite case of large coupling to WG

_{2}. Then, for the lossless coupling to WG

_{1}, i.e., for

_{1}does not shift the resonances. To understand the effect of shifting, Fig. 4(b) shows the case when

_{1}and WG

_{2}. The widths of these resonances are proportional to

_{1}coupling.

## 6. A SNAP bottle microresonator and a chain of 10 coupled microresonators: theory vs. experiment

3. M. Sumetsky, D. J. DiGiovanni, Y. Dulashko, J. M. Fini, X. Liu, E. M. Monberg, and T. F. Taunay, “Surface nanoscale axial photonics: robust fabrication of high-quality-factor microresonators,” Opt. Lett. **36**(24), 4824–4826 (2011). [CrossRef] [PubMed]

4. M. Sumetsky, K. Abedin, D. J. DiGiovanni, Y. Dulashko, J. M. Fini, E. Monberg, and E. M. Monberg, “Coupled high Q-factor surface nanoscale axial photonics (SNAP) microresonators,” Opt. Lett. **37**(6), 990–992 (2012). [CrossRef] [PubMed]

_{2}laser beam. The introduced effective radius variation is measured using the microfiber scanning method [22

22. T. A. Birks, J. C. Knight, and T. E. Dimmick, “High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment,” IEEE Photon. Technol. Lett. **12**(2), 182–183 (2000). [CrossRef]

23. M. Sumetsky and Y. Dulashko, “Radius variation of optical fibers with angstrom accuracy,” Opt. Lett. **35**(23), 4006–4008 (2010). [CrossRef] [PubMed]

_{1}in Fig. 1. The waist of the taper is translated along the SF and touches it periodically at the contact points where the transmission spectra of the taper are measured. These spectra are used for the determination of the SF radius variation. In our experiments, the resonant transmission amplitude spectra are determined with the Luna Optical Vector Analyzer (wavelength resolution 1.3 pm). The experimental data is taken along the SF at contact points spaced by 2 µm. In previous publications [3

3. M. Sumetsky, D. J. DiGiovanni, Y. Dulashko, J. M. Fini, X. Liu, E. M. Monberg, and T. F. Taunay, “Surface nanoscale axial photonics: robust fabrication of high-quality-factor microresonators,” Opt. Lett. **36**(24), 4824–4826 (2011). [CrossRef] [PubMed]

4. M. Sumetsky, K. Abedin, D. J. DiGiovanni, Y. Dulashko, J. M. Fini, E. Monberg, and E. M. Monberg, “Coupled high Q-factor surface nanoscale axial photonics (SNAP) microresonators,” Opt. Lett. **37**(6), 990–992 (2012). [CrossRef] [PubMed]

_{2}laser beam. The surface plot of the resonant transmission amplitudes for this microresonator is shown in Fig. 5(a) . It is seen that the spectral resonances are broadened and shifted due to coupling to the microfiber. Therefore, the actual spectrum of the uncoupled bottle resonator is determined in the regions where the coupling is minimized, i.e., near nodes of the field and in evanescent regions. The set of transmission amplitudes, which correspond to the microfiber positions 50 μm, 94 μm, and 120 μm and jointly contain all of such narrow resonances indicated by dashed arrows, are shown in Fig. 6(a) . These resonances are numerically fitted to the eigenvalues of the Schrödinger Eq. (2) where the effective fiber radius variation of the SNAP bottle microresonator is parameterized as a sum of Gaussian and Lorentzian shapes:The eigenvalues of Eq. (2) are found as the singularities of the bare Green’s function

_{2}laser beam spaced by 50 μm. The surface plot of the resonant transmission amplitudes for this chain is shown in Fig. 7(a). It is seen that, due to mutual coupling, the fundamental modes of microresonators form a transmission band (outlined by a dashed line) separated from other modes by a bandgap. The magnified surface plot of the fundamental transmission band is shown in Fig. 7(b). The observed spectral plots can be described theoretically by approximating the effective fiber radius variation as:where

## 7. Discussion and summary

## Appendix 1

## Transmission amplitude of a SNAP device

*q*,where

**19**(27), 26470–26485 (2011). [CrossRef] [PubMed]

*in the absence of SF when the waveguides do not couple to each other. The SF is introduced with a potential V(*

_{n}**r**) so that the T-matrix, which determines the inter-waveguide coupling due to the presence of the SF, is expressed through the Green’s function of Eq. (A1.2) as

_{1}we have:In the vicinity of the resonance

*E*which can be set to constants. In SNAP, the axial size of the waveguide/SF coupling area is small compared to the axial wavelength (i.e., to the characteristic axial variation length of the Green’s function

_{1},

20. S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. **91**(4), 043902 (2003). [CrossRef] [PubMed]

_{1}to WG

*iswhere constant*

_{n}## Appendix 2

## Calculation of the Green’s functions

*n*matching conditions at the points of coupling

_{1}is the only input and output waveguide, while all others are output only waveguides.

_{1},

_{1}and WG

_{2},

## Appendix 3

## Limitations on the transmission amplitude and coupling parameters

_{1}, the conservation of energy requires

*G*with

*C*

_{1},

*D*

_{1}, and

## Acknowledgments

## References and links

1. | M. Sumetsky, “Localization of light in an optical fiber with nanoscale radius variation,” in CLEO/Europe and EQEC 2011 Conference Digest, postdeadline paper PDA_8. |

2. | M. Sumetsky and J. M. Fini, “Surface nanoscale axial photonics,” Opt. Express |

3. | M. Sumetsky, D. J. DiGiovanni, Y. Dulashko, J. M. Fini, X. Liu, E. M. Monberg, and T. F. Taunay, “Surface nanoscale axial photonics: robust fabrication of high-quality-factor microresonators,” Opt. Lett. |

4. | M. Sumetsky, K. Abedin, D. J. DiGiovanni, Y. Dulashko, J. M. Fini, E. Monberg, and E. M. Monberg, “Coupled high Q-factor surface nanoscale axial photonics (SNAP) microresonators,” Opt. Lett. |

5. | M. Wilson, “Optical fiber microcavities reach angstrom-scale precision,” Phys. Today |

6. | M. Sumetsky, D. J. DiGiovanni, Y. Dulashko, X. Liu, E. M. Monberg, and T. F. Taunay, “Photo-induced SNAP: fabrication, trimming, and tuning of microresonator chains,” Opt. Express |

7. | F. Luan, E. Magi, T. Gong, I. Kabakova, and B. J. Eggleton, “Photoinduced whispering gallery mode microcavity resonator in a chalcogenide microfiber,” Opt. Lett. |

8. | H. G. Limberger, P.-Y. Fonjallaz, R. P. Salathé, and F. Cochet, “Compaction‐ and photoelastic‐induced index changes in fiber Bragg gratings,” Appl. Phys. Lett. |

9. | A. D. Yablon, M. F. Yan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. |

10. | A. D. Stone and A. Szafer, “What is measured when you measure a resistance?-The Landauer formula revisited,” IBM J. Res. Develop. |

11. | H. U. Baranger, R. A. Jalabert, and A. D. Stone, “Quantum-chaotic scattering effects in semiconductor microstructures,” Chaos |

12. | S. Datta, |

13. | F.-M. Dittes, “The decay of quantum systems with a small number of open channels,” Phys. Rep. |

14. | S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. Khan, C. Manolatou, and H. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B |

15. | Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

16. | M. Sumetsky and B. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express |

17. | S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, |

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

19. | M. Sumetsky, “Whispering-gallery-bottle microcavities: the three-dimensional etalon,” Opt. Lett. |

20. | S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. |

21. | A. E. Miroshnichenko, S. Flach, and Yu. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. |

22. | T. A. Birks, J. C. Knight, and T. E. Dimmick, “High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment,” IEEE Photon. Technol. Lett. |

23. | M. Sumetsky and Y. Dulashko, “Radius variation of optical fibers with angstrom accuracy,” Opt. Lett. |

24. | J. Ziman, |

**OCIS Codes**

(060.2340) Fiber optics and optical communications : Fiber optics components

(230.3990) Optical devices : Micro-optical devices

(140.3945) Lasers and laser optics : Microcavities

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: August 8, 2012

Revised Manuscript: September 11, 2012

Manuscript Accepted: September 12, 2012

Published: September 17, 2012

**Citation**

M. Sumetsky, "Theory of SNAP devices: basic equations and comparison with the experiment," Opt. Express **20**, 22537-22554 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-20-22537

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### References

- M. Sumetsky, “Localization of light in an optical fiber with nanoscale radius variation,” in CLEO/Europe and EQEC 2011 Conference Digest, postdeadline paper PDA_8.
- M. Sumetsky and J. M. Fini, “Surface nanoscale axial photonics,” Opt. Express19(27), 26470–26485 (2011). [CrossRef] [PubMed]
- M. Sumetsky, D. J. DiGiovanni, Y. Dulashko, J. M. Fini, X. Liu, E. M. Monberg, and T. F. Taunay, “Surface nanoscale axial photonics: robust fabrication of high-quality-factor microresonators,” Opt. Lett.36(24), 4824–4826 (2011). [CrossRef] [PubMed]
- M. Sumetsky, K. Abedin, D. J. DiGiovanni, Y. Dulashko, J. M. Fini, E. Monberg, and E. M. Monberg, “Coupled high Q-factor surface nanoscale axial photonics (SNAP) microresonators,” Opt. Lett.37(6), 990–992 (2012). [CrossRef] [PubMed]
- M. Wilson, “Optical fiber microcavities reach angstrom-scale precision,” Phys. Today65(2), 14–16 (2012). [CrossRef]
- M. Sumetsky, D. J. DiGiovanni, Y. Dulashko, X. Liu, E. M. Monberg, and T. F. Taunay, “Photo-induced SNAP: fabrication, trimming, and tuning of microresonator chains,” Opt. Express20(10), 10684–10691 (2012). [CrossRef] [PubMed]
- F. Luan, E. Magi, T. Gong, I. Kabakova, and B. J. Eggleton, “Photoinduced whispering gallery mode microcavity resonator in a chalcogenide microfiber,” Opt. Lett.36(24), 4761–4763 (2011). [CrossRef] [PubMed]
- H. G. Limberger, P.-Y. Fonjallaz, R. P. Salathé, and F. Cochet, “Compaction‐ and photoelastic‐induced index changes in fiber Bragg gratings,” Appl. Phys. Lett.68(22), 3069–3071 (1996). [CrossRef]
- A. D. Yablon, M. F. Yan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett.84(1), 19–21 (2004). [CrossRef]
- A. D. Stone and A. Szafer, “What is measured when you measure a resistance?-The Landauer formula revisited,” IBM J. Res. Develop.32(3), 384–413 (1988). [CrossRef]
- H. U. Baranger, R. A. Jalabert, and A. D. Stone, “Quantum-chaotic scattering effects in semiconductor microstructures,” Chaos3(4), 665–682 (1993). [CrossRef] [PubMed]
- S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University Press, UK, 1997).
- F.-M. Dittes, “The decay of quantum systems with a small number of open channels,” Phys. Rep.339(4), 215–316 (2000). [CrossRef]
- S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. Khan, C. Manolatou, and H. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B59(24), 15882–15892 (1999). [CrossRef]
- Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics62(5), 7389–7404 (2000). [CrossRef] [PubMed]
- M. Sumetsky and B. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express11(4), 381–391 (2003). [CrossRef] [PubMed]
- S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics (American Mathematical Society, Providence, 2004).
- Yu. N. Demkov and V. N. Ostrovskii, Zero-Range Potentials and their Applications in Atomic Physics (Plenum, New York, 1988).
- M. Sumetsky, “Whispering-gallery-bottle microcavities: the three-dimensional etalon,” Opt. Lett.29(1), 8–10 (2004). [CrossRef] [PubMed]
- S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett.91(4), 043902 (2003). [CrossRef] [PubMed]
- A. E. Miroshnichenko, S. Flach, and Yu. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys.82(3), 2257–2298 (2010). [CrossRef]
- T. A. Birks, J. C. Knight, and T. E. Dimmick, “High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment,” IEEE Photon. Technol. Lett.12(2), 182–183 (2000). [CrossRef]
- M. Sumetsky and Y. Dulashko, “Radius variation of optical fibers with angstrom accuracy,” Opt. Lett.35(23), 4006–4008 (2010). [CrossRef] [PubMed]
- J. Ziman, Elements of Advanced Quantum Theory (Cambridge University Press, London, 1969).

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