## Radiation loss of a nanotaper: Singular Gaussian beam model

Optics Express, Vol. 15, Issue 4, pp. 1480-1490 (2007)

http://dx.doi.org/10.1364/OE.15.001480

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### Abstract

This paper presents a model of a subwavelength diameter adiabatic microfiber taper (nanotaper), which allows an asymptotically accurate solution of the wave equation. The evanescent field near the nanotaper is expressed through a Gaussian beam having a singularity at the nanotaper axis. For certain values of parameters of the nanotaper, when it has a swell in the middle and narrows down to zero at the infinity, the nanotaper is lossless. For other values, when the nanotaper has a biconical shape, it exhibits an exponentially small radiation loss, which is determined as a tunneling rate through an effective parabolic potential barrier. The latter case represents an exceptional example of the radiation loss being distributed along the length of an adiabatic nanotaper rather than being localized near focal circumferences in the evanescent field region.

© 2007 Optical Society of America

## 1. Introduction

1. L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature ,**426**,816–819 (2003). [CrossRef] [PubMed]

*r*. Pure adiabatic tapering assumes slow dependence

*r*(

*z*) on the coordinate along the microfiber axis,

*z*, and conserves the cylindrical symmetry. Radiation loss in bent uniform microfibers is understood quite well. Adiabatically bent microfibers allow us to find the propagating mode in a curvilinear coordinate system localized along the fiber axis [9,10

10. W. L. Kath and G A. Kriegsmann, “Optical tunnelling: radiation losses in bent fibre-optic waveguides,” IMA J. Appl. Math. **41**,85–103(1988). [CrossRef]

*z*[9].

11. M. Sumetsky, “How thin can a microfiber be and still guide light?,” Opt. Lett. **31**,870–872 (2006). [CrossRef] [PubMed]

## 2. General properties of adiabatic nanotapers

*β*(

*z*), is a slow function of the coordinate

*z*along the axis of NT. The propagation constant

*β*(

*z*) is close to the propagation constant of the ambient medium

*β*

_{0}:

*γ*(

*z*) is the transversal component of the propagation constant, which is relatively small,

*γ*(z)≪

*β*

_{0}. For a given

*β*

_{0}, the function

*γ*(

*z*) is uniquely determined through the NT radius

*r*(

*z*) and can be asymptotically expressed as [12

12. M. Sumetsky, “How thin can a microfiber be and still guide light? Errata,” Opt. Lett. **31**,3577–3578 (2006). [CrossRef]

*n*

_{1}is the refractive index of the NT and

*n*

_{2}is the refractive index of the ambient medium.

*locally*near focal circumferences surrounding the NT [13

13. M. Sumetsky, “Optics of tunneling from adiabatic nanotapers,” Opt. Lett. **31**,3420–3422 (2006). [CrossRef] [PubMed]

*z*and slowly expands. Also, the radiation wave exponentially vanishes along the radial directions from the NT. Therefore, for a NT of a common shape, the effective transversal potential barrier, which determines the tunneling dynamics of light, is

*infinitely wide*. For this reason, radiation from the adiabatic NT happens primarily along the

*longitudial direction*.

## 3. Regular and singular Gaussian beams

*finite width*that is adjacent to the NT and is responsible for

*distributed*radiation loss. As opposed to the NT of a common shape, this NT exhibits radiation, which is

*transversal*(i.e. taking place through the adjacent barrier) rather than longitudial to the NT. The shape of this NT is found in the following way. First, an adiabatic NT of a general shape is considered, and the corresponding adiabatic solution of the wave equation is written out in the form of Eq. (A1.5) of Appendix 1. Then, the anzatz for the solution of the wave equation away from the NT is chosen in the form similar to the form of the Gaussian beam. The details are given in the Appendix 2. This anzatz allows factorization of the solution of the wave equation into the product of a known function and a function Λ(

*ν*) that obeys the etalon ordinary differential equation with respect to the effective transversal coordinate

*ν*:

*ν*is expressed through the cylindrical coordinates (

*ρ*,

*z*) along the NT as follows:

*a*,

*b*, and

*c*are free parameters. Function

*P*(

*ρ*,

*z*) = λ

^{2}(

*ν*)/

*ρ*

^{2}determines the distribution of the electromagnetic field power outside of the NT. In order to determine the regular Gaussian beam, one demands that

*P*(

*ρ*,

*z*) is finite for

*ρ*→ 0 and

*P*(

*ρ*,

*z*) → 0 for

*ρ*→ ∞. The latter conditions lead to the equations

*b*=

*a*

^{2}/4 and λ(

*ν*) = exp(-

*aν*

^{2}/4) which yield the familiar form of a Gaussian beam. As opposed to this regular Gaussian beam, here we look for a solution of the wave equation that does have finite asymptotic near the center of the beam. Instead, our solution matches the asymptotics of the adiabatic solution near the NT which behaves as a function of

*ρ*as

*K*

_{0}(

*γ*(

*z*)

*ρ*) (see Appendix 1). Therefore, it is singular for

*ρ*→ 0 . It is shown in Appendix 2, that matching the adiabatic solution with the solution defined with Eqs.(3)-(5) is only possible if the shape of NT corresponds to the transversal propagation constant

*L*, and the value of the transversal propagation constant in the middle of NT,

*γ*

_{0}, are expressed through the parameters of the surrounding electromagnetic field as follows:

*γ*

^{-}(

*z*) as well as the corresponding shape of the NT,

*r*

^{-}(

*z*), which can be determined from Eq. (2), have the shape shown in Fig. 1(a). This shape shows up as the shape of a thread drawn out of a liquid material reservoir (see Ref. [14

14. L. Tong, L. Hu, J. Zhang, J. Qiu, Q. Yang, J. Lou, Y. Shen, J. He, and Z. Ye, “Photonic nanowires directly drawn from bulk glasses,” Opt. Express **14**,82–87 (2006). [CrossRef] [PubMed]

*γ*

^{+}(

*z*) and the corresponding shape of the nanoswell,

*r*

^{+}(

*z*) have the shape shown in Fig. 1(b).

## 4. Transmission properties of a NT and nanoswell supporting singular Gaussian beams

*r*

^{-}(

*z*) and

*r*

^{+}(

*z*), respectively. The considered problem can be better understood with illustrations shown in Fig. 2 and 3. Fig. 2 compares the radiation loss in a bent microfiber (a) and in a NT of shape

*r*

^{-}(

*z*) (b). Fig. 3 compares the lossless propagation of a regular Gaussian beam (a) and of a beam supported by a nanoswell

*r*

^{+}(

*z*)(b).

10. W. L. Kath and G A. Kriegsmann, “Optical tunnelling: radiation losses in bent fibre-optic waveguides,” IMA J. Appl. Math. **41**,85–103(1988). [CrossRef]

*r*

^{-}(

*z*) shown in Fig. 1(a). Figure 2(b1) shows the corresponding behavior of the effective transverse dielectric constant, which is determined by Eq. (3) and is proportional to

*a*-

*bν*

^{2}. In this case,

*a*< 0 and

*b*< 0 and the NT is surrounded by the barrier (classically forbidden) region

*ν*< (

*a*/

*b*)

^{1/2}. In this region, the field exponentially decreases along the radial direction. The radiation loss is determined by the outgoing wave in the classically allowed region

*ν*> (

*a*/

*b*)

^{1/2}. Calculation of the flux density along the radial direction (see Appendix 3) yields the following expression for the attenuation constant of the NT:

*z*→ ±

*L*, where the NT radius becomes large and Eq.(8) fails. The exponent in Eq.(8) is independent of

*z*, which indicates the uniform transparency of the effective potential barrier surrounding the NT. The latter follows from separation of variables in the wave equation, possibility of introduction of the effective transversal coordinate

*ν*, and the etalon Eq. (3).

*r*

^{+}(

*z*) (b). As opposed to the NT shown in Fig. 2(b), Fig. 3 illustrates the cases when the parameter

*b*in Eq. (3) is positive,

*b*> 0 . Then at

*ν*, which is large enough, the dielectric constant is negative in both Fig. 3(a1) and Fig. 3(b1) and the electromagnetic field density exponentially decreases with growth of the distance from the nanoswell axis (Figs. 3(a), (a2) and Figs. 3(b) and (b1)). For this reason, propagation of the evanescent field along the nanoswell is similar to the propagation of a regular Gaussian beam. In particular, it is lossless in the approximation considered. However, the field in the immediate neighborhood of a nanoswell behaves more singular than the field of a regular Gaussian beam (compare Figs. 3(a2) and (b2)). It is known that tapers can be lossless in special cases (see [15

15. A.D. Capobianco, M. Midrio, C.G. Someda, and S. Curtarolo, “Lossless tapers, Gaussian beams, free-space modes: Standing waves versus through-flowing waves,” Opt. Quantum Electron. **32**,1161–1173 (2000). [CrossRef]

*r*

^{+}(

*z*) presents an interesting example of a lossless adiabatic taper which has a swell in the middle and, as follows from Eq. (2), for large

*z*, decreases logarithmically as 1/√ln(∣

*z*∣/

*L*) to zero. At first sight, the absence of losses in this taper contradicts the existence of a threshold radius of a NT, below which the propagation of the fundamental mode is impossible [11

11. M. Sumetsky, “How thin can a microfiber be and still guide light?,” Opt. Lett. **31**,870–872 (2006). [CrossRef] [PubMed]

11. M. Sumetsky, “How thin can a microfiber be and still guide light?,” Opt. Lett. **31**,870–872 (2006). [CrossRef] [PubMed]

## 5. Discussion and summary

13. M. Sumetsky, “Optics of tunneling from adiabatic nanotapers,” Opt. Lett. **31**,3420–3422 (2006). [CrossRef] [PubMed]

*ν*=

*ρ*/σ(

*z*) = (

*a*/

*b*)

^{1/2}rather than a circumference (see Eqs. (3), (4), and (5)). It is natural to suppose that for a NT, whose shape is slightly different from

*r*

^{-}(

*z*), the caustic, though becomes complex, will be still situated in proximity to real space. The radiation loss in the latter case will be still smoothed out along the length of a NT. In particular, there may exist a situation when the local attenuation constant of NT is independent on the coordinate

*z*, similar to the attenuation constant of a microfiber with constant bending radius. For the NT

*r*

^{-}(

*z*), the attenuation constant is not uniform along the NT due to the z-dependence of the pre-exponent factor in Eq. (8). However, the transparency of the introduced effective parabolic potential barrier is independent of

*z*.

*r*

^{+}(

*z*) is lossless in the considered approximation. However, the radiation loss of this nanoswell of a higher-order of smallness may be determined with the higher-order terms which were ignored in our derivation based on the first-order approximation of the parabolic equation method [16]. The latter is basically the paraxial approximation. Similarly, a regular Gaussian beam, which is the solution of the wave equation in the paraxial approximation, is lossless along the transversal directions. However, taking into the account the higher-order terms beyond the paraxial approximation it can be shown that a Gaussian beam may exhibit extremely small transversal radiation losses, which grow with simultaneous decreasing of parameters

*a*and

*b*in Eq. (3) [17]. The latter effects are beyond the scope of this paper.

## Appendix 1.

## The fundamental mode in the vicinity of an adiabatic NT

*n*

_{1}and

*n*

_{2}, respectively. The radius of the NT,

*r*(

*z*), is assumed to be an adiabatically slow function of the coordinate

*z*along the fiber axis. Also, it is assumed that

*r*(

*z*) is significantly less than the characteristic radiation wavelength,

*β*

_{0}

*r*(

*z*)≪1, where

*β*

_{0}= 2

*πn*

_{2}/

*λ*is the propagation constant and

*λ*is the radiation wavelength in free space. In this situation, at the distances from the NT

*ρ*≫

*r*(

*z*) the fundamental mode is linearly polarized and can be described by a scalar wave equation:

*z*) are used. The condition

*β*

_{0}

*r*(

*z*)≪1 implies smallness of the local radial component of the propagation constant outside the NT,

*γ*(

*z*), compared to the longitudinal component,

*β*(

*z*), i.e.

*γ*(

*z*) is determined through the NT radius

*r*(

*z*) by Eq. (2). In particular, for a glass NF in air,

*n*

_{1}= 1.45 and

*n*

_{2}= 1 and

*K*

_{0}(

*x*) is the modified Bessel function of the second kind. For

*γ*(

*z*)

*ρ*≫ 1 the asymptotics of Eq. (A1.5) is:

*ρ*≪

*ρ*-

_{L}*L*/

_{γ}*β*

_{0}, where

*L*-∣

*γ*/(

*dγ*/

*dz*)∣ is the characteristic length of the NT nonuniformity [13

13. M. Sumetsky, “Optics of tunneling from adiabatic nanotapers,” Opt. Lett. **31**,3420–3422 (2006). [CrossRef] [PubMed]

*γ*/

*β*

_{0}≪ 1 this vicinity is relatively small. Furthermore, the radiation loss of a NT is predominantly determined by the behavior of solution in the region

*ρ*≳

*ρ*where Eq. (A1.5) fails [13

_{L}**31**,3420–3422 (2006). [CrossRef] [PubMed]

*ρ*≪

*ρ*. The semiclassical approach developed in Ref. [13

_{L}**31**,3420–3422 (2006). [CrossRef] [PubMed]

## Appendix 2.

## Singular Gaussian beam model

*γ*(

*z*) ≪

*β*, the parabolic equation method [16,17] allows us to find asymptotic solutions of Eq.(A1.1) localized near the fiber taper axis,

*z*in the form:

*ν*) satisfies the equation

*A*

_{11},

*A*

_{12},

*a*, and

*b*. In the main text, Eq. (5) is equivalent to Eq. (A2.2). However, in Eq. (5) the coefficient

*A*

_{12}has been eliminated by translation along the axis

*z*.

*ρ*→ 0 and tend to zero for

*ρ*→ ∞. In this case, solution of Eq. (A2.3) should be finite for

*ν*→ 0 and tends to zero for

*ν*→ ∞. Then Λ(

*ν*) can be expressed through the generalized Laguerre polynomials. Alternatively, in our situation, solution (A2.1) should be matched with solution (A1.6), which is singular at

*ρ*→ 0 . Due to the cylindrical symmetry of Eq. (A1.6) we have to set

*m*= 0 in Eq. (A2.1). Then function Ψ(

*ρ*,

*φ*,

*z*) is independent of

*φ*and, for simplicity, will be written as Ψ(

*ρ*,

*z*). Solutions of Eq. (A2.3) can be exactly expressed through the confluent hypergeometric functions. Matching the solution (A2.1) with the evanescent wave (A1.6) can be performed only if

*a*< 0. The corresponding asymptotic evanescent solution of Eq. (A2.3), which is valid in the classically forbidden (barrier) region,

*bν*

_{2}-

*a*> 0 , and for -1/

*a*≪

*ν*, is

*ν*≪∣

*a*/

*b*∣

^{1/2}with Eq. (A1.6) shows that these two solutions can be matched if and only if

*ν*>

*ν*=(

_{turn}*a*/

*b*)

^{1/2}, Eq. (A2.1) should contain only the waves that are outgoing along the axis

*ρ*. In order to satisfy this condition, one has to consider a linear combination of the solution defined by Eq. (A2.6) and a solution which is similar to Eq. (A2.6) but exponentially grows along the axis

*ρ*(i.e. has a positive sign in front of ∫

^{ν}

_{0}…). The latter solution is exponentially small in the region close to the NT taper, and does not affect satisfaction of the boundary condition. Direct application of the matching rules [18] near the turning point

*ν*=

*ν*

_{turn}yields the following asymptotic form of the solution in the classically allowed region

*ν*>

*ν*

_{turn}:

## Appendix 3.

## Radiation loss of a biconical taper *r*^{-}(*z*)

*z*is

*α*(

*z*) is determined from the equation:

## References

1. | L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature , |

2. | G. Brambilla, V. Finazzi, and D. J. Richardson, “Ultra-low-loss optical fiber nanotapers,” Opt. Express , |

3. | S. G. Leon-Saval, T. A. Birks, W. J. Wadsworth, and P. St.J. Russell, “Guidance properties of low-contrast photonic bandgap fibres,” Opt. Express , |

4. | K. J. Vahala, “Optical microcavities,” Nature , |

5. | M. Sumetsky, “Optical fiber microcoil resonator,” Opt. Express , |

6. | M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, and D. J. DiGiovanni, “The Microfiber Loop Resonator: Theory, Experiment, and Application,” IEEE J. Lightwave Technol. , |

7. | M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, and J. W. Nicholson, “Probing optical microfiber nonuniformities at nanoscale,” Opt. Lett. |

8. | X. Jiang, Q. Yang, G. Vienne, Y. Li, L. Tong, J. Zhang, and L. Hu, “Demonstration of microfiber knot laser,” Appl. Phys. Lett. |

9. | A. W. Snyder and J. D. Love, |

10. | W. L. Kath and G A. Kriegsmann, “Optical tunnelling: radiation losses in bent fibre-optic waveguides,” IMA J. Appl. Math. |

11. | M. Sumetsky, “How thin can a microfiber be and still guide light?,” Opt. Lett. |

12. | M. Sumetsky, “How thin can a microfiber be and still guide light? Errata,” Opt. Lett. |

13. | M. Sumetsky, “Optics of tunneling from adiabatic nanotapers,” Opt. Lett. |

14. | L. Tong, L. Hu, J. Zhang, J. Qiu, Q. Yang, J. Lou, Y. Shen, J. He, and Z. Ye, “Photonic nanowires directly drawn from bulk glasses,” Opt. Express |

15. | A.D. Capobianco, M. Midrio, C.G. Someda, and S. Curtarolo, “Lossless tapers, Gaussian beams, free-space modes: Standing waves versus through-flowing waves,” Opt. Quantum Electron. |

16. | V. M. Babič and V. S. Buldyrev, |

17. | M. Sumetskii, “Tunnel effect at the boundary of state stability: optical cavity and highly excited hydrogen atom in a magnetic field,” Sov. Phys. JETP , |

18. | J. Heading, |

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

**OCIS Codes**

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

(190.0190) Nonlinear optics : Nonlinear optics

(230.7370) Optical devices : Waveguides

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: December 13, 2006

Revised Manuscript: February 4, 2007

Manuscript Accepted: February 5, 2007

Published: February 19, 2007

**Citation**

M. Sumetsky, "Radiation loss of a nanotaper: Singular Gaussian beam model," Opt. Express **15**, 1480-1490 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-4-1480

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### References

- L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003). [CrossRef] [PubMed]
- G. Brambilla, V. Finazzi, and D. J. Richardson, "Ultra-low-loss optical fiber nanotapers," Opt. Express 12, 2258-2263 (2004). [CrossRef] [PubMed]
- S. G. Leon-Saval, T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, "Guidance properties of low-contrast photonic bandgap fibres," Opt. Express 12, 2864-2869 (2004). [CrossRef] [PubMed]
- K. J. Vahala, "Optical microcavities," Nature 424, 839-846 (2003). [CrossRef] [PubMed]
- M. Sumetsky, "Optical fiber microcoil resonator," Opt. Express 12, 2303-2316 (2004). [CrossRef] [PubMed]
- M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, and D. J. DiGiovanni, "The Microfiber Loop Resonator: Theory, Experiment, and Application," J. Lightwave Technol. 24, 242-250 (2006). [CrossRef]
- M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, and J. W. Nicholson, "Probing optical microfiber nonuniformities at nanoscale," Opt. Lett. 31, 2393-2395 (2006). [CrossRef] [PubMed]
- X. Jiang, Q. Yang, G. Vienne, Y. Li, L. Tong, J. Zhang, L. Hu, "Demonstration of microfiber knot laser," Appl. Phys. Lett. 89, Art. 143513 (2006) [CrossRef]
- A. W. Snyder and J. D. Love, Optical waveguide theory (Chapman and Hall, London, 1983).
- W. L. Kath and G A. Kriegsmann, "Optical tunnelling: radiation losses in bent fibre-optic waveguides," IMA J. Appl. Math. 41, 85-103 (1988). [CrossRef]
- M. Sumetsky, "How thin can a microfiber be and still guide light?" Opt. Lett. 31, 870-872 (2006). [CrossRef] [PubMed]
- M. Sumetsky, "How thin can a microfiber be and still guide light? Errata," Opt. Lett. 31, 3577-3578 (2006). [CrossRef]
- M. Sumetsky, "Optics of tunneling from adiabatic nanotapers," Opt. Lett. 31, 3420-3422 (2006). [CrossRef] [PubMed]
- L. Tong, L. Hu, J. Zhang, J. Qiu, Q. Yang, J. Lou, Y. Shen, J. He, and Z. Ye, "Photonic nanowires directly drawn from bulk glasses," Opt. Express 14, 82-87 (2006). [CrossRef] [PubMed]
- A. D. Capobianco, M. Midrio, C. G. Someda, and S. Curtarolo, "Lossless tapers, Gaussian beams, free-space modes: Standing waves versus through-flowing waves," Opt. Quantum Electron. 32, 1161-1173 (2000). [CrossRef]
- V. M. Babič and V. S. Buldyrev, Short-wavelength diffraction theory (Springer, Berlin, 1991).
- M. Sumetskii, "Tunnel effect at the boundary of state stability: optical cavity and highly excited hydrogen atom in a magnetic field," Sov. Phys. JETP, 67, 49-59 (1988).
- J. Heading, Phase Integral Methods (New York, Wiley, 1962).
- L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 1965, 2nd edition).

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