## Concentrating evanescent waves: Systematic analyses of properties of the needle beam in three-medium dielectric cylindrical waveguide |

Optics Express, Vol. 19, Issue 20, pp. 18795-18806 (2011)

http://dx.doi.org/10.1364/OE.19.018795

Acrobat PDF (1682 KB)

### Abstract

Needle beam is a guided beam with nanoscale beam size and significant power propagating in core area of a three-layer dielectric waveguide. Systematical numerical analyses of properties of the needle beam are presented. Properties of the fundamental mode of the needle beam, including field distribution, power distribution, and power concentration, are calculated for different waveguide parameters. It is shown that there is an optimum value of normalized frequency for maximum power concentration. Concentrated power is higher if the refractive index difference between the core and the middle layer is higher.

© 2011 OSA

## 1. Introduction

1. V. Bondarenko and Y. Zhao, “Needle beam”: Beyond-diffraction-limit concentration of field and transmitted power in dielectric waveguide,” Appl. Phys. Lett. **89**(14), 141103 (2006). [CrossRef]

2. V. Bondarenko and Y. Zhao, “Addendum: The 'needle beam': Beyond-diffraction-limit concentration of field and transmitted power in dielectric waveguide,” Appl. Phys. Lett. **91**(8), 089903 (2007). [CrossRef]

3. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science **297**(5582), 820–822 (2002). [CrossRef] [PubMed]

8. S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B **67**(20), 205402 (2003). [CrossRef]

9. H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. **66**(7-8), 163–182 (1944). [CrossRef]

3. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science **297**(5582), 820–822 (2002). [CrossRef] [PubMed]

7. K. Y. Kim, Y. K. Cho, H. S. Tae, and J. H. Lee, “Light transmission along dispersive plasmonic gap and its subwavelength guidance characteristics,” Opt. Express **14**(1), 320–330 (2006). [CrossRef] [PubMed]

10. J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. **22**(7), 475–477 (1997). [CrossRef] [PubMed]

8. S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B **67**(20), 205402 (2003). [CrossRef]

11. Q. F. Xu, V. R. Almeida, R. R. Panepucci, and M. Lipson, “Experimental demonstration of guiding and confining light in nanometer-size low-refractive-index material,” Opt. Lett. **29**(14), 1626–1628 (2004). [CrossRef] [PubMed]

12. V. R. Almeida, Q. F. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. **29**(11), 1209–1211 (2004). [CrossRef] [PubMed]

13. L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature **426**(6968), 816–819 (2003). [CrossRef] [PubMed]

1. V. Bondarenko and Y. Zhao, “Needle beam”: Beyond-diffraction-limit concentration of field and transmitted power in dielectric waveguide,” Appl. Phys. Lett. **89**(14), 141103 (2006). [CrossRef]

2. V. Bondarenko and Y. Zhao, “Addendum: The 'needle beam': Beyond-diffraction-limit concentration of field and transmitted power in dielectric waveguide,” Appl. Phys. Lett. **91**(8), 089903 (2007). [CrossRef]

_{01}and TM

_{01}modes were considered in [1

1. V. Bondarenko and Y. Zhao, “Needle beam”: Beyond-diffraction-limit concentration of field and transmitted power in dielectric waveguide,” Appl. Phys. Lett. **89**(14), 141103 (2006). [CrossRef]

2. V. Bondarenko and Y. Zhao, “Addendum: The 'needle beam': Beyond-diffraction-limit concentration of field and transmitted power in dielectric waveguide,” Appl. Phys. Lett. **91**(8), 089903 (2007). [CrossRef]

## 2. Governing Equations and Mode Conditions

_{2>}n

_{1}and n

_{2}>n

_{3}.The relationship between n

_{1}and n

_{3}is not fixed. Since the waveguide is cylindrically symmetric, we use the cylindrical coordinate system in our analyses. The field components are E

_{r}, E

_{φ}, E

_{z}, H

_{r}, H

_{φ}, H

_{z}. Here, z is the propagation direction. It is well known that wave equations for the z components are [14]where ∇

^{2}is the Laplacian operator,

*k = 2π /*λ (λ is the wavelength). Solutions to Eq. (1) take the formwhere β is the propagation constant,

*l*= 0, 1, 2, 3, …,

*w*is the eigenmode frequency, and

*Ψ(r)*has different forms in each of three layers in the waveguide: In Eqs. (3) to (5), p

^{2}= β

^{2}-n

_{1}

^{2}k

_{0}

^{2}, h

^{2}= n

_{2}

^{2}k

_{0}

^{2}-β

^{2}, q

^{2}= β

^{2}-n

_{3}

^{2}k

_{0}

^{2},

*I*and

_{l}(x), J_{l}(x), Y_{l}(x),*K*are the Modified Bessel function of the first kind, Bessel function of the first kind, Bessel functions of the second kind, and Modified Bessel function of the second kind, respectively, of order

_{l}(x)*l*, and

*a*and

_{1}, a_{2}, b, c, d_{1},*d*are arbitrary constants.

_{2}*a*, we consider boundary conditions at r = 0 and r→∞. Fields have to be finite at

_{1}, a_{2}, b, c, d_{1}, d_{2}*r*= 0 and r→∞. Therefore,

*a*and

_{2}*d*must be zero. Thus, the fields in the core medium (Region I), the middle medium (Region II), and the outside medium (Region III) are described by I, J + Y, K, respectively, and we call the wave I-(J + Y)-K profile. Equations (3-5) can now be written as Here we have used a different set of notations for arbitrary constants.

_{1}_{r}, E

_{φ}, H

_{r}, and H

_{φ}. In the core area (r < r

_{1}), they are

_{r}, E

_{φ}, H

_{r}, and H

_{φ}in the middle layer (r

_{1}<r <r

_{2}) are

_{2}), they are

_{1}= n

_{1}

^{2}, ε

_{2}= n

_{2}

^{2}, ε

_{3}= n

_{3}

^{2}, and

_{µ}is the permeability of the medium. All the eigenmodes propagating in the three-medium waveguide must satisfy the boundary conditions that E

_{z}, H

_{z}, E

_{φ}, and H

_{φ}should be continuous at the I/II boundary (r = r

_{1}) and II/III boundary (r = r

_{2}). This leads to the corresponding eight equations. The eight equations can be expressed by the following matrix form

_{i}, B

_{i}, C

_{i}, D

_{i}(i = 1, 2) in Eq. (20), the determinant of the above 8 × 8 matrix must vanish. We will use this requirement to obtain mode conditions. To solve the mode conditions, we use the parameters V, η, θ, defined as V

^{2}= k

_{0}

^{2}r

_{2}

^{2}(n

_{2}

^{2}-n

_{3}

^{2}), η

^{2}= (n

_{2}

^{2}- n

_{1}

^{2})/ (n

_{2}

^{2}- n

_{3}

^{2}) and θ = r

_{1}/ r

_{2}[1

**89**(14), 141103 (2006). [CrossRef]

**91**(8), 089903 (2007). [CrossRef]

_{2}, n

_{2}, and n

_{3}. θ is the ratio between r

_{1}and r

_{2}. If θ = 0, we have a conventional two-medium step-index optical fiber. Letting the determinant of M be zero under fixed values of V, η, and θ, we can calculate different hr

_{2}values, which correspond to different eigenmodes. Then we can calculate other characteristic constants of these modes, such as cut off frequency and normalized propagation constant. We use Matlab software in our calculations. It should be noted that Eq. (20) can be simplified analytically first and then solved numerically to obtain the similar results of mode conditions [15

15. H. Ito, K. Sakaki, T. Nakata, W. Jhe, and M. Ohtsu, “Optical-Potential for Atom Guidance in a Cylindrical-Core Hollow-Fiber,” Opt. Commun. **115**(1-2), 57–64 (1995). [CrossRef]

_{r}, E

_{φ}, E

_{z}, H

_{r}, H

_{φ}, H

_{z}components. The eigenvalues resulting from matrix equation (Eq. (20)) lead to the two classes of solutions corresponding to the conventionally designated EH or HE modes. When

*l*= 0, HE and EH modes become TE and TM modes, respectively.

*k*

_{0}and normalized frequency V with fixed waveguide parameters, θ = 0.729, n

_{1}= 1.00, n

_{2}= 3.48, and n

_{3}= 1.48. With n

_{1}= 1, we have a waveguide with a small hole in the core area. It appears from Fig. 2 that for V<2.5, only the fundamental HE

_{11}mode can propagate, which is the single-mode waveguide condition. It should be noted that the cut off frequency of different mode depends on the ratio of r

_{1}and r

_{2}, as well as n

_{1}, n

_{2}, and n

_{3}. This is different from two-medium step-index fiber, whose cut off frequency of different modes is decided by the core size and the refractive indices of the waveguide. In the following sections, we will use the HE

_{11}mode as an example and analyze its fields and power characteristics.

## 3. Field and Power Characteristics of HE_{11} Mode

_{1}in HE

_{11}mode). After obtained β/

*k*

_{0}of an eigenmode and set input signal, we can calculate normalized amplitudes of all fields. Then the Poynting vector and power can be numerically obtained.

_{11}mode, we set A

_{1}= 1. Then we solve for the values of A

_{2}, B

_{1}, C

_{1}, B

_{2}, C

_{2}, D

_{1}, and D

_{2}of different fields using Eq. (20). After the field functions (Eqs. (6) to (19)) are obtained, the power can be calculated. In the cylindrical coordinate system, the time-averaged Poynting vector along the waveguide is expressed by

_{core}), middle layer (P

_{mid}) and outside layer (P

_{out}) are given by

_{r}, E

_{φ}, E

_{z}, H

_{r}, H

_{φ}, and H

_{z}in the core medium are presented in Fig. 3a to Fig. 3f with different n

_{2}and other fixed waveguide parameters, θ = 0.729, n

_{1}= 1, and n

_{3}= 1.48. Again we consider a waveguide with a hole in the core area (n

_{1}= 1). It should be noted that there is a π/2 phase difference between two sets of fields (E

_{z}, E

_{φ}, and H

_{r}) and (H

_{z}, E

_{r}, and H

_{φ}) because the amplitude coefficients of E

_{z}, E

_{φ}, and H

_{r}are real numbers and those of H

_{z}, E

_{r}, and H

_{φ}are imaginary numbers.

_{2}is larger. Figure 5 shows different power percentage of HE

_{11}mode with different V and fixed values of θ, n

_{1}, n

_{2}and n

_{3}. Compared to TE

_{01}mode in Refs. 1

**89**(14), 141103 (2006). [CrossRef]

**91**(8), 089903 (2007). [CrossRef]

_{11}mode has similar behavior when V changes. It also shows that there is an optimal V for maximum P

_{1}. Here the optimal V is about 3.8, which corresponds to a maximum power percentage of 30% in core medium when θ = 0.729, n

_{1}= 1, n

_{2}= 3.48, and n

_{3}= 1.48.

_{2}. With fixed θ value in Fig. 6a, a decrease in V corresponds to the decrease of r

_{2}, and hence r

_{1}. Decreasing in r

_{1}causes the increase in power density, but decrease in total volume of core region. As a results, there is an optimal V for maximum power percentage in the core region, i.e., V (and, with fixed θ, hence r

_{1}) cannot be too large or too small for maximum power concentration. The optimal V value becomes larger when n

_{2}changes from 3.48 to 1.75.

_{2}leads to larger power percentage in the core medium.

## 4. Discussions

_{core}is larger for larger θ up to certain value. Note that θ is determined by the diameters of core and middle layer. Therefore, larger θ corresponds to relatively larger core size. If θ is too large, it is no longer a nanoscale effect.

_{2}) leads to higher power percentage in the core medium. This can be explained based on the fact that the needle beam is an effect of evanescent wave at the boundary between core and middle layers [1

**89**(14), 141103 (2006). [CrossRef]

**91**(8), 089903 (2007). [CrossRef]

_{11}. For high power concentration in a small area, we have also examined power distribution of other modes. Table 1 lists power percentage in core medium and middle medium for different eigenmodes of the waveguide with V = 20, θ = 0.729, n

_{1}= 1, n

_{2}= 3.48, and n

_{3}= 1.48. From Table 1, we can see that power percentage of TE and HE modes in core medium are larger than that of TM and EH modes. This is a typical result of boundary conditions of dielectric waveguides.

## 5. Conclusions

## References and links

1. | V. Bondarenko and Y. Zhao, “Needle beam”: Beyond-diffraction-limit concentration of field and transmitted power in dielectric waveguide,” Appl. Phys. Lett. |

2. | V. Bondarenko and Y. Zhao, “Addendum: The 'needle beam': Beyond-diffraction-limit concentration of field and transmitted power in dielectric waveguide,” Appl. Phys. Lett. |

3. | H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science |

4. | R. Gordon, “Angle-dependent optical transmission through a narrow slit in a thick metal film,” Phys. Rev. B |

5. | F. J. García-Vidal, L. Martin-Moreno, E. Moreno, L. K. S. Kumar, and R. Gordon, “Transmission of light through a single rectangular hole in a real metal,” Phys. Rev. B |

6. | H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. |

7. | K. Y. Kim, Y. K. Cho, H. S. Tae, and J. H. Lee, “Light transmission along dispersive plasmonic gap and its subwavelength guidance characteristics,” Opt. Express |

8. | S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B |

9. | H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. |

10. | J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. |

11. | Q. F. Xu, V. R. Almeida, R. R. Panepucci, and M. Lipson, “Experimental demonstration of guiding and confining light in nanometer-size low-refractive-index material,” Opt. Lett. |

12. | V. R. Almeida, Q. F. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. |

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

14. | A. Yariv, |

15. | H. Ito, K. Sakaki, T. Nakata, W. Jhe, and M. Ohtsu, “Optical-Potential for Atom Guidance in a Cylindrical-Core Hollow-Fiber,” Opt. Commun. |

**OCIS Codes**

(060.2270) Fiber optics and optical communications : Fiber characterization

(130.2790) Integrated optics : Guided waves

(230.7370) Optical devices : Waveguides

(240.0240) Optics at surfaces : Optics at surfaces

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: July 14, 2011

Revised Manuscript: August 26, 2011

Manuscript Accepted: August 29, 2011

Published: September 12, 2011

**Citation**

Fangli Qin and Yang Zhao, "Concentrating evanescent waves: Systematic analyses of properties of the needle beam in three-medium dielectric cylindrical waveguide," Opt. Express **19**, 18795-18806 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-20-18795

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### References

- V. Bondarenko and Y. Zhao, “Needle beam”: Beyond-diffraction-limit concentration of field and transmitted power in dielectric waveguide,” Appl. Phys. Lett.89(14), 141103 (2006). [CrossRef]
- V. Bondarenko and Y. Zhao, “Addendum: The 'needle beam': Beyond-diffraction-limit concentration of field and transmitted power in dielectric waveguide,” Appl. Phys. Lett.91(8), 089903 (2007). [CrossRef]
- H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science297(5582), 820–822 (2002). [CrossRef] [PubMed]
- R. Gordon, “Angle-dependent optical transmission through a narrow slit in a thick metal film,” Phys. Rev. B75(19), 193401 (2007). [CrossRef]
- F. J. García-Vidal, L. Martin-Moreno, E. Moreno, L. K. S. Kumar, and R. Gordon, “Transmission of light through a single rectangular hole in a real metal,” Phys. Rev. B74(15), 153411 (2006). [CrossRef]
- H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett.96(9), 097401 (2006). [CrossRef] [PubMed]
- K. Y. Kim, Y. K. Cho, H. S. Tae, and J. H. Lee, “Light transmission along dispersive plasmonic gap and its subwavelength guidance characteristics,” Opt. Express14(1), 320–330 (2006). [CrossRef] [PubMed]
- S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B67(20), 205402 (2003). [CrossRef]
- H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev.66(7-8), 163–182 (1944). [CrossRef]
- J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett.22(7), 475–477 (1997). [CrossRef] [PubMed]
- Q. F. Xu, V. R. Almeida, R. R. Panepucci, and M. Lipson, “Experimental demonstration of guiding and confining light in nanometer-size low-refractive-index material,” Opt. Lett.29(14), 1626–1628 (2004). [CrossRef] [PubMed]
- V. R. Almeida, Q. F. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett.29(11), 1209–1211 (2004). [CrossRef] [PubMed]
- L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature426(6968), 816–819 (2003). [CrossRef] [PubMed]
- A. Yariv, Optical Electronics in Modern Communications (Oxford University Press, New York; Oxford, 1997).
- H. Ito, K. Sakaki, T. Nakata, W. Jhe, and M. Ohtsu, “Optical-Potential for Atom Guidance in a Cylindrical-Core Hollow-Fiber,” Opt. Commun.115(1-2), 57–64 (1995). [CrossRef]

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