## The phase shift of light scattering at sub-wavelength dielectric structures |

Optics Express, Vol. 21, Issue 5, pp. 5957-5967 (2013)

http://dx.doi.org/10.1364/OE.21.005957

Acrobat PDF (1049 KB)

### Abstract

We present a new theoretical analysis for the light scattering at sub-wavelength dielectric structures. This analysis can provide new intuitive insights into the phase shift of the scattered light that cannot be obtained from the existing approaches. Unlike the traditional analytical (e.g. Mie formalism) and numerical (e.g. FDTD) approaches, which simulate light scattering by rigorously matching electromagnetic fields at boundaries, we consider sub-wavelength dielectric structures as leaky resonators and evaluate the light scattering as a coupling process between incident light and leaky modes of the structure. Our analysis indicates that the light scattering is fundamentally dictated by the eigenvalue of the leaky modes. It indicates that the upper limit for the scattering efficiency of a cylindrical cylinder in free space is *4n*, where *n* is the refractive index. It also indicates that the phase shift of the forward scattered light at dielectric structures can only cover half of the phase space [0, 2π], but backward scattering can provide a full phase coverage.

© 2013 OSA

## 1. Introduction

4. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature **391**(6668), 667–669 (1998). [CrossRef]

5. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. **8**(9), 758–762 (2009). [CrossRef] [PubMed]

6. S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. **101**(4), 047401 (2008). [CrossRef] [PubMed]

7. S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nat. Photonics **3**(7), 388–394 (2009). [CrossRef]

9. Z. W. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science **315**(5819), 1686–1686 (2007). [CrossRef] [PubMed]

10. A. Alù and N. Engheta, “Cloaking a Sensor,” Phys. Rev. Lett. **102**(23), 233901 (2009). [CrossRef] [PubMed]

11. W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics **1**(4), 224–227 (2007). [CrossRef]

12. Y. M. Liu and X. Zhang, “Metamaterials: a new frontier of science and technology,” Chem. Soc. Rev. **40**(5), 2494–2507 (2011). [CrossRef] [PubMed]

13. L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. **8**(8), 643–647 (2009). [CrossRef] [PubMed]

16. O. L. Muskens, S. L. Diedenhofen, B. C. Kaas, R. E. Algra, E. P. A. M. Bakkers, J. G. Rivas, and A. Lagendijk, “Large photonic strength of highly tunable resonant nanowire materials,” Nano Lett. **9**(3), 930–934 (2009). [CrossRef] [PubMed]

13. L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. **8**(8), 643–647 (2009). [CrossRef] [PubMed]

15. Y. Yu, V. E. Ferry, A. P. Alivisatos, and L. Y. Cao, “Dielectric core-shell optical antennas for strong solar absorption enhancement,” Nano Lett. **12**(7), 3674–3681 (2012). [CrossRef] [PubMed]

*4n*, where

*n*is the refractive index. It also indicates that the phase shift in forward scattered light at dielectric structures can only cover half of the phase space [0, 2π], but backward scattering can provide a full phase coverage.

## 2. Calculations of leaky modes

### 2.1 One-dimensional (1D) cylinder

*r, ϕ, z*) as illustrated in Fig. 1 , the electric field in the z direction

*E*

_{z}can be generally written as, where

*C*is a constant to be determined,

*β*is the propagation constant in the z direction,

*J*

_{m}() and

*H*

_{m}() are the

*m*th order of Bessel function of the first kind and Hankel function of the first kind, respectively,

*r*

_{0}is the radius of the cylinder,

*κ*and

*γ*are wave vectors in the transverse direction inside and outside the cylinder as

*n*is the refractive index of the materials, and

*k*is the wave vector in the environment that is assumed to be free space. Equations (1) and (2) can be re-written in terms of odd and even modes,

*H*

_{z}can be written as

*E*

_{ϕ}and

*H*

_{ϕ}can be derived from

*E*

_{z}and

*H*

_{z}as where

*α*=

*κ*(

*γ*) for the field inside (outside) the cylinder,

*ω*is the frequency,

*μ*and

*ε*are the permeability and permittivity, respectively. By matching the boundary conditions at the cylinder/environment interface, we may have We only consider the transverse mode that has

*β*= 0 (and thus

*κ*=

*nk*,

*γ*=

*k*), i.e. no propagation along the cylinder. For transverse electric (TE) modes,

*C*= 0,For transverse magnetic (TM) modes,

*C’*= 0

### 2.2 Zero-dimensional (0D) sphere

*r, θ, ϕ*), (Fig. 2 ) and can be written as From

*D*and

*D’*are constant coefficients, and refer to

*odd*and

*even*modes, and

*j*

_{m}() and

*h*

_{m}() are the

*m*th order of spherical Bessel function of the first kind and spherical Hankel function of the first kind, respectively.

*θ*and

*ϕ*, and their expressions can be found out in typical textbooks on light scattering [2, 17]. Because both

*Riccati-Bessel*functions

*nkr*

_{0}(

*nkr*

_{0}=

*N*

_{real}-

*N*

_{imag}

*i*). These complex values are eigenvalues of the leaky modes. We have previously calculated and tabulated the eigenvalue for typical leaky modes [14

14. Y. Yu and L. Y. Cao, “Coupled leaky mode theory for light absorption in 2D, 1D, and 0D semiconductor nanostructures,” Opt. Express **20**(13), 13847–13856 (2012). [CrossRef] [PubMed]

*N*

_{real}indicates the condition for the resonance with leaky modes, and the imaginary part

*N*

_{imag}refers to the radiative leakage of the electromagnetic energy stored in the leaky mode. For materials without intrinsic absorption loss, this imaginary part indicates spectral width of the leaky mode resonance.

## 3. Coupled leaky mode theory for the light scattering at 1D and 0D dielectric structures

*b*

_{ml}in Eq. (32). The leaky modes with different mode number

*m*are orthogonal to each other and thus may interact with incident light independently. Therefore, the scattering efficiency from a structure with multiple leaky modes

*Q*

_{sca}can be obtained by simply adding up the contribution from the leaky modes with different

*m*as

*b*

_{m}defined here has an opposite sign as the scattering coefficient defined in Mie Theory [2]), where

*Re*means the real part. The scattering efficiency

*b*

_{m}includes contributions from a group of leaky modes with the same mode number

*m*but different order number

*l*(we refer them as group

*m*leaky modes for the convenience of discussion) as

*ω*

_{ml}is relatively close to the incident frequency

*ω*(- π <(

*ω*

_{ml}–

*ω*).

*nr*

_{0}/c< π), as illustrated in Fig. 4 . This is because the coupling efficiency of incident light to a leaky mode exponentially decreases with the difference between the incident frequency and the eigenfrequency (

*ω*

_{ml}–

*ω*) [14

14. Y. Yu and L. Y. Cao, “Coupled leaky mode theory for light absorption in 2D, 1D, and 0D semiconductor nanostructures,” Opt. Express **20**(13), 13847–13856 (2012). [CrossRef] [PubMed]

*b*

_{m}of the group

*m*with an easier way. For an arbitrary incident frequency

*ω*, we consider

*b*

_{m}equal to one single-mode coefficient

*b*

_{ml}if the incident frequency

*ω*is very close to the eigenfrequency of a leaky mode TM

_{ml}(typically, - 0.4 <(

*ω*

_{ml}–

*ω*).

*nr*

_{0}/c < 0.4). In this on-resonance case, we ignore the contribution from all other leaky modes in the group

*m*. We consider

*b*

_{m}equal to (

*b*

_{ml}+

*b*

_{m,l + 1})/2 if

*ω*is not very close to any eigenfrequency, where the subscripts

*ml*and

*m,l + 1*refer to the two leaky modes nearest to

*ω*. Figure 5(a) -5(b) indicates that the scattering coefficient

*b*

_{m}constructed from

*b*

_{ml}nicely matches the scattering coefficient calculated from Mie formalism as

*Q*

_{sca}calculated using our model shows excellent consistence with the result calculated using the Mie formalism (Fig. 5(c)). Additionally, we calculated the scattering coefficient for 0D spheres, and again found it matching the results calculated using the Mie formalism (Fig. 6(a) -6(b)).

*4n*. From the above analysis, we can find that the collective contribution to the scattering efficiency from all the modes with the same mode number

*m*and different order number

*l*is −2

*Re*(

*b*

_{ml})/

*kr*

_{0}or -

*Re*(

*b*

_{ml}+

*b*

_{m,l + 1})/

*kr*

_{0}, which is no larger than 2/

*kr*

_{0}. The overall scattering efficiency

*Q*

_{sca}is just a simple add-up of the contribution from the modes with different mode number

*m*. As illustrated in Fig. 4, for an arbitrary incident frequency

*ω*, the mode number

*m*that can be involved in the light scattering bear a simple relationship with the normalized parameter of the scattering system,

*m*≤

*nkr*

_{0}. By considering the dual degeneracy of typical leaky modes (even and odd modes are degenerate), we can find out the number of leaky modes that can be involved into light scattering is no larger than 2

*nkr*

_{0}. Therefore, the scattering efficiency

*Q*

_{sca}is no larger than

*4n*.

*ω*can be written as

*m*> 0. For the scattered light at far field (

*kr*>>1), we can use the asymptotic approximation of the Hankel function (

*e*indicates the optical path of the light passing without scattering (for forward scattering) or reflected from a perfect mirror (for backward scattering),

^{ikr}*B*is a real number, and

*θ*

_{S}- π/4 is the phase shift in the scattered light.

*B*and

*θ*

_{S}are determined by the sum of scattering coefficients as

*b*

_{ml}can never be larger than 0. Therefore, the real part of the sum

*θ*

_{S}in a range of [π/2, 3π/2]. But backward scattering can provide the phase shifts of [0, 2π] due to the involvement of the term (−1)

*. While this notion is illustrated with cylinders, the underlying rationale can generally apply to other one-dimensional (1D) structures with non-circular cross-section or zero-dimensional (0D) dielectric structures. Therefore, this conclusion on the phase coverage of backward and forward scatterings generally holds for 1D and 0D dielectric structures with arbitrary shapes.*

^{m}## 4. Conclusions

## Acknowledgments

## References and links

1. | P. W. Barber and R. K. Chang, eds., |

2. | C. F. Bohren and D. R. Huffman, |

3. | A. Taflove and S. C. Hagness, |

4. | T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature |

5. | N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. |

6. | S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. |

7. | S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nat. Photonics |

8. | N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science |

9. | Z. W. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science |

10. | A. Alù and N. Engheta, “Cloaking a Sensor,” Phys. Rev. Lett. |

11. | W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics |

12. | Y. M. Liu and X. Zhang, “Metamaterials: a new frontier of science and technology,” Chem. Soc. Rev. |

13. | L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. |

14. | Y. Yu and L. Y. Cao, “Coupled leaky mode theory for light absorption in 2D, 1D, and 0D semiconductor nanostructures,” Opt. Express |

15. | Y. Yu, V. E. Ferry, A. P. Alivisatos, and L. Y. Cao, “Dielectric core-shell optical antennas for strong solar absorption enhancement,” Nano Lett. |

16. | O. L. Muskens, S. L. Diedenhofen, B. C. Kaas, R. E. Algra, E. P. A. M. Bakkers, J. G. Rivas, and A. Lagendijk, “Large photonic strength of highly tunable resonant nanowire materials,” Nano Lett. |

17. | J. D. Jackson, |

18. | R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A |

19. | Z. C. Ruan and S. H. Fan, “Temporal coupled-mode theory for Fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C |

20. | Z. Ruan and S. Fan, “Super-scattering of light from subwavelength nano-structures,” Phys. Rev. Lett. |

21. | H. A. Haus, |

22. | A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. |

**OCIS Codes**

(260.5740) Physical optics : Resonance

(160.4236) Materials : Nanomaterials

(290.5825) Scattering : Scattering theory

**ToC Category:**

Scattering

**History**

Original Manuscript: November 22, 2012

Revised Manuscript: February 18, 2013

Manuscript Accepted: February 21, 2013

Published: March 4, 2013

**Citation**

Yiling Yu and Linyou Cao, "The phase shift of light scattering at sub-wavelength dielectric structures," Opt. Express **21**, 5957-5967 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-5957

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### References

- P. W. Barber and R. K. Chang, eds., Optical Effects Associated with Small Particles (World Scientific, 1988).
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).
- T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998). [CrossRef]
- N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater.8(9), 758–762 (2009). [CrossRef] [PubMed]
- S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett.101(4), 047401 (2008). [CrossRef] [PubMed]
- S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nat. Photonics3(7), 388–394 (2009). [CrossRef]
- N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science308(5721), 534–537 (2005). [CrossRef] [PubMed]
- Z. W. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315(5819), 1686–1686 (2007). [CrossRef] [PubMed]
- A. Alù and N. Engheta, “Cloaking a Sensor,” Phys. Rev. Lett.102(23), 233901 (2009). [CrossRef] [PubMed]
- W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1(4), 224–227 (2007). [CrossRef]
- Y. M. Liu and X. Zhang, “Metamaterials: a new frontier of science and technology,” Chem. Soc. Rev.40(5), 2494–2507 (2011). [CrossRef] [PubMed]
- L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater.8(8), 643–647 (2009). [CrossRef] [PubMed]
- Y. Yu and L. Y. Cao, “Coupled leaky mode theory for light absorption in 2D, 1D, and 0D semiconductor nanostructures,” Opt. Express20(13), 13847–13856 (2012). [CrossRef] [PubMed]
- Y. Yu, V. E. Ferry, A. P. Alivisatos, and L. Y. Cao, “Dielectric core-shell optical antennas for strong solar absorption enhancement,” Nano Lett.12(7), 3674–3681 (2012). [CrossRef] [PubMed]
- O. L. Muskens, S. L. Diedenhofen, B. C. Kaas, R. E. Algra, E. P. A. M. Bakkers, J. G. Rivas, and A. Lagendijk, “Large photonic strength of highly tunable resonant nanowire materials,” Nano Lett.9(3), 930–934 (2009). [CrossRef] [PubMed]
- J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1998).
- R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A75(5), 053801 (2007). [CrossRef]
- Z. C. Ruan and S. H. Fan, “Temporal coupled-mode theory for Fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C114(16), 7324–7329 (2010). [CrossRef]
- Z. Ruan and S. Fan, “Super-scattering of light from subwavelength nano-structures,” Phys. Rev. Lett.105(1), 013901 (2010). [CrossRef] [PubMed]
- H. A. Haus, Wave and Fields in Optoelectronics (Prentice-Hall, 1984).
- A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett.36, 321–322 (2000).

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