## Cylinder light concentrator and absorber: theoretical description |

Optics Express, Vol. 18, Issue 16, pp. 16646-16662 (2010)

http://dx.doi.org/10.1364/OE.18.016646

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### Abstract

We present a detailed theoretical description of a broadband omnidirectional light concentrator and absorber with cylinder geometry. The proposed optical “trap” captures nearly all the incident light within its geometric cross-section, leading to a broad range of possible applications – from solar energy harvesting to thermal light emitters and optoelectronic components. We have demonstrated that an approximate lamellar black-hole with a moderate number of homogeneous layers, while giving the desired ray-optical performance, can provide absorption efficiencies comparable to those of ideal devices with a smooth gradient in index.

© 2010 OSA

## 1. Introduction

1. V. M. Shalaev, W. S. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. **30**(24), 3356–3358 (2005). [CrossRef]

8. E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. **95**(4), 041106 (2009). [CrossRef]

8. E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. **95**(4), 041106 (2009). [CrossRef]

8. E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. **95**(4), 041106 (2009). [CrossRef]

## 2. The system

**95**(4), 041106 (2009). [CrossRef]

_{.}

## 3. Ray-optical description

*m*is the (conserved) angular momentum,

*p*equal to −1, 1, 2, and 3 respectively; the trivial case of the beam propagating in free-space (

## 4. The wave optical description

*ω*. We start with the monochromatic Maxwell’s equations

*c*,

*e*and

*h*, which are further on collectively denoted as

*f*.

### 4.1 Scalar wave equations and separation of variables

*L*is given by

*ε*, while it vanishes for any TE-wave (

*r*,

*ϕ*,

*z*), linked to the Cartesian coordinates (

*x*,

*y*,

*z*) through

### 4.2 TE and TM solutions for ε ( r ) = const

### 4.3 TE and TM solutions for the ‘quadratic decay’ of permittivity, r 2 ε ( r ) = const

### 4.4 Boundary conditions and the mode matching

*f*describes the scattered field (i.e. does not include the incident wave), it should obey the Sommerfeld radiation condition,

### 4.5 Incident light: the Gaussian beams and plane wave

*not*an exact solution of Maxwell’s equations in the free space, but instead corresponds to the paraxial approximation [13]. To avoid the approximate nature of this approach and resulting inaccuracy, we consider that the field, while following the Gaussian profile at the waist cross section, is in fact the exact solution of the Helmholtz wave equation with the given direction of propagation [14

14. G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. **69**(4), 575 (1979). [CrossRef]

*y*-plane waveswhere the amplitudes

## 5. Layered Systems

### 5.1 Cascading the cylindrical layers

*i*

^{th}layer can be represented as a sum of angular harmonics,

*m*

^{th}mode, similar to that shown in (14) presented in a more general form,where for the simplicity of notation (and from now on in this section) we suppress the index

*m*. To satisfy the condition (22) in the core layer we put

16. V. Kildishev, U. K. Chettiar, Z. Jacob, V. M. Shalaev, and E. E. Narimanov, “Materializing a binary hyperlens design,” Appl. Phys. Lett. **94**(7), 071102 (2009). [CrossRef]

*l*boundary conditions has the form:

*m*

^{th}mode of the field by using (32); the solution to the entire problem in the

*i*

^{th}layer is given as a sum of the angular momentum modes,

### 5.2 Scattering and absorption efficiencies

*total field*

*m*

^{th}mode of the total field is given by,orso that the total field (40) is decomposed into a plane wave illumination sourceand the scattered fieldwhere coefficients

*W*is the total power transmitted by the incident wave through

### 5.3 The ideal electromagnetic black hole

*p*= 2) electromagnetic black hole, see (1), (2), (3). It is a three layered system with a gradient-index shell and absorbing core, for which the formalism of Section 5.1 can be applied to calculate the exact full-wave solution. Fig. 3 shows an example simulation of an ideal black hole with

*A*is obtained from the boundary conditions,

*X*, we use the identity

### 5.4 The lamellar electromagnetic black hole

*l.*The device is illuminated by a plane wave with free-space wavelength

*l*equal to 3, 5, 9, and 17. Scattering and absorption efficiencies are given in (49)-(50), the latter,

## 6. Semiclassical description

*p*= 2) electromagnetic black hole, introduced in Section 2.

*m*has physical meaning of the product of the (fixed) wavenumber and the impact parameter – see Fig. 2 – the sum over

*m*to

*,*

*r*

_{s}= 20 µm, and

## 7. Summary

## Acknowledgments

## References and links

1. | V. M. Shalaev, W. S. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. |

2. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

3. | Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express |

4. | A. Salandrino and N. Engheta, “Far-Field Subdiffraction Optical Microscopy Using Metamaterial Crystals: Theory and Simulations,” Phys. Rev. B |

5. | N. M. Litchinitser and V. M. Shalaev, “Metamaterials: transforming theory into reality,” J. Opt. Soc. Am. B |

6. | N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. |

7. | T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics |

8. | E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. |

9. | L. D. Landau, and E. M. Lifshitz, Mechanics, Pergamon Press, Oxford, (1976). |

10. | R. K. Luneburg, Mathematical Theory of Optics, University of California Press, Berkeley, 1964, p. 12. |

11. | E. J. Post, Formal Structure of Electromagnetics: General Covariance and Electromagnetics, 1962, p. 152. |

12. | S. Gradshteyn, and I. M. Ryzhik, Tables of integrals, series and products, Academic Press, New York, CD-ROM Edition, 1994, Eq. (8).511.4. |

13. | M. Born, and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, New York, 1999). |

14. | G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. |

15. | M. Abramowitz, and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York, Dover, (1972). |

16. | V. Kildishev, U. K. Chettiar, Z. Jacob, V. M. Shalaev, and E. E. Narimanov, “Materializing a binary hyperlens design,” Appl. Phys. Lett. |

17. | S. D. Conte, and C. W. Boor, “Elementary Numerical Analysis: An Algorithmic Approach, 3rd edition,” 1980. |

18. | H. Hulst, Light scattering by small particles, Dover, 1981, p. 309. |

**OCIS Codes**

(260.0260) Physical optics : Physical optics

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: March 30, 2010

Revised Manuscript: June 19, 2010

Manuscript Accepted: June 29, 2010

Published: July 23, 2010

**Citation**

Alexander V. Kildishev, Ludmila J. Prokopeva, and Evgenii E. Narimanov, "Cylinder light concentrator and absorber: theoretical description," Opt. Express **18**, 16646-16662 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-16646

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### References

- V. M. Shalaev, W. S. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30(24), 3356–3358 (2005). [CrossRef]
- J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]
- Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006). [CrossRef] [PubMed]
- A. Salandrino and N. Engheta, “Far-Field Subdiffraction Optical Microscopy Using Metamaterial Crystals: Theory and Simulations,” Phys. Rev. B 74(7), 075103 (2006). [CrossRef]
- N. M. Litchinitser and V. M. Shalaev, “Metamaterials: transforming theory into reality,” J. Opt. Soc. Am. B 26(12), B161–B169 (2009). [CrossRef]
- N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008). [CrossRef] [PubMed]
- T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics 2(5), 299–301 (2008). [CrossRef]
- E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009). [CrossRef]
- L. D. Landau, and E. M. Lifshitz, Mechanics, Pergamon Press, Oxford, (1976).
- R. K. Luneburg, Mathematical Theory of Optics, University of California Press, Berkeley, 1964, p. 12.
- E. J. Post, Formal Structure of Electromagnetics: General Covariance and Electromagnetics, 1962, p. 152.
- S. Gradshteyn, and I. M. Ryzhik, Tables of integrals, series and products, Academic Press, New York, CD-ROM Edition, 1994, Eq. (8).511.4.
- M. Born, and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, New York, 1999).
- G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69(4), 575 (1979). [CrossRef]
- M. Abramowitz, and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York, Dover, (1972).
- V. Kildishev, U. K. Chettiar, Z. Jacob, V. M. Shalaev, and E. E. Narimanov, “Materializing a binary hyperlens design,” Appl. Phys. Lett. 94(7), 071102 (2009). [CrossRef]
- S. D. Conte, and C. W. Boor, “Elementary Numerical Analysis: An Algorithmic Approach, 3rd edition,” 1980.
- H. Hulst, Light scattering by small particles, Dover, 1981, p. 309.

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