## Ultrathin broadband nearly perfect absorber with symmetrical coherent illumination |

Optics Express, Vol. 20, Issue 3, pp. 2246-2254 (2012)

http://dx.doi.org/10.1364/OE.20.002246

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### Abstract

As highlighted by recent articles [Phys. Rev. Lett. 105, 053901 (2010) and Science 331, 889-892 (2011)], the coherent control of narrowband perfect absorption in intrinsic silicon slab has attracted much attention. In this paper, we demonstrate that broadband coherent perfect absorber (CPA) can be achieved by heavily doping an ultrathin silicon film. Two distinct perfect absorption regimes are derived with extremely broad and moderately narrow bandwidth under symmetrical coherent illumination. The large enhancement of bandwidth may open up new avenues for broadband applications. Subsequently, interferometric method is used to control the absorption coherently with extremely large contrast between the maximum and minimum absorptance. Compared with the results in literatures, the thin film CPAs proposed here show much more flexibility in both operation frequency and bandwidth.

© 2012 OSA

## 1. Introduction

1. Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. **105**(5), 053901 (2010). [CrossRef] [PubMed]

2. W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science **331**(6019), 889–892 (2011). [CrossRef] [PubMed]

3. Y. D. Chong and A. D. Stone, “Hidden black: coherent enhancement of absorption in strongly scattering media,” Phys. Rev. Lett. **107**(16), 163901 (2011). [CrossRef] [PubMed]

4. 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]

7. M. Pu, C. Hu, M. Wang, C. Huang, Z. Zhao, C. Wang, Q. Feng, and X. Luo, “Design principles for infrared wide-angle perfect absorber based on plasmonic structure,” Opt. Express **19**(18), 17413–17420 (2011). [CrossRef] [PubMed]

## 2. Principle

*d*and complex refractive index

*n*is illuminated by two coherent beams on both sides at normal incidence. Since the problem is much easier than that in nanostructured surfaces [5

5. 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]

*T*matrix) or scatter matrix (

*S*matrix) [8]. The output beams (

*C*and

*D*) can be expressed in terms of the input beams (

*A*and

*B*) through the scatter matrix.

*S*matrix can be written as:where

*r*and

*t*are the reflection and transmission coefficients of a single beam of

*A*or

*B*:

*C*=

*D*= 0). Due to the mirror symmetry of the system we consider, coherent perfect absorption can only be realized for symmetrical inputs (

*A*=

*B*,

*r*+

*t*= 0) or antisymmetrical inputs (

*A*= -

*B*,

*r*-

*t*= 0). In both conditions, the magnitude of reflection and transmission are equal, implying that the CPAs are exactly beamsplitters when illuminated by a single beam. Using Eqs. (2) and (3), the CPA condition for normal incidence can be obtained:

1. Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. **105**(5), 053901 (2010). [CrossRef] [PubMed]

*kd*>>1. The bandwidth is defined as the frequency width between the maximum absorption and adjacent minimum absorption and characterized by

## 3. Thin film CPA

*d*is extremely thin (

*d*<<λ, |

*nkd*|<<1). In this case, the left side of Eq. (4) becomes

*nkd*| is very small, only plus sign in the right side (symmetric mode) should be chosen and the real and imaginary parts of the refractive index (

*nkd*|<<1 becomes

*kd*<<1 and the required refractive index must be much larger than unit (

*n*>>1). Different from the general CPA condition depicted in Eq. (4), the CPA condition for ultrathin film is explicit. Obviously, proper complex refractive index must be chosen to obtain the coherent perfect absorption at a specific frequency. Due to the extremely low quality factor of the thin film, such absorption may be wideband. However, material with specific dispersion characteristics should be used to obtain a broadband CPA since the required complex refractive index is frequency dependent. In the following, we will show that metal is just the natural material for broadband CPAs.

^{2}/Vs,

^{2}/Vs,

^{−3},

*c*is the speed of light in vacuum. This characteristic length is just the so-called Woltersdorff thickness [12

12. W. Woltersdorff, “Über die optischen Konstanten dünner Metallschichten im langwelligen Ultrarot,” Z. Phys. **91**(3-4), 230–252 (1934). [CrossRef]

*d*=

*d*, the maximum absorption is 0.5, while the reflection and transmission are 0.25, respectively. For

_{w}*d*<

*d*most of the energy is transmitted; For

_{w}*d*>

*d*most of it is reflected.

_{w}14. In the impedance theory, the thin film CPA can be approximated as a resistive sheet with *E*/*J*, is -*Z*_{0}/2, which is just in opposite to the thin film CPA condition. Such a radiation can be thought as the time reversed process of the broadband CPA, although the infinite oscillating current sheet is not applicable in practical applications.

^{−3}. The corresponding plamon frequency and scattering time are 7.0e14rad/s and 8.1fs. Then the Woltersdorff thickness and Plasmon thickness can be calculated as 151nm and 416nm using Eqs. (12) and (14).

1. Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. **105**(5), 053901 (2010). [CrossRef] [PubMed]

*kd*>>1. However, in the region where

*kd*<<1, the absorption is nearly frequency-independent. In Fig. 2(b), the calculated Woltersdorff thickness and Plasmon thickness are 150nm and 450nm, in good agreement with the theoretical values (151nm and 416nm). The differences arise from the approximation made in Eqs. (12) and (14).

15. Q. L. Zhou, Y. L. Shi, T. Li, B. Jin, D. M. Zhao, and C. L. Zhang, “Carrier dynamics and terahertz photoconductivity of doped silicon measured by femtosecond pump-terahertz probe spectroscopy,” Sci. China, Ser. G **52**(12), 1944–1948 (2009). [CrossRef]

## 4. Coherent control of absorption

*n*, the output for each channel with phase modulated input

*n*is the refractive index satisfying Eq. (4) [1

**105**(5), 053901 (2010). [CrossRef] [PubMed]

2. W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science **331**(6019), 889–892 (2011). [CrossRef] [PubMed]

16. J. Kim, R. Jonathan, B. V. Sharma, J. G. Fujimoto, F. X. Kärtner, V. Scheuer, and G. Angelow, “Ultrabroadband beam splitter with matched group-delay dispersion,” Opt. Lett. **30**(12), 1569–1571 (2005). [CrossRef] [PubMed]

*l*is the path difference between the two arms. In this condition, the output will be frequency dependent with bandwidth of

*l*is increased from 0 to 60μm. As Δ

*l*is not very large, the bandwidth is still broadband.

*l*is chosen to be near 500μm here to illustrate the narrowband property (Fig. 5 ). The absorption curves for symmetrical and antisymmetrical inputs behave as the upper and lower bounds of the coherent absorption. The maximum absorption occurs around 27THz, with a nonzero minimum absorption. As shown in the inset, the absorption curve is periodic with narrow bandwidth of 0.6THz, similar with the intrinsic silicon CPA [2

2. W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science **331**(6019), 889–892 (2011). [CrossRef] [PubMed]

*l*is tuned from 500μm to 500 ± 5.56μm.

**331**(6019), 889–892 (2011). [CrossRef] [PubMed]

18. G. Nimtz and U. Panten, “Broad band electromagnetic wave absorbers designed with nano-metal films,” Ann. Phys. **19**(1-2), 53–59 (2010). [CrossRef]

*I*

_{0}), are dependent on the refractive index. In principle, a larger refractive index will result a larger absorption contrast.

*kd*>>1, the required refractive index is predominantly real. If the refractive index is large enough, the total absorption can be reduced to near zero. However, the refractive index for normal dielectric material is limited. For instance, the refractive index for intrinsic silicon is 3.6 + 0.0008i at

*λ*= 1μm while the minimum absorption is as large as 0.268. Although the intensity contrast can be increased with a nonuniform system as stated in [2

**331**(6019), 889–892 (2011). [CrossRef] [PubMed]

*kd*<<1, the real and imaginary parts of the complex refractive index are much larger. As stated by Eq. (7), the refractive index and contrast are determined by

*kd*. As

*kd*increases, the minimum absorption will increase and the absorption contrast will decrease. In our simulations,

*kd*is 0.0082 for 150nm thick doped silicon at 2.5THz, with a contrast larger than

*kd*is 0.2545 and the contrast becomes only 50.

## 5. Conclusion

*kd*will result larger contrast between maximum and minimum absorption, which is useful for applications such as transducers, modulators, or optical switches.

## Acknowledgments

## References and links

1. | Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. |

2. | W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science |

3. | Y. D. Chong and A. D. Stone, “Hidden black: coherent enhancement of absorption in strongly scattering media,” Phys. Rev. Lett. |

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

5. | 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 |

6. | M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B |

7. | M. Pu, C. Hu, M. Wang, C. Huang, Z. Zhao, C. Wang, Q. Feng, and X. Luo, “Design principles for infrared wide-angle perfect absorber based on plasmonic structure,” Opt. Express |

8. | B. E. A. Saleh and M. C. Teich, |

9. | S. Nashima, O. Morikawa, K. Takata, and M. Hangyo, “Measurement of optical properties of highly doped silicon by terahertz time domain reflection spectroscopy,” Appl. Phys. Lett. |

10. | R. A. Falk, “Near IR Absorption in Heavily Doped Silicon-An Empirical Approach,” in |

11. | B. V. Zeghbroeck, |

12. | W. Woltersdorff, “Über die optischen Konstanten dünner Metallschichten im langwelligen Ultrarot,” Z. Phys. |

13. | M. Dressel and G. Gruner, |

14. | In the impedance theory, the thin film CPA can be approximated as a resistive sheet with |

15. | Q. L. Zhou, Y. L. Shi, T. Li, B. Jin, D. M. Zhao, and C. L. Zhang, “Carrier dynamics and terahertz photoconductivity of doped silicon measured by femtosecond pump-terahertz probe spectroscopy,” Sci. China, Ser. G |

16. | J. Kim, R. Jonathan, B. V. Sharma, J. G. Fujimoto, F. X. Kärtner, V. Scheuer, and G. Angelow, “Ultrabroadband beam splitter with matched group-delay dispersion,” Opt. Lett. |

17. | E. D. Palik, |

18. | G. Nimtz and U. Panten, “Broad band electromagnetic wave absorbers designed with nano-metal films,” Ann. Phys. |

**OCIS Codes**

(310.3915) Thin films : Metallic, opaque, and absorbing coatings

(160.3918) Materials : Metamaterials

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Thin Films

**History**

Original Manuscript: November 16, 2011

Revised Manuscript: December 22, 2011

Manuscript Accepted: December 23, 2011

Published: January 17, 2012

**Citation**

Mingbo Pu, Qin Feng, Min Wang, Chenggang Hu, Cheng Huang, Xiaoliang Ma, Zeyu Zhao, Changtao Wang, and Xiangang Luo, "Ultrathin broadband nearly perfect absorber with symmetrical coherent illumination," Opt. Express **20**, 2246-2254 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-2246

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### References

- Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett.105(5), 053901 (2010). [CrossRef] [PubMed]
- W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science331(6019), 889–892 (2011). [CrossRef] [PubMed]
- Y. D. Chong and A. D. Stone, “Hidden black: coherent enhancement of absorption in strongly scattering media,” Phys. Rev. Lett.107(16), 163901 (2011). [CrossRef] [PubMed]
- 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. Photonics2(5), 299–301 (2008). [CrossRef]
- M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B79(3), 033101 (2009). [CrossRef]
- M. Pu, C. Hu, M. Wang, C. Huang, Z. Zhao, C. Wang, Q. Feng, and X. Luo, “Design principles for infrared wide-angle perfect absorber based on plasmonic structure,” Opt. Express19(18), 17413–17420 (2011). [CrossRef] [PubMed]
- B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed., (Wiley, 2007).
- S. Nashima, O. Morikawa, K. Takata, and M. Hangyo, “Measurement of optical properties of highly doped silicon by terahertz time domain reflection spectroscopy,” Appl. Phys. Lett.79(24), 3923–3925 (2001). [CrossRef]
- R. A. Falk, “Near IR Absorption in Heavily Doped Silicon-An Empirical Approach,” in Proceedings of the 26th ISTFA, 2000.
- B. V. Zeghbroeck, Principles of Semiconductor Devices (Boulder, 1997).
- W. Woltersdorff, “Über die optischen Konstanten dünner Metallschichten im langwelligen Ultrarot,” Z. Phys.91(3-4), 230–252 (1934). [CrossRef]
- M. Dressel and G. Gruner, Electrodynamics of Solids: Optical Properties of Electrons in Matter (Cambridge, New York, 2002).
- In the impedance theory, the thin film CPA can be approximated as a resistive sheet with Z=1/(dwσ0)=Z0/2 as the thickness of the slab is much smaller than the skin depth. Here, σ0=ωp2τε0 is the AC conductivity and Z0=μ0/ε0 is the impedance of vacuum. Then consider the radiation property of an infinite oscillating current sheet in xy plane. Assuming that the current is J→=Ksin(ωt)x→, the electric field at z = 0 can be written as: E→=−0.5μ0cKsin(ωt)x→. The effective sheet impedance, defined as E/J, is -Z0/2, which is just in opposite to the thin film CPA condition. Such a radiation can be thought as the time reversed process of the broadband CPA, although the infinite oscillating current sheet is not applicable in practical applications.
- Q. L. Zhou, Y. L. Shi, T. Li, B. Jin, D. M. Zhao, and C. L. Zhang, “Carrier dynamics and terahertz photoconductivity of doped silicon measured by femtosecond pump-terahertz probe spectroscopy,” Sci. China, Ser. G52(12), 1944–1948 (2009). [CrossRef]
- J. Kim, R. Jonathan, B. V. Sharma, J. G. Fujimoto, F. X. Kärtner, V. Scheuer, and G. Angelow, “Ultrabroadband beam splitter with matched group-delay dispersion,” Opt. Lett.30(12), 1569–1571 (2005). [CrossRef] [PubMed]
- E. D. Palik, Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985).
- G. Nimtz and U. Panten, “Broad band electromagnetic wave absorbers designed with nano-metal films,” Ann. Phys.19(1-2), 53–59 (2010). [CrossRef]

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