## Gain-assisted Hybrid-superlens Hyperlens for Nano Imaging |

Optics Express, Vol. 20, Issue 20, pp. 22953-22960 (2012)

http://dx.doi.org/10.1364/OE.20.022953

Acrobat PDF (1869 KB)

### Abstract

We propose an innovative active imaging device named gain-assisted hybrid-superlens hyperlens and examine its resolving power theoretically. This semi-cylindrical device consists of a core of semi-cylindrical super-lens and a half cylindrical outer shell of hyperlens. Both the superlens and hyperlens parts of the device are appropriately designed multi-layered metal-dielectric structures having indefinite eigenvalues of dielectric tensors. The dielectric layers of the hyperlens are doped with Coumarin, which play the role of gain medium. The gain medium is analyzed thoroughly using a generic four-level system model, and the permittivity of the gain medium is extracted from this analysis for simulating the imaging characteristics of the device. According to our simulation at wavelength of 365 nm, an excellent resolution power much better than the diffraction limit value can be achieved.

© 2012 OSA

## 1. Introduction

18. N. Papasimakis, Z. Q. Luo, Z. X. Shen, F. De Angelis, E. Di Fabrizio, A. E. Nikolaenko, and N. I. Zheludev, “Graphene in a photonic metamaterial,” Opt. Express **18**(8), 8353–8359 (2010). [CrossRef] [PubMed]

19. C. Kurter, P. Tassin, L. Zhang, T. Koschny, A. P. Zhuravel, A. V. Ustinov, S. M. Anlage, and C. M. Soukoulis, “Classical analogue of electromagnetically induced transparency with a metal-superconductor hybrid metamaterial,” Phys. Rev. Lett. **107**(4), 043901 (2011). [CrossRef] [PubMed]

20. A. Fang, Th. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B **79**(24), 241104 (2009). [CrossRef]

21. S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. **105**(12), 127401 (2010). [CrossRef] [PubMed]

**E**|<<10

^{4}). It is not necessary using numerical calculations, such as retrieval method, to define the permittivity of the gain medium when this condition is satisfied. With our equation, the permittivity of the four-level active medium can be calculated directly, and it is used for simulating the imaging characteristics of this active device. Our simulation at wavelength 365 nm reveals that an excellent resolution power much better than the diffraction limit value can indeed be achieved when the gain layers are used.

## 2. Principle

20. A. Fang, Th. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B **79**(24), 241104 (2009). [CrossRef]

*r*is the pumping rate,

*τ*is the relaxation time or lifetime between the

_{ij}*i*th and

*j*th energy level,

*ω*is the optical absorption frequency, and

_{a}*N*is the total electron density. This relation will be used later.

_{21}~τ

_{43}~10

^{−14}sec. Conversely, the relaxation time τ

_{32}(~10

^{−12}sec.) for the transition from energy level 3 to 2 is much longer than τ

_{21}and τ

_{43}, making the population inversion possible. Subtracting Eq. (3) from Eq. (2) and Eq. (1) from Eq. (4), defining

*N*,

_{1}>>N_{4}*N*,

_{3}>> N_{2}*N ~N*+

_{1}*N*+

_{3}~N_{1}*ΔΝ*, we getwhere

20. A. Fang, Th. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B **79**(24), 241104 (2009). [CrossRef]

*is the bandwidth of the atomic transition, and*

_{a}*σ*is the coupling strength of polarization to the external electric field. In this gain medium we assume that the external sinusoidal electric field coupled to the four-level atomic system has instantaneous amplitude

_{a}**|E|**sin(

*ω*) and the instantaneous amplitude of the induced polarization is

_{α}t**|P|**sin(

*ω*), where

_{a}t + δ*δ*is the phase delay of the later with respect to the former, and

**|E|**and

**|P|**are the absolute magnitudes of them, respectively. In our model we also assume the condition,

*ω*>>

_{a}*γ*and under this condition Eq. (7) can be approximated asAs shown in Eq. (8), the non-linear term gives the first-order perturbation correction to the population numbers. Nevertheless, no well-developed method can be readily implemented to solve this differential equation for finding its exact solution. Though we are not able to solve it exactly, approximate solution is still possible to obtain if the electric field is weak enough. Hereafter we assume that the weak field condition is fulfilled, thus the nonlinear term on the right hand side of Eq. (8) can be dropped. The extracted effective permittivity under this assumption is still of Lorentz form, written as

_{a}**79**(24), 241104 (2009). [CrossRef]

^{4}or higher orders, the non-linear effect cannot be neglected. This argument had already been mentioned in [20

**79**(24), 241104 (2009). [CrossRef]

## 3. Simulation result

8. B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B **74**(11), 115116 (2006). [CrossRef]

9. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B **74**(7), 075103 (2006). [CrossRef]

^{1}its absorption wavelength is in the violet-blue region and there is indeed laser operating at the same wavelength [21

21. S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. **105**(12), 127401 (2010). [CrossRef] [PubMed]

^{2}it is almost non-toxin to the human body since Coumarin and its derivatives are commonly used as food additive and ingredient in perfume, and

^{3}Coumarin can be easily doped into PMMA to a high concentration and its knowledge for being gain medium is well-established [22]. To fit our designing parameters, Coumarin 2 (7-(Ethylamino)-4,6-dimethylcoumarin) is chosen to be the gain material because its absorption wavelength (λ

_{a}= 365 nm) is readily acquirable using commercial laser products. We also assume that the PMMA has been doped with Coumarin 2 to the concentration 1.0 × 10

^{−2}M [23

23. G. Somasundaram and A. Ramalingam, “Gain studies of Coumarin 1 dye-doped polymer laser,” J. Lumin. **90**(1-2), 1–5 (2000). [CrossRef]

^{26}per cubic meter, the lifetime τ

_{32}is 4.2 × 10

^{−9}second [24

24. H. E. Zimmerman, J. H. Penn, and C. W. Carpenter, “Evaluation of single-photon-counting measurements of excited-state lifetimes,” Proc. Natl. Acad. Sci. U.S.A. **79**(6), 2128–2132 (1982). [CrossRef] [PubMed]

_{a}is 10

^{−8}C

^{2}/kg [20

**79**(24), 241104 (2009). [CrossRef]

^{9}. According to these chosen parameters, the real and imaginary parts of the effective permittivity of pure Coumarin 2 are shown in Fig. 2(a) . Based on the results shown in Fig. 2, the permittivity of Coumarin 2 at λ = 365 nm is 0.9057 + 94.5550i. As has been mentioned before, the concentration of Coumarin 2 in PMMA is 1.0 × 10

^{−2}M, corresponding to a ratio of weight (denoted as f) about 0.002. This value is derived from the data that the density of PMMA is 1.35 g/cm

^{3}and the molecular weight of Coumarin 2 is 217.26 g/mol. The dielectric constant of the gain medium layer thus becomes 2.222 + 0.1887i. The commercial solver COMSOL Multiphysics

^{TM}3.5a based on finite element method (FEM) is utilized insimulating the two-dimensional structure, and perfectly matched layers surrounding the simulation region are used. Transverse magnetic (TM) polarized incident light is considered with the incident electric field (x-direction) being perpendicular to the axis of the cylindrical structure (along the y-direction), as shown in Fig. 1. Figure 2(b) plots the isofrequency dispersion relations for the superlens and hyperlens structures of the hybrid-superlens hyperlens at the incident wavelength of 365 nm.

8. B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B **74**(11), 115116 (2006). [CrossRef]

12. X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. **7**(6), 435–441 (2008). [CrossRef] [PubMed]

## 4. Conclusion

## Acknowledgment

## References and links

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

2. | D. P. Tsai and W. C. Lin, “Probing the near fields of the super-resolution near-field optical structure,” Appl. Phys. Lett. |

3. | D. P. Tsai, C. W. Yang, W. C. Lin, F. H. Ho, H. J. Huang, M. Y. Chen, T. F. Tseng, C. H. Lee, and C. J. Yeh, “Dynamic aperture of near-field super resolution structures,” Jpn. J. Appl. Phys. |

4. | W. C. Liu, C. Y. Wen, K. H. Chen, W. C. Lin, and D. P. Tsai, “Near-field images of the AgOx-type super-resolution near-field structure,” Appl. Phys. Lett. |

5. | T. C. Chu, D. P. Tsai, and W. C. Liu, “Readout contrast beyond diffraction limit by a slab of random nanostructures,” Opt. Express |

6. | K. P. Chiu, K. F. Lai, and D. P. Tsai, “Application of surface polariton coupling between nano recording marks to optical data storage,” Opt. Express |

7. | D. O. S. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express |

8. | B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B |

9. | A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B |

10. | W. Cai, D. A. Genov, and V. M. Shalaev, “Superlens based metal-dielectric composites,” Phys. Rev. B |

11. | D. Schurig and D. R. Smith, “Sub-diffraction imaging with compensating bilayers,” New J. Phys. |

12. | X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. |

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

14. | H. Lee, Z. Liu, Y. Xiong, C. Sun, and X. Zhang, “Development of optical hyperlens for Imaging below the diffraction limit,” Opt. Express |

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

16. | S. Schwaiger, M. Bröll, A. Krohn, A. Stemmann, C. Heyn, Y. Stark, D. Stickler, D. Heitmann, and S. Mendach, “Rolled-up three-dimensional metamaterials with a tunable plasma frequency in the visible regime,” Phys. Rev. Lett. |

17. | I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science |

18. | N. Papasimakis, Z. Q. Luo, Z. X. Shen, F. De Angelis, E. Di Fabrizio, A. E. Nikolaenko, and N. I. Zheludev, “Graphene in a photonic metamaterial,” Opt. Express |

19. | C. Kurter, P. Tassin, L. Zhang, T. Koschny, A. P. Zhuravel, A. V. Ustinov, S. M. Anlage, and C. M. Soukoulis, “Classical analogue of electromagnetically induced transparency with a metal-superconductor hybrid metamaterial,” Phys. Rev. Lett. |

20. | A. Fang, Th. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B |

21. | S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. |

22. | F. J. Duarte and L. W. Hillman, “Dye laser principles with applications” (1990), See appendix. |

23. | G. Somasundaram and A. Ramalingam, “Gain studies of Coumarin 1 dye-doped polymer laser,” J. Lumin. |

24. | H. E. Zimmerman, J. H. Penn, and C. W. Carpenter, “Evaluation of single-photon-counting measurements of excited-state lifetimes,” Proc. Natl. Acad. Sci. U.S.A. |

25. | B. H. Cheng, Y. Z. Ho, Y. C. Lan, and D. P. Tsai, “Optical hybrid-superlens-hyperlens for superresolution imaging,” IEEE J. Sel. Top. Quantum Electron. (submitted). |

**OCIS Codes**

(100.6640) Image processing : Superresolution

(110.0180) Imaging systems : Microscopy

(160.1190) Materials : Anisotropic optical materials

(160.3380) Materials : Laser materials

(160.3918) Materials : Metamaterials

**ToC Category:**

Microscopy

**History**

Original Manuscript: July 30, 2012

Revised Manuscript: September 13, 2012

Manuscript Accepted: September 15, 2012

Published: September 21, 2012

**Virtual Issues**

Vol. 7, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Yao Ting Wang, Bo Han Cheng, You Zhe Ho, Yung-Chiang Lan, Pi-Gang Luan, and Din Ping Tsai, "Gain-assisted Hybrid-superlens Hyperlens for Nano Imaging," Opt. Express **20**, 22953-22960 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-20-22953

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### References

- J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85(18), 3966–3969 (2000). [CrossRef] [PubMed]
- D. P. Tsai and W. C. Lin, “Probing the near fields of the super-resolution near-field optical structure,” Appl. Phys. Lett.77(10), 1413–1415 (2000). [CrossRef]
- D. P. Tsai, C. W. Yang, W. C. Lin, F. H. Ho, H. J. Huang, M. Y. Chen, T. F. Tseng, C. H. Lee, and C. J. Yeh, “Dynamic aperture of near-field super resolution structures,” Jpn. J. Appl. Phys.39(Part 1, No. 2B), 982–983 (2000). [CrossRef]
- W. C. Liu, C. Y. Wen, K. H. Chen, W. C. Lin, and D. P. Tsai, “Near-field images of the AgOx-type super-resolution near-field structure,” Appl. Phys. Lett.78(6), 685–687 (2001). [CrossRef]
- T. C. Chu, D. P. Tsai, and W. C. Liu, “Readout contrast beyond diffraction limit by a slab of random nanostructures,” Opt. Express15(1), 12–23 (2007). [CrossRef] [PubMed]
- K. P. Chiu, K. F. Lai, and D. P. Tsai, “Application of surface polariton coupling between nano recording marks to optical data storage,” Opt. Express16(18), 13885–13892 (2008). [CrossRef] [PubMed]
- D. O. S. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express13(6), 2127–2134 (2005). [CrossRef] [PubMed]
- B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B74(11), 115116 (2006). [CrossRef]
- A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B74(7), 075103 (2006). [CrossRef]
- W. Cai, D. A. Genov, and V. M. Shalaev, “Superlens based metal-dielectric composites,” Phys. Rev. B72(19), 193101 (2005). [CrossRef]
- D. Schurig and D. R. Smith, “Sub-diffraction imaging with compensating bilayers,” New J. Phys.7, 162 (2005). [CrossRef]
- X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater.7(6), 435–441 (2008). [CrossRef] [PubMed]
- Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express14(18), 8247–8256 (2006). [CrossRef] [PubMed]
- H. Lee, Z. Liu, Y. Xiong, C. Sun, and X. Zhang, “Development of optical hyperlens for Imaging below the diffraction limit,” Opt. Express15(24), 15886–15891 (2007). [CrossRef] [PubMed]
- Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315(5819), 1686 (2007). [CrossRef] [PubMed]
- S. Schwaiger, M. Bröll, A. Krohn, A. Stemmann, C. Heyn, Y. Stark, D. Stickler, D. Heitmann, and S. Mendach, “Rolled-up three-dimensional metamaterials with a tunable plasma frequency in the visible regime,” Phys. Rev. Lett.102(16), 163903 (2009). [CrossRef] [PubMed]
- I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science315(5819), 1699–1701 (2007). [CrossRef] [PubMed]
- N. Papasimakis, Z. Q. Luo, Z. X. Shen, F. De Angelis, E. Di Fabrizio, A. E. Nikolaenko, and N. I. Zheludev, “Graphene in a photonic metamaterial,” Opt. Express18(8), 8353–8359 (2010). [CrossRef] [PubMed]
- C. Kurter, P. Tassin, L. Zhang, T. Koschny, A. P. Zhuravel, A. V. Ustinov, S. M. Anlage, and C. M. Soukoulis, “Classical analogue of electromagnetically induced transparency with a metal-superconductor hybrid metamaterial,” Phys. Rev. Lett.107(4), 043901 (2011). [CrossRef] [PubMed]
- A. Fang, Th. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B79(24), 241104 (2009). [CrossRef]
- S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett.105(12), 127401 (2010). [CrossRef] [PubMed]
- F. J. Duarte and L. W. Hillman, “Dye laser principles with applications” (1990), See appendix.
- G. Somasundaram and A. Ramalingam, “Gain studies of Coumarin 1 dye-doped polymer laser,” J. Lumin.90(1-2), 1–5 (2000). [CrossRef]
- H. E. Zimmerman, J. H. Penn, and C. W. Carpenter, “Evaluation of single-photon-counting measurements of excited-state lifetimes,” Proc. Natl. Acad. Sci. U.S.A.79(6), 2128–2132 (1982). [CrossRef] [PubMed]
- B. H. Cheng, Y. Z. Ho, Y. C. Lan, and D. P. Tsai, “Optical hybrid-superlens-hyperlens for superresolution imaging,” IEEE J. Sel. Top. Quantum Electron. (submitted).

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