## Sub-diffraction negative and positive index modes in mid-infrared waveguides

Optics Express, Vol. 16, Issue 21, pp. 16404-16409 (2008)

http://dx.doi.org/10.1364/OE.16.016404

Acrobat PDF (753 KB)

### Abstract

We characterize a strongly anisotropic waveguide consisting of alternating 80 nm layers of n^{+}-InGaAs and i-AlInAs on InP substrate. A strong increase in the transverse magnetic (TM) reflection at *λ*=8.4 *µ*m corresponds to a characteristic low-order mode cutoff for the left-handed waveguide. The subsequent decrease of TM reflection at λ=11.5 *µ*m represents the onset of right-handed no-cutoff light guiding. Good qualitative agreement is found when the experimental results are compared to finite element and transfer-matrix frequency domain simulations.

© 2008 Optical Society of America

## 1. Introduction

1. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science **292**, 77–79 (2001). [CrossRef] [PubMed]

6. J. Yao, Z. Liu, Y. Liu, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical Negative Refraction in Bulk Metamaterials of Nanowires,” Science **321**, 930 (2008). [CrossRef] [PubMed]

3. D. Smith and Schurig, “Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors,” Phys. Rev. Lett. **90**, 077405 (2003). [CrossRef] [PubMed]

5. A. J. Hoffman, L. Alekseyev, S. S. Howard, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. **6**, 946–950 (2007). [CrossRef] [PubMed]

8. J. Elser, A. A. Govyadinov, I. Avrustky, I. Salakhutdinov, and V. A. Podolskiy, “Plasmonic nanolayer composites: coupled plasmon polaritons, effective-medium response, and subdiffraction light manipulation,” J. Nanomaterials , 2007, 79469 (2007). [CrossRef]

10. Z. Jacob, L. V. Alekseyev, and E. E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express **14**, 8247–8256 (2006). [CrossRef] [PubMed]

11. V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B **71**, 201101 (2005). [CrossRef]

8. J. Elser, A. A. Govyadinov, I. Avrustky, I. Salakhutdinov, and V. A. Podolskiy, “Plasmonic nanolayer composites: coupled plasmon polaritons, effective-medium response, and subdiffraction light manipulation,” J. Nanomaterials , 2007, 79469 (2007). [CrossRef]

*k*

^{2}

_{z}+

*k*

^{2}

_{y}=

*ε*

^{(e|o)}

*ν*

^{(e|o)}

*k*

^{2}where

*ε*

^{(e)}=

*ε*

_{⊥},

*ε*

^{(o)}=

*ε*

_{‖},

*ν*

^{(e|o)}=1-

*χ*

^{(e|o)}2/(

*ε*

_{∥}

*k*

^{2}) [11

11. V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B **71**, 201101 (2005). [CrossRef]

*k*

_{y},

*k*

_{z}, and

*k*are the y-component, z-component, and magnitude of the wave vector respectively;

*ε*

_{‖}=

*ε*

_{yy}=

*ε*

_{zz}and

*ε*

_{⊥}=

*ε*

_{xx}are the components of the permittivity tensor as shown in Fig. 1;

*χ*is a modal parameter of the waveguide that is proportional to mode order and is inversely proportional to the mode confinement scale; it can be solved analytically for simple geometries or numerically for more complicated structures; and (e) or (o) designate the extraordinary and ordinary waves respectively. The modal index,

*n*, is given by

*n*=(

*εν*)

^{1/2}[11

11. V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B **71**, 201101 (2005). [CrossRef]

*ν*and

*ε*have different signs, the waves decay exponentially. Modes with positive

*ν*and

*ε*have positive n and modes with negative

*ν*and

*ε*have negative

*n*. For modes with negative

*n*, the phase velocity is in a direction opposite to the wave propagation and the waveguide behaves as a complete two-dimensional analog of three-dimensional negative index materials [12

12. V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. **10**, 509 (1968). [CrossRef]

13. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**, 3966–3969 (2000). [CrossRef] [PubMed]

14. D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging,” J. Opt. Soc. Am. B **23**, 391 (2006). [CrossRef]

15. J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett **90**, 191109 (2007). [CrossRef]

*ε*

_{⊥}is negative for a large bandwidth in the mid-infrared and

*ν*can be negative if

*χ*is sufficiently large. Since larger

*χ*corresponds to higher order modes, this condition results in a cutoff for low order modes. In our experiment, we demonstrate this cutoff by showing an increase in reflection at wavelengths where the sample transitions into the region of strong anisotropy where

*ε*

_{⊥}<0 and

*ε*

_{‖}>0.

## 2. Material and experimental setup

_{0.53}Ga

_{0.47}As and Al

_{0.48}In

_{0.52}As. In total, the epitaxial layer was 20

*µ*m thick. The InGaAs layers were highly Si doped, measured as

*n*

_{d}=7.9×10

^{18}cm

^{-3}, to provide a plasma resonance of free electrons at mid-infrared wavelengths. For the measurements, the epitaxial layer serves as an anisotropic waveguide with air and InP cladding. To determine the optical anisotropy, the sample was first characterized using reflection and transmission measurements as described in a previous publication [5

5. A. J. Hoffman, L. Alekseyev, S. S. Howard, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. **6**, 946–950 (2007). [CrossRef] [PubMed]

*ε*

_{⊥}and

*ε*

_{‖}versus wavelength obtained using the effective medium approximation for the particular doping level,

*n*

_{d}, measured during characterization.

^{2}piece of the wafer. For the top 1.0 x 1.0 cm

^{2}portion of the sample, the epitaxial layer was removed using a HBr:HNO

_{3}:H

_{2}O (1:1:10) etch. The sample was then cleaved down the center to yield two 0.5×2.0 cm

^{2}pieces with smooth facets. The bottom inset of Fig. 1 depicts the facet of a sample used in the experiments.

*µ*m diameter spot using a 6 inch focal length ZnSe lens, and collected on a cooled HgCdTe (MCT) detector. The top portion of the sample (InP only) was located at the focal point of the focusing lens, as shown in the top inset of Fig. 1, the measured reflected light was maximized by translating the sample perpendicular to the incident beam, and spectra for the TM and transverse electric (TE) polarizations were collected. The sample was then translated vertically so that the spectra for the metamaterial and substrate could be collected for the same angle. The sample was first positioned such that the collected light was a maximum for the TM polarization. It was then shifted such that the power for the TM polarization was half the value previously recorded, thus locating the point of peak intensity closer to the metamaterial. This measurement was repeated for several angles; each spectrum was averaged over 400 scans.

## 3. Results

### 3.1 Experimental results

*R*

_{InP+meta}, to the part with the epitaxial layer removed,

*R*

_{InP}. The reflection data for several angles are plotted in Fig. 2(a). The curves are offset because a comparison of the absolute value of the ratio is difficult due to the sensitivity of the location of the sample relative to the center of the incident beam. The relative changes in reflection versus wavelength for a single angle are preserved however.

*λ*=8.4

*µ*m, there is a sharp increase in the TM reflection ratio. This increase in the ratio is due to a sharp increase in the reflection off the metamaterial layer which is due to the sudden change in its permittivity tensor. At this wavelength,

*ε*

_{⊥}<0 and the lower order modes cannot propagate inside of the waveguide and are thus expelled [8

8. J. Elser, A. A. Govyadinov, I. Avrustky, I. Salakhutdinov, and V. A. Podolskiy, “Plasmonic nanolayer composites: coupled plasmon polaritons, effective-medium response, and subdiffraction light manipulation,” J. Nanomaterials , 2007, 79469 (2007). [CrossRef]

*λ*=11

*µ*m there is a sharp increase in the TE reflection and a small drop in the TM reflection, corresponding to the change in the sign of both

*ε*

_{⊥}and

*ε*

_{‖}. As result of this change in the anisotropy of the metamaterial, the waveguide becomes opaque to all TE modes, increasing the reflection. The simultaneous drop in the TM reflection around this wavelength signifies the onset of cutoff-less propagation of right-handed modes; however, the large impedance mismatch between the metamaterial waveguide and vacuum at this wavelength significantly reduces the observed effect.

### 3.2 Numerical calculations

16. www.comsol.com

5. A. J. Hoffman, L. Alekseyev, S. S. Howard, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. **6**, 946–950 (2007). [CrossRef] [PubMed]

*n*=3.1. Reflection as a function of wavelength, incident angle and beam width was studied for the system. Figure 2(b) shows the results of the calculations for a 30

*µ*m diameter beam.

*λ*=8.4

*µ*m there is a large increase in the TM reflection ratio. Also at

*λ*~11

*µ*m there is a dip in the TM ratio and an increase in the TE ratio. The magnitude of variation in reflectivity strongly depends on the beam diameter with wider beams yielding lower contrast. This effect is responsible for the quantitative discrepancy between numerical simulations and experimental results.

*λ*=11.5

*µ*m the signs of the components of the effective permittivity flip, and the waveguide begins transmitting all its TM modes, regardless of the mode number. At the same wavelength all TE modes cut off. This regime, known as positive-index photonic funnel [8

*λ*=10.2

*µ*m. The

*m*=1 mode is a leaky right-handed wave:

*ν*>0, the real part of

*n*is smaller than the index of the cladding, and the imaginary part of

*n*is large. The exponentially growing field profile is a valid mathematical solution of the Maxwell equations for such a mode in an infinite planar waveguide. The exponential increase of the field originates from the leaky character of the mode, accompanied with its exponential decay along the y-coordinate [17]. This solution, however, is not a true physical solution to the Maxwell equations. The non-normalizable leaky modes cannot be excited in the real finite-sized structure; a spectrum of normalizable open-waveguide modes with real values of

*k*

_{x}will be excited instead [17].

*m*=5 mode is the first mode that has sub-wavelength field variation, characterized by

*ν*<0; however, |

*n*| is smaller than the index of the cladding material and thus the mode is also leaky. The first bound mode with

*ν*<0 is the

*m*=9 mode.

^{th}order mode as a function of wavelength is shown in Fig. 4(b). Both negative and positive index confined (|n|>3.1) propagation is clearly seen for

*λ*>10.7

*µ*m. In the right-handed regime,

*λ*>11.5

*µ*m, the waveguide essentially does not have a cut-off and may support a number of low-index modes in addition to their high-index counterparts. Here, the limits on mode propagation are governed by the relatively high-index cladding and from deviations of the metamaterial response from the effective medium theory predictions [15

15. J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett **90**, 191109 (2007). [CrossRef]

^{(e)2}>k

^{2}.

*λ*

_{o}/|

*n*| that determines the diffraction limit of planar-waveguide optics [8

18. J. Elser, R. Wangber, V. A. Podolskiy, and E. E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. **89**, 261102 (2006). [CrossRef]

## 4. Conclusion

*λ*=8.4

*µ*m that is characteristic of a sudden sign change in the anisotropy of the metamaterial waveguide and is accompanied by the onset of negative-index propagation of higher order modes. At

*λ*=11.5

*µ*m the TM reflectivity drops, while TE reflectivity increases, signifying the onset of no-cutoff waveguide propagation. The behavior of the reflectivity and properties of the waveguide modes is confirmed using FE and TMM calculations. Results of numerical solutions are in very good agreement with experimental results. This work was supported in part by PCCM (NSF-ERC), NSF (grant #ECCS-0724763), ONR (grant #N00014-07-1-0457), and ARO. We also acknowledge valuable contributions by E. E. Narimanov and L. Alekseyev in the early phases of this work.

## References and links

1. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science |

2. | V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics |

3. | D. Smith and Schurig, “Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors,” Phys. Rev. Lett. |

4. | D. R. Smith, S. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. |

5. | A. J. Hoffman, L. Alekseyev, S. S. Howard, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. |

6. | J. Yao, Z. Liu, Y. Liu, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical Negative Refraction in Bulk Metamaterials of Nanowires,” Science |

7. | J. Elser and V. A. Podolskiy, “Scattering free plasmonic optics with anisotropic metamaterials,” Phys. Rev. Lett. |

8. | J. Elser, A. A. Govyadinov, I. Avrustky, I. Salakhutdinov, and V. A. Podolskiy, “Plasmonic nanolayer composites: coupled plasmon polaritons, effective-medium response, and subdiffraction light manipulation,” J. Nanomaterials , 2007, 79469 (2007). [CrossRef] |

9. | A. A. Govyadinov and V. A. Podolskiy, “Subdiffraction light propagation in fibers with anisotropic dielectric cores,” Phys. Rev. B. 75, 155108 (2006). |

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

11. | V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B |

12. | V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. |

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

14. | D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging,” J. Opt. Soc. Am. B |

15. | J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett |

16. | |

17. | V. V. Shevchenko, |

18. | J. Elser, R. Wangber, V. A. Podolskiy, and E. E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(160.6000) Materials : Semiconductor materials

(230.7370) Optical devices : Waveguides

(160.3918) Materials : Metamaterials

**ToC Category:**

Metamaterials

**History**

Original Manuscript: August 29, 2008

Revised Manuscript: September 25, 2008

Manuscript Accepted: September 25, 2008

Published: September 29, 2008

**Citation**

Anthony J. Hoffman, Viktor A. Podolskiy, Deborah L. Sivco, and Claire Gmachl, "Sub-diffraction negative and positive index modes in mid-infrared waveguides," Opt. Express **16**, 16404-16409 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-21-16404

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### References

- R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292, 77 - 79 (2001). [CrossRef] [PubMed]
- V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photonics 1, 41 - 48 (2007). [CrossRef]
- D. R. Smith and D. Schurig, "Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors," Phys. Rev. Lett. 90, 077405 (2003). [CrossRef] [PubMed]
- D. R. Smith, S. Schurig, J. J. Mock, P. Kolinko, and P. Rye, "Partial focusing of radiation by a slab of indefinite media," Appl. Phys. Lett. 84, 2244 - 2246 (2004). [CrossRef]
- A. J. Hoffman, L. Alekseyev, S. S. Howard, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, "Negative refraction in semiconductor metamaterials," Nat. Mater. 6, 946 - 950 (2007). [CrossRef] [PubMed]
- J. Yao, Z. Liu, Y. Liu, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, "Optical Negative Refraction in Bulk Metamaterials of Nanowires," Science 321, 930 (2008). [CrossRef] [PubMed]
- J. Elser and V. A. Podolskiy, "Scattering free plasmonic optics with anisotropic metamaterials," Phys. Rev. Lett. 100, 066402 (2008). [CrossRef] [PubMed]
- J. Elser, A. A. Govyadinov, I. Avrustky, I. Salakhutdinov, and V. A. Podolskiy, "Plasmonic nanolayer composites: coupled plasmon polaritons, effective-medium response, and subdiffraction light manipulation," J. Nanomaterials, 2007, 79469 (2007). [CrossRef]
- A. A. Govyadinov and V. A. Podolskiy, "Subdiffraction light propagation in fibers with anisotropic dielectric cores," Phys. Rev. B. 75, 155108 (2006).
- Z. Jacob, L. V. Alekseyev, and E. E. Narimanov, "Optical Hyperlens: Far-field imaging beyond the diffraction limit," Opt. Express 14, 8247 - 8256 (2006). [CrossRef] [PubMed]
- V. A. Podolskiy and E. E. Narimanov, "Strongly anisotropic waveguide as a nonmagnetic left-handed system," Phys. Rev. B 71, 201101 (2005). [CrossRef]
- V. G. Veselago, "Electrodynamics of substances with simultaneously negative values of sigma and mu," Sov. Phys. Usp. 10, 509 (1968). [CrossRef]
- J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966 - 3969 (2000). [CrossRef] [PubMed]
- D. R. Smith and J. B. Pendry, "Homogenization of metamaterials by field averaging," J. Opt. Soc. Am. B 23, 391 (2006). [CrossRef]
- J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, "Nonlocal effects in effective-medium response of nanolayered metamaterials," Appl. Phys. Lett 90, 191109 (2007). [CrossRef]
- www.comsol.com
- V. V. Shevchenko, Continuous transitions in open waveguides (Golem Press, Boulder, 1971).
- J. Elser, R. Wangber, V. A. Podolskiy, and E. E. Narimanov, "Nanowire metamaterials with extreme optical anisotropy," Appl. Phys. Lett. 89, 261102 (2006). [CrossRef]

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