## Graphical retrieval method for orthorhombic anisotropic materials |

Optics Express, Vol. 18, Issue 16, pp. 17009-17019 (2010)

http://dx.doi.org/10.1364/OE.18.017009

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### Abstract

We apply the equivalent theory to orthorhombic anisotropic materials and provide a general unit-cell design criterion for achieving a length-independent retrieval of the effective material parameters from a single layer of unit cells. We introduce a graphical retrieval method and phase unwrapping techniques. The graphical method utilizes the linear regression technique. Our method can reduce the uncertainty of experimental measurements and the ambiguity of phase unwrapping. Moreover, the graphical method can simultaneously determine the bulk values of the six effective material parameters, permittivity and permeability tensors, from a single layer of unit cells.

© 2010 Optical Society of America

## 1. Introduction

1. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science **308**, 534–537 (2005). [CrossRef] [PubMed]

3. S. Feng and K. Halterman, “Parametrically shielding electromagnetic fields by nonlinear metamaterials,” Phys. Rev. Lett. **100**, 063901 (2008). [CrossRef] [PubMed]

4. F. Capolino Ed. *Theory and phenomena of metamaterials* (CRC Press, Taylor and Francis Group, New York, 2009). [CrossRef]

5. J. Valentine, S. Zhang, T. Zentgraf, E. U. Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature (London) **455**, 376–380 (2008). [CrossRef] [PubMed]

8. V. M. Shalaev, W. 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**, 3356–3358 (2005). [CrossRef]

5. J. Valentine, S. Zhang, T. Zentgraf, E. U. Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature (London) **455**, 376–380 (2008). [CrossRef] [PubMed]

9. A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, “Wave propagation retrieval method for metamaterials: Unambiguous restoration of effective parameters,” Phys. Rev. B **80**, 193101 (2009). [CrossRef]

## 2. Equivalent theory

11. D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B **65**, 195104 (2002). [CrossRef]

14. X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, Jr., and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E **70**, 016608 (2004). [CrossRef]

*n*= 1,2,⋯. The scalar terms

*ε*and

_{nj}*µ*(

_{nj}*j*=

*x,y,z*) are complex. Consider a monochromatic wave of frequency

*ω*with time dependence exp(

*−iωt*) propagates inside the orthorhombic anisotropic materials. In each layer we have

*k*

_{0}=

*ω*/

*c*. If the plane of incidence is one of the crystal planes, the TE and TM polarizations are decoupled.

15. L. I. Epstein, “The design of optical filters,” J. Opt. Soc. Am. **42**, 806–810 (1952). [CrossRef]

*AB*. We can then construct a symmetric unit cell by cascading two unit cells as

*ABBA*, which is equivalent to a single homogeneous layer according to Herpin’s theorem. This process is illustrated in Fig. 1. Applying this methodology, a symmetric unit-cell layer can often be constructed regardless the number of sub-layers and complexity of each sub-layer in the original unit cell. The permittivities and permeabilities retrieved from the symmetric unit cell, which is composed of two original asymmetric unit cells, will represent the bulk material parameters. Thus, a length-independent description can be achieved.

**represents the characteristic matrix of one symmetric unit cell. According to Herpin’s equivalent theorem, it can be replaced by an equivalent single layer. Assume the x-z plane is the plane of incidence. For TM mode, i.e.**

*M***= (0,**

*H**H*,0) and

_{y}**= (**

*E**E*,0,

_{x}*E*), we have

_{z}*ψ*is the equivalent phase thickness of the symmetric unit cell; 𝒵

_{e}_{e}is the equivalent impedance. If the material contains

*N*-layer symmetric unit cells, the characteristic matrix of the material is given by

_{e}with the negative admittance −𝒴

_{e}. Here the 𝒵

_{e}and 𝒴

_{e}are, respectively, the generalized impedance and admittance because they include incidence angle, i.e., Eq. (4) is valid for both normal and oblique incidence. They are given by

*ε*,

_{x}*ε*,

_{y}*ε*) and (

_{z}*µ*,

_{x}*µ*,

_{y}*µ*) are, respectively, the effective permittivities and permeabilities of the equivalent layer. As shown in Eq. (4), the total phase of a

_{z}*N*-layer system equals

*N*times the phase

*ψ*of the single layer; whereas the impedance 𝒵

_{e}_{e}or admittance 𝒴

_{e}is independent of the number of layers. These properties imply that the material parameters retrieved from the

*N*layers of unit cells are the same as those retrieved from one layer of unit cells. In other words, the bulk metamaterial parameters can be predicted from a single symmetric unit cell. Note that the characteristic matrix of a N-layer asymmetric unit cells does not have the nice properties as those in Eq. (4). As a consequence, the retrieved material parameters will be dependent on the number of unit cells along the propagation direction. Hence, the bulk material parameters cannot be predicted from one unit-cell layer. In other words, a length-independent description cannot be achieved for asymmetric unit cells.

## 3. Graphical retrieval method

11. D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B **65**, 195104 (2002). [CrossRef]

*ABA*. Let the

*z*-direction perpendicular to the plane of the layers. Assume each layer is orthorhombic anisotropic with the principal axes parallel in each direction and the effective permittivity and permeability tensors are given by Eq. (1). Let the x-z plane be the plane of incidence. The characteristic matrix of TM mode is defined as

*ψ*=

_{e}*K*and the

_{e}d*K*is the effective propagation constant. The subscript 1 and 2 refer to the layer A and B, respectively. The 𝒵

_{e}_{1}is the generalized impedance of the equivalent layer A. The z-component wave vector

*k*

_{1z}and

*k*

_{2z}are determined from the dispersion relations in Eq. (5). The

*d*

_{1}is twice the thickness of layer A; and the

*d*

_{2}is the thickness of layer B. The period of the symmetric unit cell is

*d*=

*d*

_{1}+

*d*

_{2}. Other parameters in Eq. (8) are given by

*d*≪

*λ*, after tedious derivation, Eq. (8) can be simplified to

*p*=

*x,y*. Equation (11) is a simplification of the most general linear case treated by Lakhtakia [16

16. A. Lakhtakia, “Constraints on effective constitutive parameters of certain bianisotropic laminated composite materials,” Electromagnetics **29**508–514 (2009). [CrossRef]

_{i}= 𝒵

_{o}, combining Eqs. (7) and (10), we obtained the retrieval formula:

*θ*is the incidence angle. The

*ε*and

_{b}*µ*are, respectively, the background relative permittivity and permeability. The retrieval formulas in Eq. (12) provide four straight lines (

_{b}*Y*,

_{M}*Y*,

_{m}*Y*,

_{E}*Y*vs.

_{e}*X*), two for each polarization. The

*Y*and

_{M}*Y*represent the dispersion lines for TM and TE polarizations, respectively. The

_{E}*Y*and

_{m}*Y*are the corresponding impedance and admittance lines. These straight lines are easy to implement experimentally. After measuring the scattering parameters at several incidence angles and plotting the data according to Eqs. (12)–(14), use linear regression technique to calculate the slopes and Y-intercepts of the four lines. From the slopes and Y-intercepts, the six effective material parameters,

_{e}*and*ε ¯

_{j}*(*µ ¯

_{j}*j*=

*x,y,z*), can be retrieved simultaneously. Let

*Y*

^{0}

_{M}and

*Y*

^{0}

_{m}represent the Y-intercepts of the two lines in TM polarization,

*S*and

_{M}*S*the corresponding slopes of the lines; whereas

_{m}*Y*

^{0}

_{E},

*Y*

^{0}

_{e},

*S*, and

_{E}*S*are the corresponding quantities for TE polarization. Thus,

_{e}*n*) > 0, ℑ(

_{m}*n*) > 0, and the real part of impedance and admittance greater than zero, i.e. ℜ(

_{e}*Z*) > 0, and ℜ(

*Y*) > 0 for passive media [11

11. D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B **65**, 195104 (2002). [CrossRef]

## 4. Resolving phase branch

*m*. The inverse cosine cos

^{−1}(·) represents the principal value: 0 ≤ cos

^{−1}(·) ≤

*π*. Knowing only the cosine value, the phase angle cannot be determined. One need the information of sine, which can be obtained from the imaginary part of the cosine as following:

*φ*and

_{r}*φ*are real. For passive medium,

_{i}*φ*> 0, and thus sinh

_{i}*ϕ*> 0. Therefore, the ± sign in front of the inverse cosine in Eq. (14) can be resolved from the imaginary part of

_{i}*A*:

*m*. This part can be very confusing when negative refractive index might be involved. Notice that among the four retrieval lines in Eq. (12), only the dispersion lines,

*Y*~

_{M}*X*and

*Y*~

_{E}*X*, are functions of the phase branch

*m*, the impedance and admittance lines are independent of

*m*. Employing this property, we provide three methods that might help to resolve the correct phase branch

*m*.

*Method-1, Using*ε ¯

^{2}

_{x}

*and*µ ¯

^{2}

_{x}: For the TM polarization in Eq. (12), the

ε ¯

^{2}

_{x}can be obtained either from the intercepts as

µ ¯

^{2}

_{x}can be obtained from either

*m*is the one that minimizes the absolute value of the differences, i.e.

*Method-2, Using*

*and*

*Method-3, Using*ε ¯ µ ¯ ε ¯ µ ¯ : The third method is to use

_{z}_{y}and_{y}_{z}*for TM and*ε ¯ µ ¯

_{z}_{y}*for TE as criteria to select the correct phase branch. Then, the algorithm becomes*ε ¯ µ ¯

_{y}_{z}*m*predicted by the first two methods is always consistent for all the frequencies in the several examples we tested. The third method predicts the same result as the first two for most of the frequencies, but sometimes it can be different by ±1 at the edge of phase transition frequencies in resonant regimes. Note that before applying Eqs. (18), (19), or (20) to resolve the correct phase branch, the ± sign in front of the inverse cosine in Eq. (14) must be determined first. From the several examples we tested, using above branch-resolving techniques, there is no need to recourse adjacent frequencies in determining the correct phase branch. Recoursing adjacent frequencies can become confusing when both positive and negative refractive indices are present in the same frequency band.

## 5. Discussion

*X*variable in Eq. (12) is real; whereas the slops and Y-intercepts are usually complex. The real and imaginary parts of the retrieval lines should be plotted separately as shown in Fig. 2 which was calculated from the scattering matrix. Drude model is used for the effective material parameters of the equivalent layers A and B,

*f*= 30THz,

_{ep}*f*= 20THz, and

_{mp}*γ*= 3THz. The

*ε*and

_{x}*µ*are described by Eq. (21) with resonances at

_{x}*f*= 20THz and

_{er}*f*= 25THz for the layer A and at

_{mr}*f*= 35THz and

_{er}*f*= 37THz for the layer B. The other parameters for A:

_{mr}*ε*=

_{y}*ε*− 0.3,

_{x}*ε*=

_{z}*ε*+ 2,

_{x}*µ*=

_{y}*µ*− 0.5, and

_{x}*µ*= 1. The other parameters for B:

_{z}*ε*=

_{y}*ε*− 0.8,

_{x}*ε*=

_{z}*ε*− 0.5,

_{x}*µ*=

_{y}*µ*+ 0.2, and

_{x}*µ*=

_{z}*µ*− 0.6. The thickness is 240nm for the layer A and 320nm for the layer B. Thus, the period of the unit cell (

_{x}*ABA*) is 800nm. Figure 3 shows the correct phase branch

*m*predicted for each frequency in the regime of interest for the thickness of one (a) and six (b) periods. In our example since

*d*≪

*λ*, for one unit-cell thickness, most of the frequencies are within the fundamental branch

*m*= 0 except for the regime of negative index of refraction [see Fig. 4(a)] where

*m*= −1. For the six-period thickness, the phase branch jumps to

*m*= 1 in the frequency range 18 ~ 20THz due to the high valves of the positive refractive index in this regime[see Fig. 4(a)].

17. M. J. Roberts, S. Feng, M. Moran, and L. Johnson, “Effective permittivity near zero in nanolaminates of silver and amorphous polycarbonate,” J. Nanophoton. **4**, 043511 (2010). [CrossRef]

*ε*≠ 0 and

_{xy}*µ*≠ 0), and thus the x and y components of the electromagnetic fields will be coupled. The dispersion relations in Eq. (5) will no longer be valid. Without the proper modification, the current parameter retrieval scheme cannot be applied to this scenario. The original classification of the left-handed and right-handed electromagnetic materials can cause confusion with chiral materials which are important class of electromagnetic materials [18

_{xy}18. M. W McCall, A. Lakhtakia, and W. S Weiglhofer, “The negative index of refraction demystified,” Eur. J. Phys. **23**, 353–359 (2002). [CrossRef]

## 6. Conclusions

## Acknowledgments

## References and links

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

2. | J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. |

3. | S. Feng and K. Halterman, “Parametrically shielding electromagnetic fields by nonlinear metamaterials,” Phys. Rev. Lett. |

4. | F. Capolino Ed. |

5. | J. Valentine, S. Zhang, T. Zentgraf, E. U. Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature (London) |

6. | N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. |

7. | S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. |

8. | V. M. Shalaev, W. 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. |

9. | A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, “Wave propagation retrieval method for metamaterials: Unambiguous restoration of effective parameters,” Phys. Rev. B |

10. | A. Herpin, “Calcul du pouvoir réflecteur d’un système stratifié quelconque,” Compt. Rend. |

11. | D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B |

12. | T. Koschny, P. Markoš, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials,” Phys. Rev. B |

13. | C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B |

14. | X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, Jr., and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E |

15. | L. I. Epstein, “The design of optical filters,” J. Opt. Soc. Am. |

16. | A. Lakhtakia, “Constraints on effective constitutive parameters of certain bianisotropic laminated composite materials,” Electromagnetics |

17. | M. J. Roberts, S. Feng, M. Moran, and L. Johnson, “Effective permittivity near zero in nanolaminates of silver and amorphous polycarbonate,” J. Nanophoton. |

18. | M. W McCall, A. Lakhtakia, and W. S Weiglhofer, “The negative index of refraction demystified,” Eur. J. Phys. |

**OCIS Codes**

(310.3840) Thin films : Materials and process characterization

(160.3918) Materials : Metamaterials

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Metamaterials

**History**

Original Manuscript: June 7, 2010

Revised Manuscript: June 21, 2010

Manuscript Accepted: July 6, 2010

Published: July 26, 2010

**Citation**

Simin Feng, "Graphical retrieval method for
orthorhombic anisotropic materials," Opt. Express **18**, 17009-17019 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-17009

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### References

- N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-diffraction-limited optical imaging with a silver superlens," Science 308, 534-537 (2005). [CrossRef] [PubMed]
- J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nat. Mater. 8, 568-571 (2009). [CrossRef] [PubMed]
- S. Feng and K. Halterman, "Parametrically shielding electromagnetic fields by nonlinear metamaterials," Phys. Rev. Lett. 100, 063901 (2008). [CrossRef] [PubMed]
- F. Capolino, ed., Theory and phenomena of metamaterials (CRC Press, Taylor and Francis Group, New York, 2009). [CrossRef]
- J. Valentine, S. Zhang, T. Zentgraf, E. U. Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 376-380 (2008). [CrossRef] [PubMed]
- N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, "Three-dimensional photonic metamaterials at optical frequencies," Nat. Mater. 7, 31-37 (2008). [CrossRef]
- S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett. 95, 137404 (2005). [CrossRef] [PubMed]
- V. M. Shalaev,W. 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, 3356-3358 (2005). [CrossRef]
- A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, "Wave propagation retrieval method for metamaterials: Unambiguous restoration of effective parameters," Phys. Rev. B 80, 193101 (2009). [CrossRef]
- A. Herpin, "Calcul du pouvoir réflecteur d’un système stratifié quelconque," Compt. Rend. 225, 182-183 (1947).
- D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002). [CrossRef]
- T. Koschny, P. Markoš, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, "Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials," Phys. Rev. B 71, 245105 (2005). [CrossRef]
- C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, "Retrieving effective parameters for metamaterials at oblique incidence," Phys. Rev. B 77, 195328 (2008). [CrossRef]
- X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, Jr., and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Phys. Rev. E 70, 016608 (2004). [CrossRef]
- L. I. Epstein, "The design of optical filters," J. Opt. Soc. Am. 42, 806-810 (1952). [CrossRef]
- A. Lakhtakia, "Constraints on effective constitutive parameters of certain bianisotropic laminated composite materials," Electromagnetics 29,508-514 (2009). [CrossRef]
- M. J. Roberts, S. Feng, M. Moran, and L. Johnson, "Effective permittivity near zero in nanolaminates of silver and amorphous polycarbonate," J. Nanophoton. 4, 043511 (2010). [CrossRef]
- M. W McCall, A. Lakhtakia, and W. S Weiglhofer, "The negative index of refraction demystified," Eur. J. Phys. 23, 353-359 (2002). [CrossRef]

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