## Negative refraction and backward wave in pseudochiral mediums: illustrations of Gaussian beams |

Optics Express, Vol. 21, Issue 3, pp. 2657-2666 (2013)

http://dx.doi.org/10.1364/OE.21.002657

Acrobat PDF (1534 KB)

### Abstract

We investigate the phenomena of negative refraction and backward wave in pseudochiral mediums, with illustrations of Gaussian beams. Due to symmetry breaking intrinsic in pseudochiral mediums, there exist two elliptically polarized eigenwaves with different wave vectors. As the chirality parameter increases from zero, the two waves begin to split from each other. For a wave incident from vacuum onto a pseudochiral medium, negative refraction may occur for the right-handed wave, whereas backward wave may appear for the left-handed wave. These features are illustrated with Gaussian beams based on Fourier integral formulations for the incident, reflected, and transmitted waves. Negative refraction and backward wave are manifest, respectively, on the energy flow in space and wavefront movement in time.

© 2013 OSA

## 1. Introduction

6. W. S. Weiglhofer and A. Lakhtakia, *Introduction to Complex Mediums for Optics and Electromagnetics* (SPIE, 2003). [CrossRef]

7. P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. **37**, 259–263 (2003). [CrossRef]

9. H. Chen, S. Xu, and J. Li, “Negative reflection of waves at planar interfaces associated with a uniaxial medium,” Opt. Lett. **34**, 3283–3285 (2009). [CrossRef] [PubMed]

10. J. B. Pendry, “A chiral route to negative refraction,” Science **306**, 1353–1355 (2004). [CrossRef] [PubMed]

12. S. Tretyakov, A. Sihvola, and L. Jylha, “Backward-wave regime and negative refraction in chiral composites,” Photonics Nanostruct. **3**, 107–115 (2005). [CrossRef]

13. Q. Cheng and T. J. Cui, “Negative refractions in uniaxially anisotropic chiral media,” Phys. Rev. B **73**, 113104 (2006). [CrossRef]

14. S. A. Tretyakov, C. R. Simovski, and M. Hudlicka, “Bianisotropic route to the realization and matching of backward-wave metamaterial slabs,” Phys. Rev. B **75**, 153104 (2007). [CrossRef]

15. B. R. Horowitz and T. Tamir, “Lateral Displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am. **61**, 586–594 (1971). [CrossRef]

## 2. Basic equations

### 2.1. Eigenwaves

*complex*mediums [6

6. W. S. Weiglhofer and A. Lakhtakia, *Introduction to Complex Mediums for Optics and Electromagnetics* (SPIE, 2003). [CrossRef]

*ε*,

*μ*,

*ξ*, and

*ζ*are in general tensors of complex quantities. In particular, the magneto-electric tensors for pseudochiral mediums have the following forms: with

*γ*being a real quantity. Consider the case where the anisotropy of the permittivity and permeability be small compared with the effect of chirality, and the material parameters be approximated by

*ε*=

*ε*

*I*and

*μ*=

*μ*

*I*, where

*I*is the identity tensor. This is the case when the magnetoelectric coupling is much more significant than the dielectric-magnetic property of the medium. The dispersion relation for pseudochiral mediums is given as (see Appendix A for details) where

*ε*> 0,

*μ*> 0, and

*γ*

^{2}<

*ε μ*. Therefore, 0 <

*γ̃*< 1 and 0 <

*ρ*< 1. For a single frequency

*ω*, there exist two wave numbers

*k*= |

^{±}**k**

*| in the pseudochiral medium.*

^{±}*xy*plane be an interface between vacuum and a pseudochiral medium. Consider a wave incident from vacuum with the wave vector lying on the

*xz*plane. For a given tangential wave vector component

*k*, there are two solutions for the normal wave vector component [cf. Eq. (4)]: where

_{x}***denotes the complex conjugate, are given as The angles of wave vectors

**k**

*and Poynting vectors 〈*

^{±}**S**

*〉 with respect to the interface normal (the*

^{±}*z*axis) are given, respectively, as For a wave incident from vacuum onto a pseudochiral medium at an angle of incidence

*θ*, negative refraction will occur for the REP wave (

*ϕ*

^{+}< 0°) when

*θ*<

*θ*

_{NR}, where is the threshold angle for negative refraction. Note that when |

*k*| =

_{x}*γ̃h*

_{0}, the time-averaged Poynting vector 〈

**S**

^{+}〉 is directed to the interface normal (

*σ*

^{+}= 0) [cf. Eq. (8)]. On the other hand, backward wave will occur for the LEP wave (

*θ*

^{−}> 90°) when

*θ*>

*θ*

_{BW}, where is the threshold angle for backward wave. Note that when |

*k*| =

_{x}*ρh*

_{0}, the wave vector

**k**

^{−}is oriented along the interface (

### 2.2. Gaussian beams

*xy*plane is an interface between vacuum on the left (

*z <*0) and a pseudochiral medium on the right (

*z*> 0), characterized by the permittivity

*ε*, permeability

*μ*, and the chirality parameter

*γ*. Consider a Gaussian beam incident from vacuum, with the beam center making an angle

*θ*with respect to the interface normal, as schematically shown in Fig. 1. Let the

*xz*plane be the plane of incidence and

*k*be the wave vector component along the interface. The Gaussian beam with the center located at

_{x}*x*=

*x*

_{0}and

*z*= −

*h*is well approximated by the Fourier integral on

*k*as [15

_{x}15. B. R. Horowitz and T. Tamir, “Lateral Displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am. **61**, 586–594 (1971). [CrossRef]

*w*

_{0}is the waist size of the Gaussian beam. Based on this formulation, the incident beams are formulated as where

*p*- or

*s*-polarized incident waves,

*R*are the reflection coefficients, given as

^{p,s}*r*,

_{pp}*r*,

_{ps}*r*, and

_{sp}*r*are listed in Appendix B.

_{ss}*k*[cf. Eq. (4)], corresponding to the REP and LEP waves. The transmitted beams are formulated as where

^{±}*T*are the transmission coefficients, given as

^{±}*t*

_{+p},

*t*

_{+s},

*t*

_{−p}, and

*t*

_{−s}are listed in Appendix B.

*R*are the reflection coefficients, given as

^{±}*r*

_{++},

*r*

_{+−,}

*r*

_{−+}, and

*r*

_{−−}are listed in Appendix B. The corresponding transmitted beams have the same forms as Eqs. (18) and (19), except that the transmission coefficients are changed to

*t*

_{++},

*t*

_{+−},

*t*

_{−+}, and

*t*

_{−−}are listed in Appendix B.

### 2.3. Fourier transforms

*z*= 0),

*ϕ*(

*k*) is the function of

_{x}*k*that contains the components of basis vectors (

_{x}*R*,

^{p,s}*R*,

^{±}*T*),

^{±}*q*=

_{z}*k*(−

_{z}*k*) for the incident (reflected) beam, and

_{z}*f*

_{0}(

*x*)] is to obtain the spectral components of the fields at the interface, and IFT is to restore these components to the physical quantity

*f*(

*x,z*), taking into account the wave propagation in the

*z*direction and the influence of the interface (reflection and transmission). In practice, the calculations of FT and IFT can be significantly accelerated by employing the FFT algorithm.

*I*= |〈

**S**〉|, where

**E**and total magnetic field

**H**in vacuum or the pseudochiral medium.

## 3. Results and discussion

*p*-polarized (TM) Gaussian beam incident from vacuum onto a pseudochiral medium with

*ε/ε*

_{0}=

*ε*= 2,

_{r}*μ/μ*

_{0}=

*μ*= 1, and

_{r}*γ*= 0.2. The incident beam center makes an angle

*θ*= 20° with respect to the interface normal (the

*z*axis). There are two transmitted beams in the pseudochiral medium, one is the REP wave with a smaller angle of the beam center, and the other is the LEP wave with a larger angle. The beam centers coincide with the time-averaged Poynting vectors 〈

**S**

*〉 [cf. Eq. (8)], with*

^{±}*ϕ*

^{+}≈ 6° and

*ϕ*

^{−}≈ 21.2°. Note that in this case,

*θ > θ*

_{NR}≈ 11.5° [cf. Eq. (10)] and ordinary refraction is expected to occur. As the chirality parameter

*γ*increases, the REP beam tends to move toward the interface, whereas the LEP beam tends to move away from it. In this configuration, the reflection coefficients

*r*and

_{pp}*r*are relatively small. Compared to the incident beam, the intensity of the reflected beam is insignificant.

_{sp}*γ*= 0.8, with the other material parameters unchanged. The REP (lower) beam is located on the same side of the incident beam, with a negative angle of refraction:

*ϕ*

^{+}≈ −19.8°. Note that in this case,

*θ < θ*

_{NR}≈ 53.1° [cf. Eq. (10)] and negative refraction is expected to occur. The LEP (upper) beam has an angle of refraction:

*ϕ*

^{−}≈ 37.6°, which is larger than that in Fig. 2(a). For a

*s*-polarized incident Gaussian beam, the features stated above are similar.

*θ*= 20° onto a pseudochiral medium with the same material parameters as in Fig. 2(b). In this case, the transmission is dominated by the REP beam, which is negatively refracted with the same

*ϕ*

^{+}as in Fig. 2(b). Compared to the REP beam, the power intensity of the LEP beam is not significant (0.037%). In Fig. 3(b), the incident beam is changed to a LCP Gaussian beam. The transmission, on the other hand, is dominated by the LEP beam, which is positively refracted with the same

*ϕ*

^{−}as in Fig. 2(b). Compared to the LEP beam, the power intensity of the REP beam is not significant (0.037%).

*θ*= 0° onto a pseudochiral medium is shown in Fig. 4(a). The transmitted wave is split into two beams even at normal incidence. This feature does not appear in either isotropic chiral or anisotropic dielectric materials, and can be manifest on the time-averaged Poynting vectors 〈

**S**

^{±}〉 at

*k*= 0. In this situation,

_{x}*σ*

^{±}= ∓

*γ̃*[cf. Eq. (7)] and

*ϕ*

^{±}= arctan(∓

*γ̃*) [cf. Eq. (9)]. In the presence of chirality parameter

*γ̃*,

*ϕ*

^{+}≠

*ϕ*

^{−}at

*θ*= 0°. The effect of

*γ̃*on

*ϕ*is plotted in Fig. 4(b). The splitting angle between the two transmitted beams increases with

^{±}*γ̃*. This feature is further analyzed by examining the basis wave functions that compose the Gaussian beams: Using Eq. (5) for

*x*) = erf (

*ix*)/

*i*is the imaginary error function. In Eq. (29),

*f*(

^{±}*x,z*) attain their local maximal values along

*x*=

*±γ̃z*, which are the locations of beam centers at

*θ*= 0°.

*s*-polarized Gaussian beam incident from vacuum at

*θ*= 25° onto the pseudochiral medium with

*ε*= 1,

_{r}*μ*= 1, and

_{r}*γ̃*= 0.95. In this case, the angles of wave vectors with respect to the interface normal for the REP and LEP waves are

*θ*

^{+}≈ 31.7° and

*θ*

^{−}≈ 105.7°, respectively. The latter represents a backward wave. The wavefronts of the LEP (upper) beam move toward the interface rather than away from it, which is indicated by the orientation of

**k**

^{−}. The angles of time-averaged Poynting vectors with respect to the interface normal for the REP and LEP waves are

*ϕ*

^{+}≈ −38.8° and

*ϕ*

^{−}≈ 47.6°, respectively. The former corresponds to negative refraction. Note that the threshold angles are

*θ*

_{NR}≈ 71.8° and

*θ*

_{BW}≈ 18.2° [cf. Eqs. (10) and (11)]. Accordingly, both negative refraction and backward wave occur in the same configuration.

## 4. Concluding remarks

## A. Dispersion relation and eigenwaves

*×*

**E**=

*iω*

**B**, ∇ ×

**H**= −

*iω*

**D**and eliminate

**D**and

**B**, we may obtain the following separate equations for

**E**and

**H**fields: which are regarded as the wave equations for bianisotropic mediums. Assume that

**E**and

**H**are of the form

*e*

^{ik·r}, the wave equations are rewritten as

*M*

**E**= 0 and

*N*

**H**= 0, where with

*I*being the identity tensor. The existence of nontrivial solutions for

**E**and

**H**fields requires that |

*M*| = 0 and |

*N*| = 0. Using

**k**= (

*k*,

_{x}*k*,

_{y}*k*) and the chirality parameters (3), the zero determinant gives rise to the

_{z}*characteristic equation*: where

*dispersion relation*: The nullspace of

*M*or

*N*gives the eigenwaves as where

## B. Reflection and transmission coefficients

## Acknowledgments

## References and links

1. | J. A. Kong, “Theorems of bianisotropic media,” Proc. IEEE |

2. | A. Serdyukov, I. Semchenko, S. Tretyakov, and A. Sihvola, |

3. | S. A. Tretyakov, A. H. Sihvola, A. A. Sochava, and C. R. Simovski, “Magnetoelectric interactions in bi-anisotropic media,” J. Electromagn. Waves Appl. |

4. | M. M. I. Saadoun and N. Engheta, “A reciprocal phase shifter using novel pseudochiral or s medium,” Microwave and Optical Technology Letters |

5. | S. A. Matos, C. R. Paiva, and A. M. Barbosa, “Surface and proper leaky-modes in a lossless grounded pseudochiral omega slab,” Microw. Opt. Technol. Lett. |

6. | W. S. Weiglhofer and A. Lakhtakia, |

7. | P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. |

8. | L. Yonghua, W. Pei, Y. Peijun, X. Jianping, and M. Hai, “Negative refraction at the interface of uniaxial anisotropic media,” Opt. Comm. |

9. | H. Chen, S. Xu, and J. Li, “Negative reflection of waves at planar interfaces associated with a uniaxial medium,” Opt. Lett. |

10. | J. B. Pendry, “A chiral route to negative refraction,” Science |

11. | C. Monzon and D. W. Forester, “Negative refraction and focusing of circularly polarized waves in optically active media,” Phys. Rev. Lett. |

12. | S. Tretyakov, A. Sihvola, and L. Jylha, “Backward-wave regime and negative refraction in chiral composites,” Photonics Nanostruct. |

13. | Q. Cheng and T. J. Cui, “Negative refractions in uniaxially anisotropic chiral media,” Phys. Rev. B |

14. | S. A. Tretyakov, C. R. Simovski, and M. Hudlicka, “Bianisotropic route to the realization and matching of backward-wave metamaterial slabs,” Phys. Rev. B |

15. | B. R. Horowitz and T. Tamir, “Lateral Displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am. |

**OCIS Codes**

(120.5710) Instrumentation, measurement, and metrology : Refraction

(230.1360) Optical devices : Beam splitters

(160.1585) Materials : Chiral media

**ToC Category:**

Physical Optics

**History**

Original Manuscript: November 29, 2012

Revised Manuscript: January 9, 2013

Manuscript Accepted: January 9, 2013

Published: January 28, 2013

**Citation**

Ruey-Lin Chern and Po-Han Chang, "Negative refraction and backward wave in pseudochiral mediums: illustrations of Gaussian beams," Opt. Express **21**, 2657-2666 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-2657

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### References

- J. A. Kong, “Theorems of bianisotropic media,” Proc. IEEE60, 1036–1046 (1972). [CrossRef]
- A. Serdyukov, I. Semchenko, S. Tretyakov, and A. Sihvola, Electromagnetics of Bi-anisotropic Materials: Theory and Applications (Gordon and Breach, 2001).
- S. A. Tretyakov, A. H. Sihvola, A. A. Sochava, and C. R. Simovski, “Magnetoelectric interactions in bi-anisotropic media,” J. Electromagn. Waves Appl.12, 481–497 (1998). [CrossRef]
- M. M. I. Saadoun and N. Engheta, “A reciprocal phase shifter using novel pseudochiral or s medium,” Microwave and Optical Technology Letters5, 184–188 (1992). [CrossRef]
- S. A. Matos, C. R. Paiva, and A. M. Barbosa, “Surface and proper leaky-modes in a lossless grounded pseudochiral omega slab,” Microw. Opt. Technol. Lett.50, 814–818 (2008). [CrossRef]
- W. S. Weiglhofer and A. Lakhtakia, Introduction to Complex Mediums for Optics and Electromagnetics (SPIE, 2003). [CrossRef]
- P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett.37, 259–263 (2003). [CrossRef]
- L. Yonghua, W. Pei, Y. Peijun, X. Jianping, and M. Hai, “Negative refraction at the interface of uniaxial anisotropic media,” Opt. Comm.246, 429–435 (2005). [CrossRef]
- H. Chen, S. Xu, and J. Li, “Negative reflection of waves at planar interfaces associated with a uniaxial medium,” Opt. Lett.34, 3283–3285 (2009). [CrossRef] [PubMed]
- J. B. Pendry, “A chiral route to negative refraction,” Science306, 1353–1355 (2004). [CrossRef] [PubMed]
- C. Monzon and D. W. Forester, “Negative refraction and focusing of circularly polarized waves in optically active media,” Phys. Rev. Lett.95, 123904 (2005). [CrossRef] [PubMed]
- S. Tretyakov, A. Sihvola, and L. Jylha, “Backward-wave regime and negative refraction in chiral composites,” Photonics Nanostruct.3, 107–115 (2005). [CrossRef]
- Q. Cheng and T. J. Cui, “Negative refractions in uniaxially anisotropic chiral media,” Phys. Rev. B73, 113104 (2006). [CrossRef]
- S. A. Tretyakov, C. R. Simovski, and M. Hudlicka, “Bianisotropic route to the realization and matching of backward-wave metamaterial slabs,” Phys. Rev. B75, 153104 (2007). [CrossRef]
- B. R. Horowitz and T. Tamir, “Lateral Displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am.61, 586–594 (1971). [CrossRef]

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