## Frequency converter implementing an optical analogue of the cosmological redshift

Optics Express, Vol. 18, Issue 5, pp. 5350-5355 (2010)

http://dx.doi.org/10.1364/OE.18.005350

Acrobat PDF (852 KB)

### Abstract

According to general relativity, the frequency of electromagnetic radiation is altered by the expansion of the universe. This effect—commonly referred to as the cosmological redshift—is of utmost importance for observations in cosmology. Here we show that this redshift can be reproduced on a much smaller scale using an optical analogue inside a dielectric metamaterial with time-dependent material parameters. To this aim, we apply the framework of transformation optics to the Robertson-Walker metric. We demonstrate theoretically how perfect redshifting or blueshifting of an electromagnetic wave can be achieved without the creation of sidebands with a device of finite length.

© 2010 Optical Society of America

## 1. Introduction

9. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**, 1780–1782 (2006). [CrossRef] [PubMed]

10. U. Leonhardt, “Optical conformal mapping,” Science **312**, 1777–1780 (2006). [CrossRef] [PubMed]

13. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nature Photon. **1**, 224–227 (2007). [CrossRef]

14. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nature Mater. **8**, 568–571 (2009). [CrossRef]

15. S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, J. Pendry, M. Rahm, and A. Starr, “Scattering theory derivation of a 3D acoustic cloaking shell,” Phys. Rev. Lett. **100**, 024301 (2008). [CrossRef] [PubMed]

16. M. Farhat, S. Enoch, S. Guenneau, and A. B. Movchan, “Broadband cylindrical acoustic cloak for linear surface waves in a fluid,” Phys. Rev. Lett. **101**, 134501 (2008). [CrossRef] [PubMed]

17. S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. **100**, 123002 (2008). [CrossRef] [PubMed]

## 2. Electromagnetic analogue of the cosmological redshift

9. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**, 1780–1782 (2006). [CrossRef] [PubMed]

11. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. **8**, 247–264 (2006). [CrossRef]

*κ*= 0) can be translated into an isotropic, homogeneous dielectric with constitutive parameters

*c*is the speed of light in vacuum. Its solutions are waves with time-dependent amplitude propagating along an arbitrary direction labeled by the unit vector

**1**

_{k}:

*G*(

*x*) is an arbitrary differentiable function. To gain some insight in these traveling waves, we consider a harmonic function

*G*(

*x*) = cos(

*x*) in Fig. 1 and we plot the temporal and spatial variation of Eq. (5). We see that the wave has a well-defined wavelength, but that its frequency is changing in time. A straightforward calculation of the instantaneous frequency

*ω*

_{inst}(

*t*) = −∂

_{t}(

**k**·

**r**− |

**k**|

*c*∫

_{t0}

^{t}d

*t̃*/

*a*(

*t̃*)) reveals that the frequencies at two different times

*t*

_{1}and

*t*

_{2}satisfy

*ω*

_{inst}(

*t*

_{2})/

*ω*

_{inst}(

*t*

_{1}) =

*a*(

*t*

_{1})/

*a*(

*t*

_{2}), in agreement with the redshift formula of Eq. (2).

## 3. A frequency converter of finite size

22. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Applic. **6**, 87–95 (2008). [CrossRef]

*L*(see Fig. 2) would suit as a frequency converter. To simplify notation, we introduce the function

*ω*

_{0}= |

**k**

_{0}|

*c*, and we want to calculate the wave that is emitted in region (III). In general, each interface would give rise to transmitted and reflected waves, and one would explicitly impose continuity of the tangential components of both

**E**and

**H**. However, since the impedance

*η*= (

*μ*/

*ε*)

^{1/2}is the same on both sides of each interface, we can anticipate that there are no reflected waves. We therefore restrict our attention to the electric field

**E**, consider purely right-moving waves and impose continuity at each interface. In regions (I), (II), and (III), respectively, the electric field can be written as

*G*(

*x*) and

*H*(

*x*) are differentiable functions. The continuity of the tangential component of the electric field at

*z*= 0 yields

*G*(

*x*) for an arbitrary value of its argument

*x*:

*f*

^{−1}. In Fig. 3, we plot the evolution of the wavefronts inside the material as a function of space and time. We can now combine Eq. (7) with Eq. (11) to determine the electric field at the rightmost boundary (

*z*=

*L*) of the device:

*z*=

*L*) can be written as

*L*= ∫

_{t1}

^{t2}

*c*/

*a*(

*t*) d

*t*=

*c*(

*f*(

*t*

_{2}) −

*f*(

*t*

_{1})), where

*t*

_{1}and

*t*

_{2}indicate the time of incidence and departure of a wavefront. If we are observing at time

*t*at the rightmost boundary, then

*f*

^{−1}(

*f*(

*t*) −

*L*/

*c*) corresponds to the time when the wavefront was at the leftmost boundary. Hence, we retrieve the cosmological redshift formula,

*ω*

_{inst}(

*t*

_{2})/

*ω*

_{inst}(

*t*

_{1}) =

*a*(

*t*

_{1})/

*a*(

*t*

_{2}). The reader should also note that the amplitude is altered with a similar factor, so that the device might also be used as an optical amplifier. It will often be beneficial to create a constant frequency shift, i.e., to generate a monochromatic wave in region (III); in this case, the right hand side of Eq. (14) has to be constant. By renaming

*f*(

*t*) =

*x*, this condition immediately implies that

*a*(

*f*

^{−1}(

*x*)) = exp(

*αx*)

*p*(

*x*), where

*α*is an arbitrary (real) constant and

*p*(

*x*) is a periodic function with period

*L*/

*c*. Using Eq. (6), we observe that

*a*(

*f*

^{−1}(

*x*)) = (

*f*

^{−1})′(

*x*) and infer that the condition is satisfied when

*f*

^{−1}(

*x*) =exp(

*αx*)

*q*(

*x*) −

*β*with

*β*constant and

*q*(

*x*) periodic with period

*L*/

*c*. By inverting this relation to obtain

*f*(

*t*) and differentiating with respect to

*t*, we can calculate those

*a*(

*t*) which generate a monochromatic wave in region (III). For instance, the special case

*q*(

*x*) = 1 leads to

*f*(

*t*) = ln(

*t*+

*β*)/

*α*and thus to the linear profile

*a*(

*t*) =

*α*(

*t*+

*β*). The resulting frequency shift is then given by

## 4. Conclusion and discussion

*linear*materials, so that the superposition principle remains valid and an arbitrary wavepacket can be frequency-shifted. This frequency shift is possible due to the time evolution of its parameters, which renders it non-stationary [23

23. N. V. Budko, “Electromagnetic radiation in a time-varying background medium,” Phys. Rev. A **80**, 053817 (2009). [CrossRef]

*f*

_{0}= 100 THz by 10 GHz in a material of thickness

*L*= 2

*λ*

_{0}= 6 μm. We then find that the modulation rate of the refractive index must equal

*α*≈ 5 GHz. Since the corresponding wavelength of this modulating signal is much larger than the dimensions of the device, we can generate the permittivity variation by low-frequency electro-optic modulation. The most difficult part is the time evolution of the permeability. This can be achieved by introducing an array of split-ring resonators [24

24. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science **305**, 788–792 (2004). [CrossRef] [PubMed]

25. C. M. Soukoulis, M. Kafesaki, and E. N. Economou, “Negative-index materials: New frontiers in optics,” Adv. Mater. **18**, 1941–1952 (2005). [CrossRef]

*F*is the filling factor and

*ω*

_{LC}is the resonance frequency, which is inversely proportional to the square root of the material’s permittivity

*ε*

_{C}inside the gap of the split rings. Operating far below resonance, we can approximate the permeability by

*μ*(

*ω*) ≈ 1+

*Fω*

^{2}

*c*

^{2}

*l*

^{2}

*ε*

_{C}/(

*dw*), with

*l*,

*d*,

*w*the geometrical parameters of the split rings [25

25. C. M. Soukoulis, M. Kafesaki, and E. N. Economou, “Negative-index materials: New frontiers in optics,” Adv. Mater. **18**, 1941–1952 (2005). [CrossRef]

## Acknowledgments

## References and links

1. | T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. Konig, and U. Leonhardt, “Fiber-Optical Analog of the Event Horizon,” Science |

2. | D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nature Phys. |

3. | E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidirectional light absorber,” Appl. Phys. Lett. |

4. | C. Qiang and C. T. Jun, “An electromagnetic black hole made of metamaterials,” arXiv:0910.2159v1 [physics.optics ] (2009). |

5. | N. L. Balazs, “Effect of a gravitational field, due to a rotating body, on the plane of polarization of an electromagnetic wave,” Phys. Rev. |

6. | J. Plebanski, “Electromagnetic waves in gravitational fields,” Phys. Rev. |

7. | D. F. Felice, “On the gravitational field acting as an optical medium,” Gen. Rel. Grav. |

8. | A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Phys. |

9. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science |

10. | U. Leonhardt, “Optical conformal mapping,” Science |

11. | U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. |

12. | U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. |

13. | W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nature Photon. |

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

15. | S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, J. Pendry, M. Rahm, and A. Starr, “Scattering theory derivation of a 3D acoustic cloaking shell,” Phys. Rev. Lett. |

16. | M. Farhat, S. Enoch, S. Guenneau, and A. B. Movchan, “Broadband cylindrical acoustic cloak for linear surface waves in a fluid,” Phys. Rev. Lett. |

17. | S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. |

18. | M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express |

19. | D. Kwon and D. H. Werner, “Polarization splitter and polarization rotator designs based on transformation optics,” Opt. Express |

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

21. | S. Carroll, |

22. | M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Applic. |

23. | N. V. Budko, “Electromagnetic radiation in a time-varying background medium,” Phys. Rev. A |

24. | D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science |

25. | C. M. Soukoulis, M. Kafesaki, and E. N. Economou, “Negative-index materials: New frontiers in optics,” Adv. Mater. |

**OCIS Codes**

(060.2630) Fiber optics and optical communications : Frequency modulation

(260.2110) Physical optics : Electromagnetic optics

(350.5720) Other areas of optics : Relativity

(160.3918) Materials : Metamaterials

**ToC Category:**

Physical Optics

**History**

Original Manuscript: February 1, 2010

Manuscript Accepted: February 22, 2010

Published: February 26, 2010

**Citation**

Vincent Ginis, Philippe Tassin, Ben Craps, and Irina Veretennicoff, "Frequency converter implementing an optical analogue of the cosmological redshift," Opt. Express **18**, 5350-5355 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-5-5350

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### References

- T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. Konig, and U. Leonhardt, "Fiber-Optical Analog of the Event Horizon," Science 319, 1367-1370 (2008). [CrossRef] [PubMed]
- D. A. Genov, S. Zhang, and X. Zhang, "Mimicking celestial mechanics in metamaterials," Nature Phys. 5, 687-692 (2009). [CrossRef]
- E. E. Narimanov and A. V. Kildishev, "Optical black hole: Broadband omnidirectional light absorber," Appl. Phys. Lett. 95, 041106 (2009). [CrossRef]
- C. Qiang and C. T. Jun, "An electromagnetic black hole made of metamaterials," arXiv:0910.2159v1 [physics.optics] (2009).
- N. L. Balazs, "Effect of a gravitational field, due to a rotating body, on the plane of polarization of an electromagnetic wave," Phys. Rev. 110, 236-239 (1957). [CrossRef]
- J. Plebanski, "Electromagnetic waves in gravitational fields," Phys. Rev. 118, 1396-1408 (1960). [CrossRef]
- D. F. Felice, "On the gravitational field acting as an optical medium," Gen. Rel. Grav. 2, 347-357 (1971). [CrossRef]
- A. J. Ward and J. B. Pendry, "Refraction and geometry in Maxwell’s equations," J. Mod. Phys. 43, 773-793 (1996).
- J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
- U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006). [CrossRef] [PubMed]
- U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," New J. Phys. 8, 247-264 (2006). [CrossRef]
- U. Leonhardt and T. G. Philbin, "Transformation optics and the geometry of light," Prog. Opt. 53, 70-152 (2009).
- W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nature Photon. 1, 224-227 (2007). [CrossRef]
- J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nature Mater. 8, 568-571 (2009). [CrossRef]
- S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, J. Pendry, M. Rahm, and A. Starr, "Scattering theory derivation of a 3D acoustic cloaking shell," Phys. Rev. Lett. 100, 024301 (2008). [CrossRef] [PubMed]
- M. Farhat, S. Enoch, S. Guenneau, and A. B. Movchan, "Broadband cylindrical acoustic cloak for linear surface waves in a fluid," Phys. Rev. Lett. 101, 134501 (2008). [CrossRef] [PubMed]
- S. Zhang, D. A. Genov, C. Sun, and X. Zhang, "Cloaking of matter waves," Phys. Rev. Lett. 100, 123002 (2008). [CrossRef] [PubMed]
- M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, "Transformation-optical design of adaptive beam bends and beam expanders," Opt. Express 16, 11555-11567 (2008). [CrossRef] [PubMed]
- D. Kwon and D. H. Werner, "Polarization splitter and polarization rotator designs based on transformation optics," Opt. Express 16, 18731-18738 (2008). [CrossRef]
- Z. Jacob, L. V. Alekseyev, and E. Narimanov, "Optical hyperlens: Far-field imaging beyond the diffraction limit," Opt. Express 14, 8247-8256 (2008). [CrossRef]
- S. Carroll, Spacetime and Geometry (Addison Wesley, New York, 2003).
- M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, "Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations," Photon. Nanostruct.: Fundam. Applic. 6, 87-95 (2008). [CrossRef]
- N. V. Budko, "Electromagnetic radiation in a time-varying background medium," Phys. Rev. A 80, 053817 (2009). [CrossRef]
- D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004). [CrossRef] [PubMed]
- C. M. Soukoulis, M. Kafesaki, and E. N. Economou, "Negative-index materials: New frontiers in optics," Adv. Mater. 18, 1941-1952 (2005). [CrossRef]

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