## The temporal coherence improvement of the two-dimensional negative-index slab image

Optics Express, Vol. 14, Issue 25, pp. 12295-12301 (2006)

http://dx.doi.org/10.1364/OE.14.012295

Acrobat PDF (385 KB)

### Abstract

In this letter,we investigate the image field of the quasimonochromatic random source in the two-dimensional negative-index slab. The prominent temporal-coherence gain of the image is observed in the numerical simulations even when the frequency-filtering effects are very weak. We find that the signals originating from the source will take the different time—“group” retarded time to reach the image location along the different optical paths. Based on the new physical picture, a simple phenomenological theory is constructed to obtain the image field and demonstrate that the temporal-coherence gain is from different “group” retarded time. The phenomenological theory agrees well with the FDTD simulation and the strict Green’s function method. These results should have important impacts on the study of coherence mechanism and the design of novel devices.

© 2006 Optical Society of America

## 1. Introduction

*ε*and negative permeability

*μ*, could refocus the electro-magnetic (EM) waves [1

1. V. G. Veselago,“The electrodynamics of substances with simultaneously negative values of *ε* and *μ*,” Sov. Phys. Usp. **10**, 509 (1968). [CrossRef]

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

*superlens*which can break through or overcome the diffraction limit of conventional imaging system. After that, several other beyond-limit properties of NIM systems are found, such as the sub-wavelength cavities [22

22. Nader Engheta, “An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability,” IEEE Antennas and Wireless Propagation Lett. **1**, 10, 2002. [CrossRef]

23. Ilya V. Shadrivov, Andrey A. Sukhorukov, and Yuri S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E **67**, 057602 (2003). [CrossRef]

6. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science **292**, 77 (2001). [CrossRef] [PubMed]

13. X. S. Rao and C. K. Ong, “Amplification of evanescent waves in a lossy left-handed material slab,” Phys. Rev. B **68**, 113103 (2003); [CrossRef]

14. Michael W. Feise and Yuri S. Kivshar, “Sub-wavelength imaging with a left-handed material flat lens,” Phys. Lett. A **334**326 (2005) [CrossRef]

15. Lei Zhou and C. T. Chan, “Vortex-like surface wave and its role in the transient phenomena of meta-material focusing,” Appl. Phys. Lett. **86**, 101104 (2005). [CrossRef]

24. B. E. A. Saleh and M. C. Teich, *Fundamentals of Photonics* (John Wiley & Sons, New York, 1991). [CrossRef]

## 2. Our model and the results of FDTD

*d*.

*d*/2 in front of the slab and set as a quasi-monochromatic field which can be expressed as

*E*

_{s}(

*t*)=

*U*

_{s}(

*t*)

*exp*(-

*iω*

_{0}

*t*), where

*U*

_{s}(

*t*) is a slow-varying random function,

*ω*

_{0}=

*π*/(20

*δ*

_{t}) is the central frequency of our random source and

*δ*

_{t}=1.18×10

^{-15}

*s*is the smallest time-step in FDTD simulation. To realize the negative

*ε*and negative

*μ*, the electric polarization density

*P⃗*and the magnetic moment density

*M⃗*are phenomenologically introduced in FDTD simulation [27

27. X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. **85**, 70 (2000). [CrossRef] [PubMed]

*ε*

_{r}(

*ω*)=

*μ*

_{r}(

*ω*)=1+

*ω*

^{2}-

*iγ*). In our model,

*ω*

_{a}=1.884×10

^{13}/

*s*,

*γ*=

*ω*

_{a}/100,

*ω*

_{P}=10×

*ω*

_{a}. At

*ω*

_{0}, we have

*ε*

_{r}=

*μ*

_{r}=-1.00-

*i*0.0029. Here, we emphasize that the distance (

*d*/2=

*λ*

_{0},

*λ*

_{0}is the wavelength in vaccum corresponding to the central frequency

*ω*

_{0}) between the source and the slab is too large to excite strong evanescent modes of NIM in FDTD simulation [13

13. X. S. Rao and C. K. Ong, “Amplification of evanescent waves in a lossy left-handed material slab,” Phys. Rev. B **68**, 113103 (2003); [CrossRef]

14. Michael W. Feise and Yuri S. Kivshar, “Sub-wavelength imaging with a left-handed material flat lens,” Phys. Lett. A **334**326 (2005) [CrossRef]

16. L. Chen, S. He, and L. Shen, “Finite-size effects of a left-handed material slab on the image quality,” Phys. Rev. Lett. **92**, 107404 (2004). [CrossRef] [PubMed]

*Actually the evanescent field in our simulation can be neglected comparing with the propagating field, and what we are studying is the property dominated by the propagating field*.

*φ*

_{i}, the starting time

*t*

_{i}and the random pulse length

*t*

_{pi}, we can represent it with sin(

*ω*

_{0}(

*t*-

*t*

_{i})×

*φ*

_{i}),

*t*

_{i}<

*t*<

*t*

_{i}+

*t*

_{pi}, then our random source is composed of many such sine-wave pulses, i.e. Σ

_{i}sin(

*ω*

_{0}(

*t*-

*t*

_{i})+

*φ*

_{i}), which possesses the average pulse length tp. The sources generated by this way are a quasimonochromatic field with the central frequency

*ω*

_{0}. The larger

*t*

_{p}, the narrower its spectrum width. In the simulation, the fields of the source and the image are recorded for a duration of 4×10

^{5}

*δ*

_{t}to obtain the data for analysis.

*E*(

*ω*)|> as the

*field spectrum*(FS), where

*ω*

_{s}≃

*ω*

_{0}/20). The temporal-coherence gain of the image is observed, in the meantime the FS width of the image is also found narrower than that of the source (Δ

*ω*

_{i}<Δ

*ω*

_{s}). It is obvious that there are the frequency-filtering effects caused by the NIM dispersion, such as the frequency-dependent interface reflection and focal length. Then we reduce the FS width of the source to Δ

*ω*

_{s}≃

*ω*

_{0}/80 by increasing the pulse-length

*t*

_{p}of the source, at this moment, the reflection and the focal-length difference are very small [28]. With such source, the FS widths of the source and its image are almost the same(Δ

*ω*

_{i}≃Δ

*ω*

_{s}), the difference between them is less than 5%, which is our criterion to judge whether the frequency-filtering effects can be neglected or not. Figure 2(a) shows the

*E*(

*ω*) of a certain random source field

*E*

_{s}(

*t*) and its corresponding image field

*E*

_{i}(

*t*) obtained by FDTD simulation, it is noted that we do not perform the ensemble average on Fig. 2(a) in order to make the phenomena of no frequency-filtering clear at a glance. Even so the prominent gain of temporal coherence is still observed. In Fig. 2(b), the source field (up) and the image field (down) obtained by FDTD simulation are compared. The

*profiles*of them are genically similar, but the image profile is much smoother. The normalized temporalcoherence function

*g*

^{(1)}(

*τ*)=<

*E**(

*t*)

*E*(

*t*+

*τ*)>/<

*E**(

*t*)

*E*(

*t*)> of the source (black) and the image (red) obtained by FDTD simulation are shown in Fig. 3. The temporal coherence of the image field is obviously better than that of the source. From

*g*

^{(1)}, the image coherent time is obtained,

*τ*)|

^{2}

*dτ*=1268

*δ*

_{t}, which is about 50% longer than the source coherent time

*δ*

_{t}.

*only*the ray near a certain incident angle (such as only paraxial ray)can pass through the NIM slab are performed. The results show that there is no gain of coherence anymore and the image field profile also looks like that of the source field very much.

*Therefore, we think that the gain of temporal coherence of the NIM slab image is not from one ray corresponding to a certain incident angle, but probably from the interference between the rays corresponding to the different incident angles*.

## 3. The phenomenological theory

*nds*which determines the wave phase and the refracted “paths” of rays, i.e. ∫

*nds*=

*const*. In this sense, the NIM slab and the traditional lens have the same focusing mechanism(just ∫

_{paths}

*nds*=0 for NIM slab). Howerver, there is something special for NIM slab. Because the temporal-coherence information is in the signals —fluctuation of random field, the signal propagating picture should be essential in our study.

*We know that the optical signals propagate in the group velocity v*

_{g}

*which is always positive*. Obviously, if the path is longer (corresponding to larger incident angle), the signal will spend longer propagating time, which is called

*group retarded time*(GRT) in this Letter. So we expect that the rays corresponding to the different incident angles have different retarded time. And it is really confirmed by our FDTD simulation. The larger the incident angle, the longer the retarded time. Inside the NIM, the GRT of a path is

*θ*is the incident angle and

*v*

_{g}=

*c*/3.04 is the group velocity of NIM around

*ω*

_{0}[29]. The total GRT from source to image is

*τ*

_{r}=

*τ*

_{0}/

*cos*(

*θ*), where

*τ*

_{0}=

*d*/

*c*+

*d*/

*v*

_{g}is the GRT of the paraxial ray. Now, the new picture of a signal propagating through NIM slab is that a signal, generated at

*t*

_{s}from the source, will propagate along all the focusing paths and reach imaging point at different moment

*t*

_{s}+

*τ*

_{0}/

*cos*(

*θ*) corresponding to different paths(this is schematically shown in Fig. 1). This picture is totally different from traditional lenses, whose images don’t have obvious temporal-coherence gain because their focusing rays have the same OPL

*and similar GRT*.

*the random quasimonochromatic source in the NIM slab is the sum of all signals from different paths with different GRT*. This is the key point of SST, and then the image field

*E*

_{i}(

*t*) can be obtained:

*U*

_{0}is the normalization factor. The image field based on Eq.(1) is shown in Fig. 2(c) (up), we can see it excellently agrees with the FDTD result in Fig. 2(b)(down). To show the interference effect of different paths, we assume there are only two paths (such as

*A*and

*B*in Fig. 1). Based on Eq.(1), the image field

*τ*≃±(

*τ*is a continuous variable. So the interfering terms between the paths are responsible for the image temporal-coherence gain.

*t*

_{s}=

*t*-

*τ*

_{0}/

*cosθ*and some algebra, the relation of the temporal coherence between the image and the source can be obtained from Eq. (1):

*G*

_{s}(

*t*

_{2}-

*t*

_{1})=<

*E**

_{s}(

*t*

_{1})

*E*

_{s}(

*t*

_{2})> is the temporal-coherence function of the source. According to SST, we calculate the image coherence function

*g*

^{(1)}vs time (Fig. 3 blue) which agrees with our FDTD result (Fig. 3 red) pretty well (we will discuss the deviation later). Eq.(2) can also explain the temporal-coherence gain of the image. Even if the source field is totally temporal

*incoherent*

*G*

_{s}(

*t*

_{2}-

*t*

_{1})∝

*δ*(

*t*

_{2}-

*t*

_{1}), based on Eq.(2) we can find that

*G*

_{i}(

*τ*) is not a

*δ*-function anymore, so the image is partial temporal coherent. The product of

*h**

_{i}(

*t*

_{1})

*h*

_{i}(

*t*

_{2}+

*τ*) includes the interference between paths.

*strict*Green’s function method [15

15. Lei Zhou and C. T. Chan, “Vortex-like surface wave and its role in the transient phenomena of meta-material focusing,” Appl. Phys. Lett. **86**, 101104 (2005). [CrossRef]

*We only include the propagating field (no evanescent wave) in Green’s function*. The strict image field vs time is shown in Fig. 2(c)(down), and the image temporalcoherence function

*g*

^{(1)}(

*τ*) vs time is shown in Fig. 3 (green). In Fig. 3, we can see that the FDTD result (red) is almost the same as the strict Green’s function method (green). But the result of SST (blue) deviates from that of the Green’s function at very large

*τ*(>3000

*δ*

_{t}) which is corresponding to very long path(or very large incident angle). This is understandable since we neglect the dispersion of NIM totally and only use

*v*

_{g}(

*ω*

_{0}) in SST. For the very-large-angle rays a small index difference (from the dispersion of NIM) can lead to a large focal-length difference. So the deviation is from the focus-filtering effects. When we reduce the FS width of the source further, the deviation of SST is smaller.

*finite-long*two-dimensional NIM slab is a good example which is hard to deal by Green’s function method. In Fig. 4, we plot the coherent time

*L*of NIM slab obtained from the FDTD simulation (blue) and SST (red) respectively. They coincide with each other pretty well (the deviation reason has been discussed). The increase of

*L*can be explained simply by SST. Since the image field

*θ*>

*θ*

_{max})and their contributions to the temporal-coherence gain are missed in the short NIM slab.

*n*≃-1 NIM slab, also applicable to other NIM slabs, such as the photonic crystal slab with negative refractive index [9

9. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. **91**, 207401 (2003); [CrossRef] [PubMed]

11. S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction at media with negative refractive index,” Phys. Rev. Lett. **90**, 107402 (2003); [CrossRef] [PubMed]

21. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B **68**, 045115 (2003). [CrossRef]

## 4. Conclusion

11. S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction at media with negative refractive index,” Phys. Rev. Lett. **90**, 107402 (2003); [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | V. G. Veselago,“The electrodynamics of substances with simultaneously negative values of |

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

3. | J. B. Pendry, “Comment on :Left-handed materials do not make a perfect lens,”Phys. Rev. Lett. |

4. | D. R. Smith, D. Schurig, M. Rosenbluth, S Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on sub-diffraction imaging with a negative refractive index slab,”Appl. Phys. Lett. |

5. | G. Gomez-Santos,“Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. |

6. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science |

7. | A. Pimenov, P. Przyslupski, A Loidl, and B. Dabrowski, “Negative refraction in ferromagnet -superconductor superlattices,”Phys. Rev. Lett. |

8. | 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. |

9. | E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett. |

10. | E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Negative refraction by photonic crystals,” Nature (London) |

11. | S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction at media with negative refractive index,” Phys. Rev. Lett. |

12. | J. B. Pendry and D. R. Smith, “Comment on: Wave refraction in negative-index media: always positive and very inhomogeneous,” Phys. Rev. Lett. |

13. | X. S. Rao and C. K. Ong, “Amplification of evanescent waves in a lossy left-handed material slab,” Phys. Rev. B |

14. | Michael W. Feise and Yuri S. Kivshar, “Sub-wavelength imaging with a left-handed material flat lens,” Phys. Lett. A |

15. | Lei Zhou and C. T. Chan, “Vortex-like surface wave and its role in the transient phenomena of meta-material focusing,” Appl. Phys. Lett. |

16. | L. Chen, S. He, and L. Shen, “Finite-size effects of a left-handed material slab on the image quality,” Phys. Rev. Lett. |

17. | R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,”Phys. Rev. E |

18. | S. A. Cummer, “Simulated causal subwavelength focusing by a negative refractive index slab,” Appl. Phys. Lett. |

19. | P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,”Phys. Rev. E |

20. | R. Merlin, “Analytical solution of the almost-perfect-lens problem,” Appl. Phys. Lett. |

21. | C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B |

22. | Nader Engheta, “An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability,” IEEE Antennas and Wireless Propagation Lett. |

23. | Ilya V. Shadrivov, Andrey A. Sukhorukov, and Yuri S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E |

24. | B. E. A. Saleh and M. C. Teich, |

25. | L. Mandel and E. Wolf, |

26. | Marlan O. Scully and M. Suhail Zubairy, |

27. | X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. |

28. |
In our source frequency range, the index range is about -1- |

29. |
The “group velocity” is not a well-defined value if the working frequency ω0 is near the resonant frequency |

30. | Xunya Jiang, Wenda Han, and Peijun Yao, unpublished. |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(160.0160) Materials : Materials

(260.0260) Physical optics : Physical optics

**ToC Category:**

Metamaterials

**History**

Original Manuscript: September 11, 2006

Revised Manuscript: November 9, 2006

Manuscript Accepted: November 9, 2006

Published: December 11, 2006

**Citation**

Peijun Yao, Wei Li, Songlin Feng, and Xunya Jiang, "The temporal coherence improvement of the two-dimensional negative-index slab image," Opt. Express **14**, 12295-12301 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-25-12295

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### References

- V. G. Veselago,"The electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys. Usp. 10, 509 (1968). [CrossRef]
- J. B. Pendry, "Negative refraction makes a perfect lens,"Phys. Rev. Lett. 85, 3966 (2000). [CrossRef] [PubMed]
- J. B. Pendry,"Comment on :Left-handed materials do not make a perfect lens,"Phys. Rev. Lett. 91, 099701 (2003). [CrossRef] [PubMed]
- D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, "Limitations on subdiffraction imaging with a negative refractive index slab,"Appl. Phys. Lett. 82, 1506 (2003). [CrossRef]
- G. Gomez-Santos,"Photonic band gap from a stack of positive and negative index materials, " Phys. Rev. Lett. 90, 077401 (2003). [PubMed]
- R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction,"Science 292, 77 (2001). [CrossRef] [PubMed]
- A. Pimenov, P. Przyslupski, A. Loidl, and B. Dabrowski, "Negative refraction in ferromagnet -superconductor superlattices,"Phys. Rev. Lett. 95, 247009 (2005); [CrossRef] [PubMed]
- 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]
- E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, " Subwavelength resolution in a twodimensional photonic-crystal-based superlens," Phys. Rev. Lett. 91, 207401 (2003); [CrossRef] [PubMed]
- E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis,"Negative refraction by photonic crystals," Nature (London) 423, 604 (2003). [CrossRef] [PubMed]
- S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, "Refraction at media with negative refractive index," Phys. Rev. Lett. 90, 107402 (2003); [CrossRef] [PubMed]
- J. B. Pendry and D. R. Smith, "Comment on :Wave refraction in negative-index media: always positive and very inhomogeneous," Phys.Rev. Lett. 90, 029703 (2003). [CrossRef] [PubMed]
- X. S. Rao and C. K. Ong, "Amplification of evanescent waves in a lossy left-handed material slab," Phys. Rev. B 68, 113103 (2003); [CrossRef]
- MichaelW. Feise, Yuri S. Kivshar, "Sub-wavelength imaging with a left-handed material flat lens," Phys. Lett. A 334326 (2005) [CrossRef]
- Lei Zhou, C. T. Chan, "Vortex-like surface wave and its role in the transient phenomena of meta-material focusing, " Appl. Phys. Lett. 86, 101104 (2005). [CrossRef]
- L. Chen, S. He and L. Shen, "Finite-size effects of a left-handed material slab on the image quality," Phys. Rev. Lett. 92, 107404 (2004). [CrossRef] [PubMed]
- R. W. Ziolkowski and E. Heyman, "Wave propagation in media having negative permittivity and permeability," Phys. Rev. E 64, 056625 (2001). [CrossRef]
- S. A. Cummer, "Simulated causal subwavelength focusing by a negative refractive index slab,"Appl. Phys. Lett. 82, 1503 (2003). [CrossRef]
- P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, "Electromagnetic waves focused by a negative-index planar lens," Phys. Rev. E 67, 025602(R) (2003). [CrossRef]
- R. Merlin, "Analytical solution of the almost-perfect-lens problem," Appl. Phys. Lett. 84, 1290 (2004). [CrossRef]
- C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "Subwavelength imaging in photonic crystals," Phys.Rev. B 68, 045115 (2003). [CrossRef]
- Nader Engheta, "An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability," IEEE Antennas and Wireless Propagation Lett. 1, 10,2002. [CrossRef]
- IlyaV. Shadrivov, Andrey A. Sukhorukov, and Yuri S. Kivshar, "Guided modes in negative-refractive-index waveguides, " Phys. Rev. E 67, 057602 (2003). [CrossRef]
- B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, New York, 1991). [CrossRef]
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, Cambridge, 1995).
- MarlanO. Scully and M. Suhail Zubairy, Quantum Optics, (Cambridge University, Cambridge, 1997).
- X. Jiang and C. M. Soukoulis, "Time dependent theory for random lasers," Phys. Rev. Lett. 85, 70(2000). [CrossRef] [PubMed]
- In our source frequency range, the index range is about.1.i0.0029±(0.006+i10.5), so the difference of the focal length and the reflection are very small.
- The "group velocity" is not a well-defined value if the working frequency ω0 is near the resonant frequency ωa of the NIM. But the GRT is still well-defined.
- Xunya Jiang, Wenda Han, and Peijun Yao, unpublished.

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