## Coupled wave analysis of holographically induced transparency (HIT) generated by two multiplexed volume gratings |

Optics Express, Vol. 19, Issue 8, pp. 7094-7105 (2011)

http://dx.doi.org/10.1364/OE.19.007094

Acrobat PDF (795 KB)

### Abstract

We present a holographic system that can be used to manipulate the group velocity of light pulses. The proposed structure is based on the multiplexing of two sequential holographic volume gratings, one in transmission and the other in reflection geometry, where one of the recording beams must be the same for both structures. As in other systems such as grating induced transparency (GIT) or coupled-resonator-induced transparency (CRIT), by using the coupled wave theory it is shown that this holographic structure represents a classical analogue of the electromagnetically induced transparency (EIT). Analytical expressions were obtained for the transmittance induced at the forbidden band (spectral hole) and conditions where the group velocity was slowed down were analyzed. Moreover, the propagation of Gaussian pulses is analyzed for this system by obtaining, after further approximations, analytical expressions for the distortion of the transmitted field. As a result, we demonstrate the conditions where the transmitted pulse is slowed down and its shape is only slightly distorted. Finally, by comparing with the exact solutions obtained, the range of validity of all the analytical formulae was verified, demonstrating that the error is very low.

© 2011 OSA

## 1. Introduction

1. R. Boyd, O. Hess, C. Denz, and E. Paspalkalis, “Slow light,” J. Opt. **12**(10), 100301 (2010). [CrossRef]

2. J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis,” J. Opt. Soc. Am. B **22**, 1062–1074 (2005). [CrossRef]

3. F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics **1**(1), 65–71 (2007). [CrossRef]

4. Z. Shi, R. W. Boyd, R. M. Camacho, P. K. Vudyasetu, and J. C. Howell, “Slow-light fourier transform interferometer,” Phys. Rev. Lett. **99**(24), 240801 (2007). [CrossRef]

5. Z. M. Shi, R. W. Boyd, D. J. Gauthier, and C. C. Dudley, “Enhancing the spectral sensitivity of interferometers using slow-light media,” Opt. Lett. **32**(8), 915–917 (2007). [CrossRef] [PubMed]

*ν*) of a propagating pulse requires control over the frequency dependence of the refractive index by using strong material dispersion [6

_{G}6. R. Boyd and D. J. Gauthier, ““Slow” and “fast” light,” in *Progress in Optics*, E. Wolf, ed., (Elsevier, 2002) vol. 43, chap. 6, pp. 497–530. [CrossRef]

*et al.*[7

7. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature **397**(6720), 594–598 (1999). [CrossRef]

8. A. Kasapi, M. Jain, G. y. Yin, and S. E. Harris, “Electromagnetically induced transparency - propagation dynamics,” Phys. Rev. Lett. **74**(13), 2447–2450 (1995). [CrossRef] [PubMed]

10. L. V. Hau, “Optical information processing in Bose-Einstein condensates,” Nat. Photonics **2**(8), 451–453 (2008). [CrossRef]

11. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science **301**(5630), 200–202 (2003). [CrossRef] [PubMed]

12. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. **90**(11), 113903 (2003). [CrossRef] [PubMed]

13. R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys. Rev. A **73**(6), 063812 (2006). [CrossRef]

14. R. M. Camacho, C. J. Broadbent, I. Ali-Khan, and J. C. Howell, “All-optical delay of images using slow light,” Phys. Rev. Lett. **98**, 043902 (2007). [CrossRef] [PubMed]

15. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature **438**(7064), 65–69 (2005). [CrossRef] [PubMed]

17. M. Notomi, “Manipulating light with strongly modulated photonic crystals,” Rep. Prog. Phys. **73**(9), 096501 (2010). [CrossRef]

3. F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics **1**(1), 65–71 (2007). [CrossRef]

18. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. **24**(11), 711–713 (1999). [CrossRef]

19. D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A **69**(6), 063804 (2004). [CrossRef]

20. Z. M. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “Broadband SBS slow light in an optical fiber,” J. Lightwave Technol. **25**(1), 201–206 (2007). [CrossRef]

21. L. Thevenaz, “Slow and fast light in optical fibres,” Nat. Photonics **2**(8), 474–481 (2008). [CrossRef]

22. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today **50**(7), 36–42 (1997). [CrossRef]

19. D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A **69**(6), 063804 (2004). [CrossRef]

23. M. F. Yanik, W. Suh, Z. Wang, and S. H. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. **93**(23), 233903 (2004). [CrossRef] [PubMed]

24. K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency.” Phys. Rev. Lett. **98**(21), 213904 (2007). [CrossRef] [PubMed]

25. H. C. Liu and A. Yariv, “Grating induced transparency (GIT) and the dark mode in optical waveguides,” Opt. Express **17**(14), 11710–11718 (2009). [CrossRef] [PubMed]

## 2. Theoretical background

6. R. Boyd and D. J. Gauthier, ““Slow” and “fast” light,” in *Progress in Optics*, E. Wolf, ed., (Elsevier, 2002) vol. 43, chap. 6, pp. 497–530. [CrossRef]

*ε*(assuming zero conductivity in the medium) given by: where

**r**= (

*x,*0

*,z*),

*ε*

_{1}

*, ε*

_{2}are the dielectric constant modulations and: where

*ω*the angular frequency used in the recording steps, c is the light velocity in the free space, and

_{o}*θ*,

_{r}*θ*,

_{s}*θ*are the angles shown in Fig. 1. The wave propagation in the resulting multiplexed holographic structure is described by the Helmholtz equation: Using the coupled wave theory methodology [26] to solve Eq. (4), only three waves will be present in the material, so the total electric field inside the device will be given by the superposition of the three waves with complex amplitudes R(z), S(z) and W(z): The two superposed gratings described by the grating vectors

_{w}**K**and

_{1}**K**connect R with S and S with W respectively, so the Bragg conditions are satisfied: Assuming that

_{2}**K**is given by: where

_{R}*ω*is the angular frequency used in the reconstruction step. From Eqs. (2), (3), and (6) it follows that: Introducing the electric field E given by Eq. (5) in the Helmholtz equation (4) and using the coupled wave theory [26] approximations, we obtain the differential equation system: where: where

*β*=

*nω/c*. The boundary conditions for Eqs. (10) are

*R*(0) = 1,

*S*(

*L*) = 0 and

*W*(

*L*) = 0, where we have assumed that the thickness of the HRS is L. In order to obtain an analytical expression for the HRS amplitude transmittance t =

*R*(

*L*) near the recording frequency

*ω*(

_{o}*ω*=

*ω*+

_{o}*δ ω*,

*δ ω <<*1) and for Bragg angle reconstruction.

### 2.1. Normal incidence in reconstruction step

*θ*= 0 and at the Bragg angle in the reconstruction step

_{r}*θ*=

*θ*= 0. Assuming normal incidence parameters given by Eq. (11) can be approximated to: where all of them are functions of recording parameters (

_{r}*θ*,

_{r}*θ*,

_{w}*θ*,

_{s}*ω*) material properties (

_{o}*ε*

_{1},

*ε*

_{2}

*, n*) and only

*depends on frequency detuning (*

_{p}*δω*). Introducing approximations (12) into Eq. (10) we obtain the differential equation system given by: where we defined the parameters: being

*n*is the group index of waves S and W. It is important to note that Eq. (14) shows that the normalized group index values

_{p}*n*/

_{s}*n*and

*n*/

_{w}*n*are negatives in the range of recording angles for transmission gratings (see Fig. 2) where |

*n*|

_{w}*> n*and |

*n*|

_{s}*> n*as can be seen in Fig. 2, so the values of the group index range from −2n to −4n. Negative values of the group index show us that S and W are backward propagating waves.

25. H. C. Liu and A. Yariv, “Grating induced transparency (GIT) and the dark mode in optical waveguides,” Opt. Express **17**(14), 11710–11718 (2009). [CrossRef] [PubMed]

- The
**A**matrix coefficients given by Eq. (13), are all purely imaginary. However, in the GIT matrix*A*_{12}and*A*_{21}are reals. - Due to the holographic recording process
*A*_{12}≠*A*_{21}and*A*_{23}=*A*_{32}will be only be fulfilled if the transmission grating is recorded with symmetrical geometry. - According to Eqs. (13) all the coupling coefficients depend on group index parameters, and in the GIT matrix only diagonal elements depend on the group index.
- According to Eq. (14), in the HIT model, the group index depends on the geometry of the recoding process.

*R*(

*z*) can be expressed as: and the amplitude transmittance t: where

*x*are the three roots of the characteristic polynomial (P) of matrix

_{i}**A**in Eq. (13): and

*C*are integration constants given by: where:

_{i}*θ*=

*θ*) and at frequency

_{r}*ω*(

_{o}*δω*= 0), according to Eqs. (14),

*ξ*=

_{s}*ξ*= 0 and in this case the roots of P given by Eq. (17) are:

_{w}*x*

_{0}= 0,

*ν*

_{2}| > |

*ν*

_{1}|), therefore, taking into account Eqs. (16), (18), and (19), transmittance t can be expressed as: It may be deduced from Eq. (20) that the set of values

*t*= 1 at frequency design

*ω*. Therefore, if the HRS has a thickness

_{o}*L*=

*L*

_{1}, in the forbidden band of the reflection grating, a permitted band appears, centered at

*ω*. In the next section we will describe the process for obtaining a practicable analytical expression of transmittance near the recording frequency

_{o}*ω*

_{0}that permits us to analytically study the pulse propagation inside the HRS.

## 3. Analytical expression for the permitted band

*ω*

_{0}(

*δω <<*1), it can be assumed that the parameters

*ξ*<< 1 and

_{w}*ξ*<< 1 under the conditions that the resulting permitted band is narrow, so the roots of polynomial P can be approximated (using a first order Taylor series expansion on variables

_{s}*ξ*and

_{w}*ξ*, at

_{s}*ξ*= 0 and

_{w}*ξ*= 0) by:

_{s}*ψ*(on variables

*ξ*and

_{w}*ξ*) and finally a second order Taylor series expansion on

_{s}*ξ*and first order on

_{w}*ξ*on the argument of the root square in the denominator of Eq. (21).

_{s}*δω*variable, and a phase that is linear on

*δω*: where

*τ*is the time-delay induced by the device and

_{d}*γ*is the scale parameter which specifies the half-width at half-maximum of the Lorentzian function (the accuracy of this analytical approximation may be seen in the numerical section). Time delay and scale parameter only depend on the material properties and the recording geometry of HRS. Assuming that

*ε*

_{2}=

*αε*

_{1}, we obtain that their values are:

*τ*→ ∞ if

_{d}*α*approaches

*γ*will also approach 0. Therefore, in order to obtain the maximum time delay with non-null transmittance we are going to analyze the case in which: Introducing Eq. (28) in Eqs. (26) and (27), it follows that time delay and scale factor can be written as: In order to obtain a Lorentzian function when we introduce Eqs. (29) and (30) in Eq. (25), the condition

*γ*

^{2}

*>*1 must be fulfilled.Thus it follows from Eq. (30) that |

*η*| > 1, but obviously as demonstrated before, values closer to 1 provides higher time delays and a narrower Lorentzian transmittance function. If we introduce Eqs. (12), (14), and (28) in

## 4. Propagation through the permitted band

*W*

_{0}is the free space pulse spatial width, defined as the length from the maximum at which the pulse amplitude decreases a factor

*e*

^{−1/2}. If we introduce the free-space pulse time length as

*T*

_{0}=

*W*

_{0}

*/c*, we can obtain the frequency domain of the incident field

*(*

_{inc}*ω*) as the direct Fourier transform of the incident field (33): where

*(*

_{tr}*t*) will then be given by the inverse Fourier transform:

*(*

_{inc}*ω*) by the expression given in Eq. (34) in Eq. (36), and approximating t(

*ω*) to: it follows that the transmitted field Φ

*(*

_{tr}*t*) is given by: where we have used that

*t*(–

*ω*) =

*t*

^{*}(

*ω*). Equation (38) shows that the transmitted field is equal to a time-delayed incident field (33), distorted by a factor

*D*given by [27

_{rms}27. B. Macke and B. Segard, “Propagation of light-pulses at a negative group-velocity,” Eur. Phys. J. D **23**, 125–141 (2003). [CrossRef]

## 5. Numerical simulations

*ω*= 3.5431 × 10

_{o}^{15}

*s*

^{−1}, n= 1.5,

*ε*

_{1}= 0.0075, and

*η*= 1.01. Introducing these parameters into Eqs. (29), (30), and (31), it is easy to observe that the thickness

*L*

_{1}, which is given by equation 31 depends on group index

*n*(see Fig. 3). So it can be deduced that the reflection grating determines the HRS thickness. In the case of the time delay and scale factor both of them depend on group index

_{s}*n*,

_{s}*n*.

_{w}*L*1 = 2

*mm*corresponds to

*n*= −4.8793, which is the value that we are going to take in order to analyze an example of our system,together with

_{s}*n*= −4.8793 (which implies that the transmission grating was recorded in symmetrical geometry). For these parameters Fig. 4 shows the values of transmissivity (

_{w}*T*= |

*t*|

^{2}) obtained using the exact solution (16) and the analytical expression given by Eq. (25). As can be seen, a narrow permitted band is shown in the center of the forbidden band of the reflection grating. In the center of the figure the permitted band is zoomed, and the analytical solution is identical to the exact one, with a relative error of our approximated function lower than 0.012 %, as shown in the inset curve of Fig. 4. Figure 5 shows the phase values obtained by using exact solution (16) and the analytical expression given by Eq. (25). It may be seen that both solutions are identical in a region

*δω*lower than the region where T functions coincides (compare Figs. 4 and 5), so if we want to use our analytical approximated solution, and the results shown in Eqs. (38) and (39), we must restrict the width of the incident field in the frequency domain

*(*

_{inc}*ω*) to a region where the exact and approximated phase will be in good agreement. For the parameters previously mentioned we find, using Eqs. (29) and (30), that time delay

*τ*= 124

_{d}*ns*and

*γ*= 80.8

*MHz*. Taking into account these values, we introduced in our system a Gaussian pulse (see Eqs. (33), (34)) with parameters

*W*

_{0}= 108

*μm,T*

_{0}= 333

*ns*in order to ensure that

*D*<< 1 (see Eq. (40)). Figure 6 shows the incident pulse (brown color) and the time-delayed pulses obtained using Eq. (35) taking into account approximations (25) (blue) and (37)–(38) (pink). As can be observed both results are very similar (it is important that the pulses were not normalized for comparison), where the relative error with respect to the exact value is lower than 3% in the region of interest, as can be observed in the inset curve. The values of

_{rms}*D*obtained by using Eqs. (39) and (40) are 0.024 and 0.026 respectively, which implies that there is no significant distortion, and that the analytical expression obtained in Eq. (38) can be used.

_{rms}## 6. Conclusions

## Acknowledgments

## References and links

1. | R. Boyd, O. Hess, C. Denz, and E. Paspalkalis, “Slow light,” J. Opt. |

2. | J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis,” J. Opt. Soc. Am. B |

3. | F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics |

4. | Z. Shi, R. W. Boyd, R. M. Camacho, P. K. Vudyasetu, and J. C. Howell, “Slow-light fourier transform interferometer,” Phys. Rev. Lett. |

5. | Z. M. Shi, R. W. Boyd, D. J. Gauthier, and C. C. Dudley, “Enhancing the spectral sensitivity of interferometers using slow-light media,” Opt. Lett. |

6. | R. Boyd and D. J. Gauthier, ““Slow” and “fast” light,” in |

7. | L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature |

8. | A. Kasapi, M. Jain, G. y. Yin, and S. E. Harris, “Electromagnetically induced transparency - propagation dynamics,” Phys. Rev. Lett. |

9. | M. D. Lukin, “Colloquium: Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. |

10. | L. V. Hau, “Optical information processing in Bose-Einstein condensates,” Nat. Photonics |

11. | M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science |

12. | M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. |

13. | R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys. Rev. A |

14. | R. M. Camacho, C. J. Broadbent, I. Ali-Khan, and J. C. Howell, “All-optical delay of images using slow light,” Phys. Rev. Lett. |

15. | Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature |

16. | T. Baba, “Slow light in photonic crystals,” Nat. Photonics |

17. | M. Notomi, “Manipulating light with strongly modulated photonic crystals,” Rep. Prog. Phys. |

18. | A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. |

19. | D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A |

20. | Z. M. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “Broadband SBS slow light in an optical fiber,” J. Lightwave Technol. |

21. | L. Thevenaz, “Slow and fast light in optical fibres,” Nat. Photonics |

22. | S. E. Harris, “Electromagnetically induced transparency,” Phys. Today |

23. | M. F. Yanik, W. Suh, Z. Wang, and S. H. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. |

24. | K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency.” Phys. Rev. Lett. |

25. | H. C. Liu and A. Yariv, “Grating induced transparency (GIT) and the dark mode in optical waveguides,” Opt. Express |

26. | H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. |

27. | B. Macke and B. Segard, “Propagation of light-pulses at a negative group-velocity,” Eur. Phys. J. D |

**OCIS Codes**

(050.7330) Diffraction and gratings : Volume gratings

(090.0090) Holography : Holography

(090.4220) Holography : Multiplex holography

**ToC Category:**

Holography

**History**

Original Manuscript: February 1, 2011

Revised Manuscript: March 1, 2011

Manuscript Accepted: March 5, 2011

Published: March 29, 2011

**Citation**

Luis Carretero, Salvador Blaya, Pablo Acebal, Antonio Fimia, Roque Madrigal, and Angel Murciano, "Coupled wave analysis of holographically induced transparency (HIT) generated by two multiplexed volume gratings," Opt. Express **19**, 7094-7105 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-8-7094

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### References

- R. Boyd, O. Hess, C. Denz, and E. Paspalkalis, “Slow light,” J. Opt. 12(10), 100301 (2010). [CrossRef]
- J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis,” J. Opt. Soc. Am. B 22, 1062–1074 (2005). [CrossRef]
- F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007). [CrossRef]
- Z. Shi, R. W. Boyd, R. M. Camacho, P. K. Vudyasetu, and J. C. Howell, “Slow-light fourier transform interferometer,” Phys. Rev. Lett. 99(24), 240801 (2007). [CrossRef]
- Z. M. Shi, R. W. Boyd, D. J. Gauthier, and C. C. Dudley, “Enhancing the spectral sensitivity of interferometers using slow-light media,” Opt. Lett. 32(8), 915–917 (2007). [CrossRef] [PubMed]
- R. Boyd and D. J. Gauthier, ““Slow” and “fast” light,” in Progress in Optics , E. Wolf, ed., (Elsevier, 2002) vol. 43, chap. 6, pp. 497–530. [CrossRef]
- L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999). [CrossRef]
- A. Kasapi, M. Jain, G. y. Yin, and S. E. Harris, “Electromagnetically induced transparency - propagation dynamics,” Phys. Rev. Lett. 74(13), 2447–2450 (1995). [CrossRef] [PubMed]
- M. D. Lukin, “Colloquium: Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. 75(2), 457–472 (2003). [CrossRef]
- L. V. Hau, “Optical information processing in Bose-Einstein condensates,” Nat. Photonics 2(8), 451–453 (2008). [CrossRef]
- M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301(5630), 200–202 (2003). [CrossRef] [PubMed]
- M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90(11), 113903 (2003). [CrossRef] [PubMed]
- R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys. Rev. A 73(6), 063812 (2006). [CrossRef]
- R. M. Camacho, C. J. Broadbent, I. Ali-Khan, and J. C. Howell, “All-optical delay of images using slow light,” Phys. Rev. Lett. 98, 043902 (2007). [CrossRef] [PubMed]
- Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438(7064), 65–69 (2005). [CrossRef] [PubMed]
- T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008). [CrossRef]
- M. Notomi, “Manipulating light with strongly modulated photonic crystals,” Rep. Prog. Phys. 73(9), 096501 (2010). [CrossRef]
- A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999). [CrossRef]
- D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69(6), 063804 (2004). [CrossRef]
- Z. M. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “Broadband SBS slow light in an optical fiber,” J. Lightwave Technol. 25(1), 201–206 (2007). [CrossRef]
- L. Thevenaz, “Slow and fast light in optical fibres,” Nat. Photonics 2(8), 474–481 (2008). [CrossRef]
- S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997). [CrossRef]
- M. F. Yanik, W. Suh, Z. Wang, and S. H. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93(23), 233903 (2004). [CrossRef] [PubMed]
- K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency.” Phys. Rev. Lett. 98(21), 213904 (2007). [CrossRef] [PubMed]
- H. C. Liu and A. Yariv, “Grating induced transparency (GIT) and the dark mode in optical waveguides,” Opt. Express 17(14), 11710–11718 (2009). [CrossRef] [PubMed]
- H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2945 (1969).
- B. Macke and B. Segard, “Propagation of light-pulses at a negative group-velocity,” Eur. Phys. J. D 23, 125–141 (2003). [CrossRef]

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