## Recursion formula for reflectance and the enhanced effect on the light group velocity control of the stratified and phase-shifted volume index gratings

Optics Express, Vol. 15, Issue 5, pp. 2055-2066 (2007)

http://dx.doi.org/10.1364/OE.15.002055

Acrobat PDF (328 KB)

### Abstract

We derived a recursion formula for the reflectance of the stratified and phase-shifted volume index gratings. The characteristics of the reflectance spectra of the stratified and phase-shifted volume index gratings were studied based on the recursion formula. It is shown that narrow bandwidth transparency peaks appear within the stop-band of the reflectance spectrum of the volume index gratings due to the intervention of the homogeneous buffer layers that induce the phase-shifts between neighboring volume index gratings. The spectral positions of the transparency peaks can be shifted within the stop-band by controlling the phase-shift, i.e., the buffer layer thickness. The described properties may find applications in addressable band-pass filter, switching, wavelength division multiplexing, and de-multiplexing. The dispersion near the transparency peaks of the stratified and phase-shifted volume index grating is found to be sharply enhanced as compared to the uniform volume index gratings. Significantly enhanced control on the group velocity of light by several orders of magnitude while keeping high transmittance is demonstrated in the stratified and phase-shifted volume index grating.

© 2007 Optical Society of America

## 1. Introduction

3. R. C. Alferness, C. H. Joyner, M. D. Divino, M. J. R. Martyak, and L. L. Buhl, “Narrowband grating resonator filters in InGaAsP/InP waveguides,” Appl. Phys. Lett. **49**, 125–127 (1986). [CrossRef]

8. Ch. Martinez and P. Ferdinand, “Analysis of phase-shifted fiber Bragg gratings written with phase plate,” Appl. Opt. **38**, 3223–3228 (1999). [CrossRef]

9. S. Longhi, M. Marano, P. Laporta, O. Svelto, and M. Belmonte, “Propagation, manipulation, and control of picosecond optical pulses at 1.5 μm in fiber Bragg gratings,” J. Opt. Soc. Am B **19**, 2742–2757 (2002). [CrossRef]

10. Y. Painchaud, A. Chandonnet, and J. Lauzon, “Chirped fibre gratings produced by tilting the fibre,” Electron. Lett. **31**, 171–172 (1995). [CrossRef]

11. B. Malo, S. Thériault, D. C. Johnson, F. Bilodeau, J. Albert, and K. O. Hill, “Apodised in-fibre Bragg grating reflectors photoimprinted using a phase mask,” Electron. Lett. **31**, 223–225 (1995). [CrossRef]

12. Ch. Martinez, S. Magne, and P. Ferdinand, “Apodized fiber Bragg gratings manufactured with the phase plate process,” Appl. Opt. **41**, 1733–1740 (2002). [CrossRef] [PubMed]

13. R. V. Johnson and A. R. Tanguay, “Stratified volume holographic optical elements,” Opt. Lett. **13**, 189–191 (1988). [CrossRef] [PubMed]

16. J. J. Stankus, S. M. Silence, W. E. Moerner, and G. C. Bjorklund, “Electric-field-switchable stratified volume holograms in photorefractive polymers,” Opt. Lett. **19**, 1480–1482 (1994). [CrossRef] [PubMed]

3. R. C. Alferness, C. H. Joyner, M. D. Divino, M. J. R. Martyak, and L. L. Buhl, “Narrowband grating resonator filters in InGaAsP/InP waveguides,” Appl. Phys. Lett. **49**, 125–127 (1986). [CrossRef]

5. R. Zengerle and O. Leminger, “Phase-shifted Bragg-Grating Filters with Improved Transmission Characteristics,” J. Lightwave Technol. **13**, 2354–2358 (1995). [CrossRef]

6. L. Wei and J. w. Y. Lit, “Phase-Shifted Bragg Grating Filters with Symmetrical Structures,” J. Lightwave Tech-nol. **15**, 1405–1410 (1997). [CrossRef]

17. V. M. Petrov, C. Caraboue, J. Petter, T. Tschudi, V. V. Bryksin, and M. P. Petrov, “A dynamic narrow-band tunable optical fliter,” Appl. Phys. B **76**, 41–44 (2003). [CrossRef]

18. Y. Lai, W. Zhang, L. Zhang, J. A. R. Williams, and I. Bennion, “Optically tumable fiber grating transmission filters,” Opt. Lett. **28**, 2446–2448 (2003). [CrossRef] [PubMed]

4. G. P. Agrawal and S. Radic, “Phase-Shifted Fiber Bragg Gratings and their Application for Wavelength Demultiplexing,” IEEE Photon. Technol. Lett. **6**, 995–997 (1994). [CrossRef]

19. A. D’Orazio, M. De Sario, V. Petruzzelli, and F. Prudenzano, “Photonic band gap filter for wavelength division multiplexer,” Opt. Express **11**, 230–239 (2003). [CrossRef] [PubMed]

20. M. Yamada and K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,” Appl. Opt. **26**, 3474–3478 (1987). [CrossRef] [PubMed]

21. M. McCall, “On the application of coupled mode theory for modeling fiber Bragg gratings,” J. Lightwave Tech-nol. **18**, 236–242 (2000). [CrossRef]

25. J. J. Monzón, T. Yonte, and L. L. Sánchez-Soto, “Charcterizing the reflectance of periodic layered media,” Opt. Commun. **218**, 43–47 (2003). [CrossRef]

26. I. S. Nefedov and S. A. Tretyakov, “Photonic band gap structure containing metamaterial with negative permittivity and permeability,” Phys. Rev. E **66**, 036611–1–4 (2002). [CrossRef]

27. J. R. Birge and F. X. Kärtner, “Efficient analytic computation of dispersion from multilayer structures,” Appl. Opt. **45**, 1478–1483 (2006). [CrossRef] [PubMed]

28. D. Yevick and L. Thylén, “Analysis of gratings by the beam-propagation method,” J. Opt. Soc. Am. **72**, 1084–1089 (1982). [CrossRef]

31. Y. Tsuji, M. Koshiba, and N. Takimoto, “Finite element beam propagation method for anisotropic optical waveguides,” J. Lightwave Technol. **17**, 723–728 (1999). [CrossRef]

13. R. V. Johnson and A. R. Tanguay, “Stratified volume holographic optical elements,” Opt. Lett. **13**, 189–191 (1988). [CrossRef] [PubMed]

16. J. J. Stankus, S. M. Silence, W. E. Moerner, and G. C. Bjorklund, “Electric-field-switchable stratified volume holograms in photorefractive polymers,” Opt. Lett. **19**, 1480–1482 (1994). [CrossRef] [PubMed]

32. P. K. Kelly and M. Piket-May, “Propagation characteristics for a one-dimensional grounded finite height finite length electromagnetic crystal,” J. Lightwave Technol. **17**, 2008–2012 (1999). [CrossRef]

## 2. The structure of SPVIG

_{0}. The refractive index modulation amplitude for the grating layers is n

_{1}, and the grating wave vector along x-axis is K = 2

*π*/Λ with L being the grating spacing. Therefore the refractive index distribution of the j-th grating layer can be expressed as n

_{j}(x) = n

_{0}+ n

_{1}cos(K(x - (j -1)(D + d))).

## 3. Derivation of the recursion formula for the reflectance of SPVIG

_{1}(x) and B

_{1}(x) are the complex amplitudes of the forward and the backward propagating waves, respectively, and k

_{0}= 2

*π*n

_{0}/λ with λ being the wavelength of the incident light in vacuum. By substituting the refractive index n(x) into Eq. (1), and neglecting the terms higher than ∼ n

^{2}

_{1}, under the slowly varying amplitude approximation, we obtain two coupled differential equations for A

_{1}(x) and B

_{1}(x)

_{1}/λ is the coupling constant, Δk = 2k

_{0}-K is the momentum mismatch. The general analytical solutions for A

_{1}(x) and B

_{1}(x) can be written as

_{1}(0) is the amplitude of the incident light at x = 0, s is given by s = (κ

^{2}- (Δk/2)

^{2})

^{1/2}, and C

_{1}is a parameter determined by the boundary conditions. Under the boundary condition B

_{1}(D) = 0, the reflection coefficient of the grating layer is given by

_{N}of a N-layer SPVIG with N grating layers interleaved with N-1 buffer layers, we will derive a recursion formula for the reflection coefficient r

_{N+1}of a (N+1)-layer SPVIG. Such a recursion technique for calculation of reflection coefficient is generally used in distributed feedback laser [33].

_{j}(x) in the j-th grating layer is the summation of the forward and the backward propagation waves E

_{j}(x) = A

_{j}(x)exp(ik

_{0}(x-(j-1)(D+ d)))+B

_{j}(x)exp(-ik

_{0}(x-(j-1)(D+d))). The amplitudes of the forward and backward propagation waves in the first layer are described by Eqs. (5) and (6), regardless of the layer number of the SPVIG. In each case, the parameter C

_{1}is determined by the boundary conditions at the two interfaces of the buffer layer sandwiched between the first and the second grating layers. These boundary conditions can be expressed as

_{N}= B

_{2}(D+d)/A

_{2}(D+d), we obtain the parameter C

_{1}as a function of r

_{N}

_{N+1}of the (N+1)-layer SPVIG

_{N+1}= ∣r

_{N+1}∣

^{2}and the transmittance TN+1 can be obtained through T

_{N+1}= 1-R

_{N+1}. The recursion formula ( 11) provides a precise prediction of the characteristics of the reflectance spectra of the SPVIGs with a normal incident light. It is worthy of mention that for the case of oblique incidence the results will be different for the TE and the TM waves, however, a detailed discussion on the oblique incidence case deserves another full-length paper and is beyond the scope of this paper.

## 4. Characteristics of the reflectance spectra of SPVIGs

_{0}for the volume index grating layer, i.e., Δk = 0, the reflection coefficient of the 2-layer SPVIG can be simplified to be

_{2}is equal to zero under the condition 2k

_{0}d = (2m + 1)

*π*, where m is an integer. Therefore a transparency peak appears in the stop-band of the reflectance spectrum of the volume index grating, as shown in Fig. 2, where the reflectance spectrum of a 2-layer SPVIG is shown. Similar phenomena were also reported in phase-shifted fiber Bragg gratings [4

4. G. P. Agrawal and S. Radic, “Phase-Shifted Fiber Bragg Gratings and their Application for Wavelength Demultiplexing,” IEEE Photon. Technol. Lett. **6**, 995–997 (1994). [CrossRef]

5. R. Zengerle and O. Leminger, “Phase-shifted Bragg-Grating Filters with Improved Transmission Characteristics,” J. Lightwave Technol. **13**, 2354–2358 (1995). [CrossRef]

6. L. Wei and J. w. Y. Lit, “Phase-Shifted Bragg Grating Filters with Symmetrical Structures,” J. Lightwave Tech-nol. **15**, 1405–1410 (1997). [CrossRef]

7. F. Bakhti and P. Sansonetti, “Design and Realization of Multiple Quater-Wave Phase-Shifts UV-Written Bandpass Filter in Optical Fibers,” J. Lightwave Technol. **15**, 1433–1437 (1997). [CrossRef]

8. Ch. Martinez and P. Ferdinand, “Analysis of phase-shifted fiber Bragg gratings written with phase plate,” Appl. Opt. **38**, 3223–3228 (1999). [CrossRef]

_{0}= 1.55, n

_{1}=4×10

^{-4}, Λ=0.5μm,λ

_{0}= 1.55μm, D=3 mm and d=2.25μm, respectively. It is evident that such a transparency peak is a result of the interference between the forward waves and the backward waves reflected from the first and the second volume index grating layers. For comparison, the reflectance spectrum of a 6-mm SG with the same grating parameters is also shown in Fig. 2. A broadening effect of the stop-band of the reflectance spectrum is also found for the SPVIG due to the interleave of the buffer layer.

_{0}d= (N-1)×(2m+1)π.

_{0}the transparency peak appears periodically as a function of the phase-shift 2k

_{0}d whenever the condition 2k

_{0}d = (2m+1)

*π*is satisfied, as is shown in Fig. 5. These properties can be applicable to addressable wavelength filters, wavelength division multiplexing and de-multiplexing, and switching.

## 5. Dispersion properties of the SPVIGs and group velocity control

9. S. Longhi, M. Marano, P. Laporta, O. Svelto, and M. Belmonte, “Propagation, manipulation, and control of picosecond optical pulses at 1.5 μm in fiber Bragg gratings,” J. Opt. Soc. Am B **19**, 2742–2757 (2002). [CrossRef]

34. M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E **54**, R1078–R1081 (1996). [CrossRef]

38. S. Bette, C. Caucheteur, M. Wuilpart, P. Mégret, R. Garcia-Olcina, S. Sales, and J. Capmany, “Spectral characterization of differential group delay in uniform fiber Bragg gratings,” Opt. Express **13**, 9954–9960 (2005). [CrossRef] [PubMed]

35. S. H. Lin, K. Y. Hsu, and P. Yeh, “Experimental observation of the slowdown of optical beams by a volume-index grating in a photorefractive LiNbO_{3} crystal,” Opt. Lett. **25**, 1582–1584 (2000). [CrossRef]

35. S. H. Lin, K. Y. Hsu, and P. Yeh, “Experimental observation of the slowdown of optical beams by a volume-index grating in a photorefractive LiNbO_{3} crystal,” Opt. Lett. **25**, 1582–1584 (2000). [CrossRef]

_{p}= c/n

_{0}is the phase velocity of lights in the host medium in the absence of the volume index grating. A group index n

_{g}= c/v

_{g}of 7.5 was obtained in a 3.5-cm lithium niobate crystal with a refractive index modulation of 2.1×10

^{-5}.

*s*/κ, cosγ = Δ

*k*/2κ, and φ = Δ

*kD*- 2

*k*

_{0}d for convenience. The group velocity of lights through a 2-layer SPVIG can be obtained by differentiating the phase shift per unit length with respect to the angular frequency ω

_{0}= 1.55, D = 1.46 mm, d = 0.25 μm, Λ = 0.5 μm and n

_{1}= 4 × 10

^{-4}, respectively. For comparison, the group velocity and the transmittance of a SG with n

_{0}= 1.55, a thickness of 2D + d, Λ = 0.5 μm and n

_{1}= 4 × 10

^{-4}are also shown in Fig. 6 (b). Note that we neglect the dispersion of the refractive index n

_{0}in the calculation because what we consider here is the dispersion induced by the structure of the refractive index distribution instead of the refractive index of the material itself. This is reasonable for most optical materials such as photorefractive lithium niobate crystals and optical fibers without involvement of the nonlinear effects. It is seen that superluminal light propagation is demonstrated at/near the Bragg-matched wavelength in both cases, while the group velocity in the 2-layer SPVIG case is faster by a factor of ∼ 10 as compared to that in the SG case. Moreover, the transmittance of the superluminal lights in the 2-layer SPVIG case is larger than 80%, whereas that in the SG case is less than 5% due to the Bragg-reflection effect. Figure 7 shows the group delay τ

_{g}(defined as τ

_{g}= L/v

_{g}, where L is the total thickness of the SPVIG or the SG) as a function of the thickness D for the 2-layer SPVIGs and the SGs at the Bragg-matched wavelength λ

_{0}= 1.55 μm. The parameters for n

_{0}, d, Λ and n

_{1}are set to be 1.55, 0.25 μm, 0.5 μm and 4 × 10

^{-4}, respectively. The thickness of the SG is L = 2D + d. It is seen that the group delay in the SG case increases first but then tends to be saturated with the increase of D. The group delay in the 2-layer SPVIG case varies first slowly but then becomes negative and decreases rapidly with the increase of D. A negative group delay of ∼ 30 ns is possible with D = 5 mm. This is because the transparency peak becomes sharper and sharper; therefore, the dispersion slope becomes steeper and steeper with increasing D in the 2-layer SPVIG case. Note that the time scale for the SPVIG is nano-second while that for the SG is pico-second in Fig. 7. Figure 8 illustrates the control of the group delay through the phase variation 2k

_{0}d induced by the buffer layer. The parameters for the SPVIG is n

_{0}= 1.55, D = 2 mm, Λ = 0.5 μm, and n

_{1}= 4 × 10

^{-4}, respectively. The operating wavelength is set to be Bragg-matched at 1.55 μm. We see that, whenever the condition 2k

_{0}d = (2m + 1)π is satisfied which corresponds to the appearance of the transparency peak at the Bragg-matched wavelength, a sharp increase in the group delay is observed. These results clearly illustrate the versatility and the effectiveness of the SPVIG on the control of the group velocity of light through the design of its structure parameters. Experiments on group velocity control through a 2-layer SPVIG by using a photorefractive lithium niobate crystal are currently going on in our laboratory. The photosensitive optical fiber could be an additional good material candidate to fabricate the stratified and phase-shifted volume index gratings.

## 6. Conclusion

## Appendix

## A. Derivation of the phase shift F of the transmitted light through a 2-layer SPVIG

_{2}(x) in the second grating layer is the summation of the forward and the backward propagation waves

_{2}(x) is

_{2}(x) satisfies

_{1}(D) and B

_{1}(D) can be obtained by substituting Eqs. (7) and (10) into Eqs. (5) and (6) under the boundary condition B

_{2}(2D+d) = 0. The amplitude E

_{2}(2D+d) is then expressed as

_{2}(2D+d) is

_{0}(D+ d) induced by the first grating layer and the buffer layer, we obtain the phase shift F of the transmitted light through a 2-layer SPVIG shown by Eq. (15).

## Acknowledgments

## References and links

1. | P. Yeh, |

2. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

3. | R. C. Alferness, C. H. Joyner, M. D. Divino, M. J. R. Martyak, and L. L. Buhl, “Narrowband grating resonator filters in InGaAsP/InP waveguides,” Appl. Phys. Lett. |

4. | G. P. Agrawal and S. Radic, “Phase-Shifted Fiber Bragg Gratings and their Application for Wavelength Demultiplexing,” IEEE Photon. Technol. Lett. |

5. | R. Zengerle and O. Leminger, “Phase-shifted Bragg-Grating Filters with Improved Transmission Characteristics,” J. Lightwave Technol. |

6. | L. Wei and J. w. Y. Lit, “Phase-Shifted Bragg Grating Filters with Symmetrical Structures,” J. Lightwave Tech-nol. |

7. | F. Bakhti and P. Sansonetti, “Design and Realization of Multiple Quater-Wave Phase-Shifts UV-Written Bandpass Filter in Optical Fibers,” J. Lightwave Technol. |

8. | Ch. Martinez and P. Ferdinand, “Analysis of phase-shifted fiber Bragg gratings written with phase plate,” Appl. Opt. |

9. | S. Longhi, M. Marano, P. Laporta, O. Svelto, and M. Belmonte, “Propagation, manipulation, and control of picosecond optical pulses at 1.5 μm in fiber Bragg gratings,” J. Opt. Soc. Am B |

10. | Y. Painchaud, A. Chandonnet, and J. Lauzon, “Chirped fibre gratings produced by tilting the fibre,” Electron. Lett. |

11. | B. Malo, S. Thériault, D. C. Johnson, F. Bilodeau, J. Albert, and K. O. Hill, “Apodised in-fibre Bragg grating reflectors photoimprinted using a phase mask,” Electron. Lett. |

12. | Ch. Martinez, S. Magne, and P. Ferdinand, “Apodized fiber Bragg gratings manufactured with the phase plate process,” Appl. Opt. |

13. | R. V. Johnson and A. R. Tanguay, “Stratified volume holographic optical elements,” Opt. Lett. |

14. | G. P. Nordin, R. V. Johnson, and A. R. Tanguay, “Diffraction properties of stratified volume holographic optical elements,” J. Opt. Soc. Am. A |

15. | R. De Vré and L. Hesselink, “Analysis of photorefractive stratified volume holographic optical elements,” J. Opt. Soc. Am. B |

16. | J. J. Stankus, S. M. Silence, W. E. Moerner, and G. C. Bjorklund, “Electric-field-switchable stratified volume holograms in photorefractive polymers,” Opt. Lett. |

17. | V. M. Petrov, C. Caraboue, J. Petter, T. Tschudi, V. V. Bryksin, and M. P. Petrov, “A dynamic narrow-band tunable optical fliter,” Appl. Phys. B |

18. | Y. Lai, W. Zhang, L. Zhang, J. A. R. Williams, and I. Bennion, “Optically tumable fiber grating transmission filters,” Opt. Lett. |

19. | A. D’Orazio, M. De Sario, V. Petruzzelli, and F. Prudenzano, “Photonic band gap filter for wavelength division multiplexer,” Opt. Express |

20. | M. Yamada and K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,” Appl. Opt. |

21. | M. McCall, “On the application of coupled mode theory for modeling fiber Bragg gratings,” J. Lightwave Tech-nol. |

22. | M. A. Rodriguez, M. S. Malcuit, and J. J. Butler, “Transmission properties of refrective index-shifted Bragg gratings,” Opt. Commun. |

23. | S. Khorasani and K. Mehrany, “Differential transfer-matrix method for solution of one-dimensional linear non-homogeneous optical sturctures,” J. Opt. Soc. Am. B |

24. | S. Khorasani and A. Adibi, “New analytical approach for computation of band structure in one-dimensional periodic media,” Opt. Commun. |

25. | J. J. Monzón, T. Yonte, and L. L. Sánchez-Soto, “Charcterizing the reflectance of periodic layered media,” Opt. Commun. |

26. | I. S. Nefedov and S. A. Tretyakov, “Photonic band gap structure containing metamaterial with negative permittivity and permeability,” Phys. Rev. E |

27. | J. R. Birge and F. X. Kärtner, “Efficient analytic computation of dispersion from multilayer structures,” Appl. Opt. |

28. | D. Yevick and L. Thylén, “Analysis of gratings by the beam-propagation method,” J. Opt. Soc. Am. |

29. | L. Thylen and D. Yevick, “Beam propagation method in anisotropic media,” Appl. Opt. |

30. | L Thylen and Ch. M. Lee, “Beam-propagation method based on matrix diagonalization,” J. Opt. Soc. Am. A |

31. | Y. Tsuji, M. Koshiba, and N. Takimoto, “Finite element beam propagation method for anisotropic optical waveguides,” J. Lightwave Technol. |

32. | P. K. Kelly and M. Piket-May, “Propagation characteristics for a one-dimensional grounded finite height finite length electromagnetic crystal,” J. Lightwave Technol. |

33. | L. A. Coldren and S. W. Corzine, |

34. | M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E |

35. | S. H. Lin, K. Y. Hsu, and P. Yeh, “Experimental observation of the slowdown of optical beams by a volume-index grating in a photorefractive LiNbO |

36. | S. Zhu, N. Liu, H. Zheng, and H. Chen, “Time delay of light propagation through defect modes of one-dimensional photonic band-gap structures,” Opt. Commun. |

37. | J. Liu, B. Shi, D. Zhao, and X. Wang, “Optical delay in defective photonic bandgap structures,” J. Opt. A: Pure Appl. Opt. |

38. | S. Bette, C. Caucheteur, M. Wuilpart, P. Mégret, R. Garcia-Olcina, S. Sales, and J. Capmany, “Spectral characterization of differential group delay in uniform fiber Bragg gratings,” Opt. Express |

**OCIS Codes**

(050.2770) Diffraction and gratings : Gratings

(060.1810) Fiber optics and optical communications : Buffers, couplers, routers, switches, and multiplexers

(230.4170) Optical devices : Multilayers

(260.2030) Physical optics : Dispersion

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: November 14, 2006

Revised Manuscript: January 12, 2007

Manuscript Accepted: January 12, 2007

Published: March 5, 2007

**Citation**

Guoquan Zhang, Weiyue Che, Bin Han, and Yiling Qi, "Recursion formula for reflectance and the enhanced effect on the light group velocity control of the stratified and phase-shifted volume index gratings," Opt. Express **15**, 2055-2066 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-5-2055

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### References

- P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).
- R. C. Alferness, C. H. Joyner, M. D. Divino, M. J. R. Martyak, and L. L. Buhl, "Narrowband grating resonator filters in InGaAsP/InP waveguides," Appl. Phys. Lett. 49, 125-127 (1986). [CrossRef]
- G. P. Agrawal and S. Radic, "Phase-Shifted Fiber Bragg Gratings and their Application for Wavelength Demultiplexing," IEEE Photon. Technol. Lett. 6, 995-997 (1994). [CrossRef]
- R. Zengerle and O. Leminger, "Phase-shifted Bragg-Grating Filters with Improved Transmission Characteristics," J. Lightwave Technol. 13, 2354-2358 (1995). [CrossRef]
- L. Wei and J. W. Y. Lit, "Phase-Shifted Bragg Grating Filters with Symmetrical Structures," J. Lightwave Technol. 15, 1405-1410 (1997). [CrossRef]
- F. Bakhti and P. Sansonetti, "Design and Realization of Multiple Quater-Wave Phase-Shifts UV-Written Bandpass Filter in Optical Fibers," J. Lightwave Technol. 15, 1433-1437 (1997). [CrossRef]
- Ch. Martinez and P. Ferdinand, "Analysis of phase-shifted fiber Bragg gratings written with phase plate," Appl. Opt. 38, 3223-3228 (1999). [CrossRef]
- S. Longhi, M. Marano, P. Laporta, O. Svelto, and M. Belmonte, "Propagation, manipulation, and control of picosecond optical pulses at 1.5 μm in fiber Bragg gratings," J. Opt. Soc. Am B 19, 2742-2757 (2002). [CrossRef]
- Y. Painchaud, A. Chandonnet, and J. Lauzon, "Chirped fibre gratings produced by tilting the fibre," Electron. Lett. 31, 171-172 (1995). [CrossRef]
- B. Malo, S. Thériault, D. C. Johnson, F. Bilodeau, J. Albert, and K. O. Hill, "Apodised in-fibre Bragg grating reflectors photoimprinted using a phase mask," Electron. Lett. 31, 223-225 (1995). [CrossRef]
- Ch. Martinez, S. Magne, and P. Ferdinand, "Apodized fiber Bragg gratings manufactured with the phase plate process," Appl. Opt. 41, 1733-1740 (2002). [CrossRef] [PubMed]
- R. V. Johnson, A. R. Tanguay, "Stratified volume holographic optical elements," Opt. Lett. 13, 189-191 (1988). [CrossRef] [PubMed]
- G. P. Nordin, R. V. Johnson, A. R. Tanguay, "Diffraction properties of stratified volume holographic optical elements," J. Opt. Soc. Am. A 9, 2206-2217 (1992). [CrossRef]
- R. De Vré and L. Hesselink, "Analysis of photorefractive stratified volume holographic optical elements," J. Opt. Soc. Am. B 11, 1800-1808 (1994). [CrossRef]
- J. J. Stankus, S. M. Silence, W. E. Moerner, and G. C. Bjorklund, "Electric-field-switchable stratified volume holograms in photorefractive polymers," Opt. Lett. 19, 1480-1482 (1994). [CrossRef] [PubMed]
- V. M. Petrov, C. Caraboue, J. Petter, T. Tschudi, V. V. Bryksin, and M. P. Petrov, "A dynamic narrow-band tunable optical filter," Appl. Phys. B 76, 41-44 (2003). [CrossRef]
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