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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 5 — Mar. 5, 2007
  • pp: 2055–2066
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Recursion formula for reflectance and the enhanced effect on the light group velocity control of the stratified and phase-shifted volume index gratings

Guoquan Zhang, Weiyue Che, Bin Han, and Yiling Qi  »View Author Affiliations


Optics Express, Vol. 15, Issue 5, pp. 2055-2066 (2007)
http://dx.doi.org/10.1364/OE.15.002055


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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

In this paper, we have designed a periodically layered structure, a stratified and phase-shifted volume index grating (SPVIG), consisting of a sequence of discrete thick volume index gratings interleaved with optically homogeneous buffer layers. A recursion formula for the reflectance of the SPVIGs is derived, and its reflectance properties are studied under various conditions. As compared to a single uniform volume index grating (SG), an enhanced effect on the control of light group velocity with a high transmittance by using the SPVIGs is demonstrated.

Fig. 1. The structure of the SPVIG.

2. The structure of SPVIG

Figure 1 shows the structure of the SPVIG, in which N discrete thick volume index grating layers are interleaved with N-1 optically homogeneous buffer layers. The buffer layers provide the phase-shifts between the neighboring grating layers. The thicknesses of the volume index grating layers and the buffer layers are D and d, respectively. For the sake of simplicity, the averaged refractive indices for both the grating layers and the buffer layers are set to be n0. The refractive index modulation amplitude for the grating layers is n1, and the grating wave vector along x-axis is K = 2π/Λ with L being the grating spacing. Therefore the refractive index distribution of the j-th grating layer can be expressed as nj(x) = n0 + n1cos(K(x - (j -1)(D + d))).

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

Supposing a TE (or TM) polarized plane wave E(x,t) = E(x) exp(-iωt) is incident normally onto the SPVIG at x = 0, where E(x) is the complex amplitude of the electric field of the light and ω is the angular frequency of the light. The wave equation describing the light propagation along the x-axis is given by [1

1. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

]

2E(x)x2+[ωcn(x)]2E(x)=0,
(1)

with c being the light speed in vacuum. For the simplest case when there is only one grating layer, the solution of Eq. (1) can be expressed as

E1(x)=A1(x)eik0x+B1(x)eik0x,
(2)

where A1(x) and B1(x) are the complex amplitudes of the forward and the backward propagating waves, respectively, and k0 = 2πn0/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 ∼ n2 1, under the slowly varying amplitude approximation, we obtain two coupled differential equations for A1(x) and B1(x)

A1(x)x=B1(x)eiΔkx,
(3)

and

B1(x)x=A1(x)eiΔkx,
(4)

respectively. Where κ=πn1/λ is the coupling constant, Δk = 2k0 -K is the momentum mismatch. The general analytical solutions for A1(x) and B1(x) can be written as

A1(x)=(A1(0)cosh(sx)+C1sinh(sx))eiΔkx2,
(5)

and

B1(x)=ikeiΔkxA1(x)x,
(6)

respectively. Where A1(0) is the amplitude of the incident light at x = 0, s is given by s = (κ2 - (Δk/2)2)1/2, and C1 is a parameter determined by the boundary conditions. Under the boundary condition B1(D) = 0, the reflection coefficient of the grating layer is given by

r1=B1(0)A1(0)=sinh(sD)scosh(sD)i(Δk2)sinh(sD).
(7)

Now supposing we know the reflection coefficient rN 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 rN+1 of a (N+1)-layer SPVIG. Such a recursion technique for calculation of reflection coefficient is generally used in distributed feedback laser [33

33. L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, New York, 1995).

].

It is evident that the amplitude of the electric field Ej(x) in the j-th grating layer is the summation of the forward and the backward propagation waves Ej(x) = Aj(x)exp(ik0(x-(j-1)(D+ d)))+Bj(x)exp(-ik0(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 C1 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

A1(D)eik0d=A2(D+d),
(8)

and

B1(D)=B2(D+d)eik0d,
(9)

respectively. By combining the expression of the reflection coefficient of the N-layer SPVIG rN = B2(D+d)/A2(D+d), we obtain the parameter C1 as a function of rN

C1=rNcosh(sD)eiΔkD+2ik0d+iΔk2cosh(sD)ssinh(sD)rNsinh(sD)eiΔkD+2ik0diΔk2sinh(sD)+scosh(sD)A1(0).
(10)

By substituting Eq. (10) into Eqs.(5) and (6), we obtain the recursion formula for the reflection coefficient rN+1 of the (N+1)-layer SPVIG

rN+1=B1(0)A1(0)=srNcosh(sD)eiΔkD+2ik0d+iΔk2rNsinh(sD)eiΔkD+2ik0d+sinh(sD)rNsinh(sD)eiΔkD+2ik0diΔk2sinh(sD)+scosh(sD).
(11)

Therefore the reflectance of the (N+1)-layer SPVIG is RN+1 = ∣rN+12 and the transmittance TN+1 can be obtained through TN+1 = 1-RN+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.

Fig. 2. Reflectance spectra of a SG (red) and a 2-layer SPVIG (green). The grating parameters for both cases are Λ = 0.5 μm and n1 = 4 ×10-4. Other parameters are set to be D = 3 mm, d = 2.25 μm and n0 = 1.55, respectively. The thickness of the SG is 6 mm for comparison.

In the following section, we will discuss the characteristics of the reflectance spectra of the SPVIGs.

4. Characteristics of the reflectance spectra of SPVIGs

To study the characteristics of the reflectance spectra and to understand the effects of the buffer layers on the reflectance spectra, let us first consider the simplest SPVIG with N=2, where a buffer layer is sandwiched between two volume index grating layers. By substituting Eq. (7) into Eq. (11), we obtain the reflection coefficient for the 2-layer SPVIG

r2=sinh(sD)(1+scosh(sD)+iΔk2sinh(sD)scosh(sD)iΔk2sinh(sD)eiΔkD+i2k0d)κ2sinh2(sD)scosh(sD)iΔk2sinh(sD)eiΔkD+i2k0d+scosh(sD)iΔk2sinh(sD).
(12)

In the case with a Bragg-matched incident wavelength λ0 for the volume index grating layer, i.e., Δk = 0, the reflection coefficient of the 2-layer SPVIG can be simplified to be

r2=sinh(sD)(1+ei2k0d)κ2sinh2(sD)scosh(sD)ei2k0d+scosh(sD).
(13)

It is evident that the reflection coefficient r2 is equal to zero under the condition 2k0d = (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

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

, 6

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

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

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

]. In the simulation, the structure parameters for the 2-layer SPVIG are n0 = 1.55, n1 =4×10-4, Λ=0.5μm,λ0 = 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.

Fig. 3. The reflectance spectra of the SPVIGs with N=3 (a) and N=10 (b), respectively. The parameters for both cases are set to be n0 = 1.55, n1 = 4×10-4, D = 2 mm, Λ = 0.5 μm, and d = 2.25 μm, respectively.
Fig. 4. Shift of the transparency wavelength within the stop-band by controlling the thickness of the buffer layer in a 2-layer SPVIG. The thicknesses of the buffer layers are 2.15, 2.25 and 2.35 μm for the red, green and blue curves, respectively. Other parameters are the same for three curves: n0 = 1.55, n1 = 4×10-4, D = 3 mm, and Λ = 0.5 μm, respectively.

The reflectance spectrum of a multilayer SPVIG can be calculated easily by using the recursion formula (see Eq. (11)). Figure 3 shows the reflectance spectra of a 3-layer (a) and a 10-layer (b) SPVIGs as examples. The structure parameters are the same as those of Fig. 2 except for D = 2 mm in Fig. 3. It is found that the number of the transparency peaks within the stop-band is equal to N-1, i.e., the number of the buffer layers in the structure. Therefore, two and nine transparency peaks appear in the reflectance spectra of the 3-layer and the 10-layer SPVIGs, respectively. On the other hand, a transparency peak appears at Δk = 0 when N is an even integer, whereas this is not the case when N is an odd integer. This is because the total phase-shift induced by the buffer layers of the SPVIGs is (N-1)×2k0d= (N-1)×(2m+1)π.

Fig. 5. The periodical transparency peaks at the Bragg-matched wavelength λ0 = 1.55 μm with the variation of the phase-shift 2k0d induced by the buffer layer in a 2-layer SPVIG, where n0 = 1.55, D = 3 mm, Λ = 0.5 μm and n1 = 4×10-4, respectively.

An even integer N will result in a destructive interference, whereas an odd integer N leads to a constructive interference. It is evident that the coupled mode solution provides a clear physical insight into the formation of the band structure of the reflectance spectra.

The positions of the transparency peaks can be controlled by adjusting the phase-shift induced by the buffer layer. Figure 4 shows the possibility to shift the transparency wavelength by controlling the thickness of the buffer layer in a 2-layer SPVIG. The transparency wavelength is blue-shifted with an increase in the thickness of the buffer layer, while it is red-shifted with a decrease in the buffer layer thickness. On the other hand, at the Bragg-matched wavelength λ0 the transparency peak appears periodically as a function of the phase-shift 2k0d whenever the condition 2k0d = (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

It is well known that a periodical structure is highly dispersive [1

1. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

, 2

2. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

] and such a dispersive property can be used to control the group velocity vg of light beams [9

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

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

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]

]. Lin et al. [35

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 LiNbO3 crystal,” Opt. Lett. 25, 1582–1584 (2000). [CrossRef]

] studied the dispersive property of a SG and demonstrated the possibility to control the group velocity of light through a SG recorded in an iron-doped lithium niobate crystal. The group velocity of a SG can be expressed as [35

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 LiNbO3 crystal,” Opt. Lett. 25, 1582–1584 (2000). [CrossRef]

]

νg=νp(Δk2)2κ2cosh2(sD)(Δk2)2κ2sinh(sD)sDcosh(sD),
(14)

where vp = c/n0 is the phase velocity of lights in the host medium in the absence of the volume index grating. A group index ng = c/vg of 7.5 was obtained in a 3.5-cm lithium niobate crystal with a refractive index modulation of 2.1×10-5.

We study the dispersive properties of the SPVIG and find that the dispersion of the SPVIG is greatly enhanced as compared to that of the SG. Therefore, the SPVIG offers a great feasibility to control the group velocity of lights to a large extent through the design of the SPVIGs. In the following, we take a 2-layer SPVIG as an example without loss of the generality to illustrate the enhanced dispersion effect and its application to the control of the group velocity of light. Through a lengthy but straightforward calculation (see appendix A), we obtain the phase shift Φ of the transmitted light through a 2-layer SPVIG

Fig. 6. The group velocity and the transmittance T of the light incident normally onto a 2-layer SPVIG (a) and a SG (b), respectively. The blue curves and the green curves are for the group velocity and the transmittance, respectively. The Bragg-matched wavelength is set to be 1.55 μm. The parameters for the 2-layer SPVIG are set to be n0 = 1.55, D = 1.46 mm, d = 0.25 μm, Λ = 0.5 μm and n1 = 4×10-4, respectively. The grating parameters for the SG are the same as those of the SPVIG, and the thickness of the SG is equal to 2D+d.
Φ=2πΛD+2k0d+arctan(sinh2(sD)sinφ+sinγcosγsinh(2sD)sinh2(sD)cosφ+sin2γcosh2(sD)cos2γsinh2(sD)).
(15)

Where we set sinγ = s/κ, cosγ = Δk/2κ, and φ = ΔkD- 2k 0d 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

νg=(2D+d)(Φω)1.
(16)

Figure 6 (a) shows the numerical results of the group velocity and the transmittance T of lights through a 2-layer SPVIG with n0 = 1.55, D = 1.46 mm, d = 0.25 μm, Λ = 0.5 μm and n1 = 4 × 10-4, respectively. For comparison, the group velocity and the transmittance of a SG with n0 = 1.55, a thickness of 2D + d, Λ = 0.5 μm and n1 = 4 × 10-4 are also shown in Fig. 6 (b). Note that we neglect the dispersion of the refractive index n0 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 τg (defined as τg = L/vg, 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 λ0 = 1.55 μm. The parameters for n0, d, Λ and n1 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 2k0d induced by the buffer layer. The parameters for the SPVIG is n0 = 1.55, D = 2 mm, Λ = 0.5 μm, and n1 = 4 × 10-4, respectively. The operating wavelength is set to be Bragg-matched at 1.55 μm. We see that, whenever the condition 2k0d = (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.

Fig. 7. The group delays τg of a 2-layer SPVIG (blue curve) and a SG (green curve) versus the variation of D. The wavelength is set at Bragg-matched and is at 1.55 μm. The parameters for the 2-layer SPVIG are set to be n0 = 1.55, d = 0.25 μm, Λ = 0.5 mm and n1 = 4×10-4, respectively. The thickness of the SG is equal to 2D+d.
Fig. 8. The group delay tg of a 2-layer SPVIG versus the phase variation 2k0d induced by the buffer layer. The wavelength is set to be Bragg-matched at 1.55 μm. The parameters for the 2-layer SPVIG are set to be n0 = 1.55, D = 2 mm, Λ = 0.5 μm and n1 = 4×10-4, respectively.

6. Conclusion

In conclusion, we have obtained the recursion formula for the reflectance of the SPVIGs and studied the spectral characteristics of the SPVIGs based on the recursion formula. Transparency peaks appear within the stop-band of the volume index grating due to the interleave of the homogeneous buffer layers in the SPVIG. The spectral positions of the transparency peaks can be controlled through the phase-shift induced by the buffer layer. The dispersion slope is found to be very steep at/near the transparency peaks in the SPVIG and it can be used to control the group velocity of light to a large extent. The group delay experienced by the light through the SPVIG can be controlled by adjusting the phase shift induced by the buffer layer. Negative group delay in the order of tens of nano-seconds with high transmittance is possible with a centimeter-length SPVIG. Such properties can be applicable to addressable filter, wavelength division multiplexer and de-multiplexer, switching, and controllable optical delay/advance line. As compared to the SG, the SPVIG offers more feasibility and versatility to modify the spectral response and to control the group velocity.

Appendix

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

The amplitude of the electric field E2(x) in the second grating layer is the summation of the forward and the backward propagation waves

E2(x)=A2(x)eik0(x(D+d))+B2(x)eik0(x(D+d)),
(17)

where A2(x) is

A2(x)=(A2(D+d)cosh(s(x(D+d)))+C2sinh(s(x(D+d)))eiΔk(x(D+d))2,
(18)

and B2(x) satisfies

B2(x)=iκeiΔk(x(D+d))A2(x)(x).
(19)

By combining the boundary conditions expressed by Eqs. (8) and (9), we obtain

C2=1s(iκB1(D)eik0d+iΔk2A1(D)eik0d).
(20)

Where A1(D) and B1(D) can be obtained by substituting Eqs. (7) and (10) into Eqs. (5) and (6) under the boundary condition B2(2D+d) = 0. The amplitude E2(2D+d) is then expressed as

E2(2D+d)=A2(2D+d)eik0D,
(21)

where A2(2D+d) is

A2(2D+d)=s2κ2A1(0)AReiAImeik0deiΔkD,
(22)

with the parameters ARe and AIm being

ARe=sinh2(sD)cosφ+sin2γcosh2(sD)cos2γsinh2(sD)
(23)

and

AIm=sinh2(sD)sinφ+cosγsinγsinh(2sD),
(24)

respectively. By taking account of the propagation phase shift k0(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

This work is financially supported by the Key Project of Chinese Ministry of Education under grant 104054, the National Natural Science Foundation of China (grants 60308005, 60678021 and 10334010), the Program for New Century Excellent Talents in University under grant NCET-04-0234, the Program for Changjiang Scholars and Innovative Research Team in University, the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China under grant 704012, the Municipal International Cooperation Program of Tianjin under grant 06YFGHHZ00500, the National Basic Research Program of China under grant 2007CB307002, and CNKBRSF under grant 2006CB921703.

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

22.

M. A. Rodriguez, M. S. Malcuit, and J. J. Butler, “Transmission properties of refrective index-shifted Bragg gratings,” Opt. Commun. 177, 251-257 (2000). [CrossRef]

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 20, 91–96 (2003). [CrossRef]

24.

S. Khorasani and A. Adibi, “New analytical approach for computation of band structure in one-dimensional periodic media,” Opt. Commun. 216, 439–451 (2003). [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]

29.

L. Thylen and D. Yevick, “Beam propagation method in anisotropic media,” Appl. Opt. 21, 2751–2754 (1982). [CrossRef] [PubMed]

30.

L Thylen and Ch. M. Lee, “Beam-propagation method based on matrix diagonalization,” J. Opt. Soc. Am. A 9, 142–146 (1992). [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]

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]

33.

L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, New York, 1995).

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]

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 LiNbO3 crystal,” Opt. Lett. 25, 1582–1584 (2000). [CrossRef]

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. 174, 139–144 (2000). [CrossRef]

37.

J. Liu, B. Shi, D. Zhao, and X. Wang, “Optical delay in defective photonic bandgap structures,” J. Opt. A: Pure Appl. Opt. 4, 636–639 (2002). [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]

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

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  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]
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  21. M. McCall, "On the application of coupled mode theory for modeling fiber Bragg gratings," J. Lightwave Technol. 18, 236-242 (2000). [CrossRef]
  22. M. A. Rodriguez, M. S. Malcuit, and J. J. Butler, "Transmission properties of refrective index-shifted Bragg gratings," Opt. Commun. 177, 251-257 (2000). [CrossRef]
  23. S. Khorasani and K. Mehrany, "Differential transfer-matrix method for solution of one-dimensional linear nonhomogeneous optical sturctures," J. Opt. Soc. Am. B 20, 91-96 (2003). [CrossRef]
  24. S. Khorasani and A. Adibi, "New analytical approach for computation of band structure in one-dimensional periodic media," Opt. Commun. 216, 439-451 (2003). [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]
  29. L. Thylén and D. Yevick, "Beam propagation method in anisotropic media," Appl. Opt. 21, 2751-2754 (1982). [CrossRef] [PubMed]
  30. L Thylén and Ch. M. Lee, "Beam-propagation method based on matrix diagonalization," J. Opt. Soc. Am. A 9, 142-146 (1992). [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]
  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]
  33. L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, New York, 1995).
  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]
  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 LiNbO3 crystal," Opt. Lett. 25, 1582-1584 (2000). [CrossRef]
  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. 174, 139-144 (2000). [CrossRef]
  37. J. Liu, B. Shi, D. Zhao, and X. Wang, "Optical delay in defective photonic bandgap structures," J. Opt. A: Pure Appl. Opt. 4, 636-639 (2002). [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]

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