Grating induced transparency (GIT) and the dark mode in optical waveguides

doi:10.1364/OE.17.011710

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Grating induced transparency (GIT) and the dark mode in optical waveguides

Hsi-Chun Liu and Amnon Yariv »View Author Affiliations

Optics Express, Vol. 17, Issue 14, pp. 11710-11718 (2009)
http://dx.doi.org/10.1364/OE.17.011710

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▸ Article Outline
– Abstract
1. Introduction
2. Grating induced transparency and the Dark mode
3. Uniform GIT waveguide
4. Adiabatic GIT waveguide
5. Discussion and conclusion
– References and links

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Abstract

We propose and describe a new class of optical modes consisting of superposition of three waveguide modes which can be supported by a few-mode waveguide spatially modulated by two co-spatial gratings. These supermodes bear a close, but not exact, formal analogy to the three-level quantum states involved in EIT and its attendant slow light propagation characteristics. Of particular interest is the supermode which we call the dark mode in which, in analogy with the dark state of EIT, one of the three uncoupled waveguide modes is not excited. This mode has unique dispersion characteristics that translate into a slow light propagation which possesses high bandwidth-delay product and can form the basis for a new generation of optical resonators and lasers.

1. Introduction

Electromagnetically induced transparency (EIT) is a phenomenon that comes from quantum destructive interference between excitation pathways to the upper level of an atomic three-level system [1

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

]. The combination of absorption cancellation and strong dispersion has led to the observation of very slow and stored light [2

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]

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001). [CrossRef]

]. In these experiments, atoms are prepared in a “dark”, coherent superposition of the two lower levels, a stationary eigenstate of the system of a three-level atom and two laser fields.

Classical analogs of EIT can be established in coupled optical resonators, referenced as coupled-resonator-induced transparency (CRIT) [4

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]

], where the mode splitting is a classical counterpart of dressed states arising from ac Stark effect. Several configurations of coupled resonators to obtain EIT-like resonances have been proposed and experimentally demonstrated [4

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]

–12

Y. F. Xiao, B. K. Min, X. Jiang, C. H. Dong, and L. Yang, “Coupling Whispering-Gallery-Mode Microcavities With Modal Coupling Mechanism,” IEEE J. Quantum Electron. 44(11), 1065–1070 (2008). [CrossRef]

]. These EIT-like resonances demonstrate much narrower linewidth than those of individual resonators.

In this paper we propose to use a three-mode waveguide modulated by two co-spatial gratings as an optical analog to EIT. The three waveguide modes play the roles of the three quantum states in EIT, while the gratings are counterparts of the electromagnetic waves. Unlike CRIT, where the waveguide-resonator and inter-resonator coupling occur only at discrete points, the coupling of waveguide modes by gratings is continuous along the waveguide. This enables us to describe our system by coupled mode equations, in analogy to the Hamiltonian in the EIT system. The counterpart of the dark state in EIT is one of the supermodes of the waveguide-plus-grating system. By inserting phase shifts or apodizing the gratings, we are able to utilize the slow dark mode and produce EIT-like resonance. The structures can be a new class of optical resonators or optical delay lines.

2. Grating induced transparency and the Dark mode

Guided by the formalism of the EIT three quantum state atomic system, we consider a waveguide supporting three distinct spatial modes with complex amplitudes a, b, c, propagation constants β_a , β_b , β_c , and containing two co-spatial gratings (one connects a and b; the other connects b and c). We do not restrict the direction of propagation of each mode. In the case of weak perturbation, the eigenmodes of the waveguide-plus-grating system, the supermodes, can be taken as linear superpositions of the unperturbed modes a, b, c, which obey the coupled Eqs. (13):

\frac{d a}{d z} = - j β_{a} a + κ_{ab} \exp (- j \frac{2 π}{Λ_{ab}} z) b

\frac{d b}{d z} = - j β_{b} b + κ_{ba} + \exp (j \frac{2 π}{Λ_{ab}} z) a + κ_{bc} \exp (j \frac{2 π}{Λ_{cb}} z) c,

\frac{d c}{d z} = - j β_{c} c + κ_{cb} \exp (- j \frac{2 π}{Λ_{cb}} z) b

(1)

where Λ _ab =2π/(β _a0-β _b0) and Λ _bc =2π/(β _c0-β _b0) are the periods of the two gratings designed for a center frequency ω ₀. κ_ij (i, j=a, b, c) are the coupling coefficient connecting modes i and j. By introducing new variables A, B, C, defined by a(z)=A(z)exp(-jβ _a0z), b(z)=B(z)exp(-jβ _b0z), c(z)=C(z)exp(-jβ _c0z), we arrive at a z-independent coupling matrix:

\frac{d}{d z} [\begin{array}{c} A \\ B \\ C \end{array}] = [\begin{array}{c} - j δ_{a} & κ_{ab} & 0 \\ κ_{ba} & - j δ_{b} & κ_{bc} \\ 0 & κ_{cb} & - j δ_{c} \end{array}] \cdot [\begin{array}{c} A \\ B \\ C \end{array}],

(2)

where δ_i ≡β_i -β _i0 is the detuning of the propagation constant. In the neighborhood of ω ₀, δ_i =n_i (Δω)/c where n_i is the group index of each mode and Δω=ω-ω ₀. The eigenvectors of the matrix are the supermodes of the perturbed waveguide, while the eigenvalues are the propagation constants of the supermodes. For example, the electric field of a supermode [v_a v_b v_c ]^T with a propagation constant µ is given by exp(-iµz) [v_a exp(-iβ _a0 z) E_a (x, y)+v_b exp(-iβ _b0 z) E_b (x, y)+v_c exp(-iβ _c0 z) E_c (x, y)], where E_i (x, y) is the transverse mode profile. At center frequency ω ₀, one of the supermodes is proportional to [κ_bc 0-κ_ba ]^T, with the eigenvalue equal to 0. In this supermode, which we name the dark mode, the intermediate mode B is unexcited. For comparison, the coupled equation of EIT ignoring decay rates is [1

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

]

\frac{d}{d t} = ∣ Ψ 〉 = [\begin{array}{c} - j Δ ω & - \frac{j}{2} Ω_{p} & 0 \\ - \frac{j}{2} Ω_{p} & 0 & - \frac{j}{2} Ω_{c} \\ 0 & - \frac{j}{2} Ω_{c} & 0 \end{array}] | Ψ 〉,

(3)

where Ω _p and Ω _c are the Rabi frequencies of the probe and coupling beams, respectively, and Δω is the frequency detuning of the probe beam. Equations (2) and (3) are similar on resonance. They are respectively equations of space and time. Conservation of energy defines the relation between κ_ij and κ_ji [13

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University Express, New York, 2007).

]: If mode i and j copropagate, κ_ij =-κ_ji * ; if mode i and j counterpropagate, κ_ij =κ_ji *. Suppose A is the input forward mode, there are four different choices of propagating directions of mode B and C (Fig. 1). The formal similarity between Eqs. (2) and (3) depends on the nature of the κ_ij coefficients, which implies both modes B and C are forward propagating (Fig. 1(a)). This makes sense since in quantum mechanics each state evolves forward with time. Nevertheless, the choice which is the most interesting for our slow light phenomenology is one where mode A is forward while B and C are backward propagating (Fig. 1(d)). A short-period grating connects modes A and B (κ_ab ), while a long-period grating connects modes B and C (κ_bc ). The coupling constants obey κ_bc =-κ_cb * and κ_ab =κ_ba *. By shifting constant phases of variables A, B, and C, Eq. (2) can be rewritten as

\frac{d}{d z} [\begin{array}{c} A \\ B \\ C \end{array}] = [\begin{array}{c} - j δ_{a} & κ_{ab} & 0 \\ κ_{ab} & - j δ_{b} & j δ_{bc} \\ 0 & j κ_{bc} & - j δ_{c} \end{array}] \cdot [\begin{array}{c} A \\ B \\ C \end{array}] \equiv K \cdot [\begin{array}{c} A \\ B \\ C \end{array}],

(4)

Fig. 1. Four configurations of the directions of the three modes. The black grating (/) couples modes A and B, while the red grating (\) couples mode B and C. The gratings are short-period or long-period depending on the directions of the connected modes.

where K is defined as the coupling matrix. Now both κ_ab and κ_bc are real numbers. At Δω=0, if |κ_ab |≠|κ_bc |, the corresponding eigenvectors and eigenvalues are:

Dark mode:

\frac{1}{\sqrt{κ_{ab}^{2} + κ_{bc}^{2}}} [\begin{array}{c} κ_{bc} \\ 0 \\ j κ_{ab} \end{array}], eigenvalue = 0;

(5)

the two bright modes

\propto [\pm {\sqrt{κ_{bc}^{2} - κ_{ab}^{2}}}_{κ_{bc}}^{- j κ_{ab}}], eigenvalues = \pm j \sqrt{κ_{bc}^{2} - κ_{ab}^{2}} \equiv \pm j Δ β_{0} .

(6)

The three eigenvectors are not orthogonal to each other. They are nearly parallel when |κ_ab | and |κ_bc | are nearly equal. The two bright modes can propagate only if |κ_ab |<|κ_bc |, in which case the propagation constants ±Δβ ₀ are real. Interestingly, the dark mode is a “propagating” (non-evanescent) supermode consisting of a superposition of two counter-propagating waves, A and C, which results in a group velocity that falls between the positive group velocity of the forward wave A and the negative group velocity of the backward wave C and can be derived by perturbation theory as

v_{g, dark} = \frac{1 - α^{2}}{n_{a} + α^{2} n_{c}} . c,

(7)

where the tuning factor α≡κ_ab/κ_bc . The group velocity is reduced to 0 when α approaches 1. We focus on the region |α|<1 where the group velocity is positive. Plotting the propagation constants of the three supermodes as functions of Δω results in the band structure shown in Fig. 2 where α=0.9. In general the waveguide is not periodic, unless the ratio of Λ _ab and Λ _bc is rational. Therefore, the band structure has two sets of Brillouin zones that repeat the dispersion curves periodically. The three modes intersect at Δω=0 and anti-crossing occurs. A narrow transmission band in the center lies between two band gaps. All the three supermodes can propagate within the transmission band, while no forward-propagating mode exists within the two band gaps since the only propagating supermode consists mainly of backward modes B and C. The group velocity dispersion (GVD) of the dark mode in the center (Δω=0) is zero. The group velocity of the two bright modes is v _g,bright=-(1-α ²)·c/n in the assumption that n_a ≈n_b ≈n_c ≈n.

Fig. 2. (a) Band structures of a GIT waveguide. n_a =1.5, n_b =2.5, n_c =2. κ_ab =90/m, κbc=100/m. Dash lines are the band structure without grating perturbation. (b) Zoom-in figure of the bending region. The red dashed curve is the dark mode.

3. Uniform GIT waveguide

To take advantage of the new modes, especially the dark one, we propose two types of structures. The first is a uniform one where both gratings (κ_ab and κ_bc ) are of uniform strengths along the length L of the waveguide. If the boundary condition corresponds to a single input mode A at z=0 and no input backward modes B and C at z=L, the field at z=L is proportional to [1 0 0]^T, which decomposes into all the three supermodes. Transmission of mode A and field distribution along the structure can be obtained by back propagating the three supermodes to z=0. Unity transmission at ∆ω=0 results if the length is a multiple of

L_{min} = 2 π ⁄ \sqrt{κ_{bc}^{2} - κ_{ab}^{2}}

since 0 and

\pm \sqrt{κ_{bc}^{2} - κ_{ab}^{2}}

are the propagation constants of the three supermodes. Figures 3(a) and 3(b) show the transmission spectrum of a waveguide with α=0.9, κ_bc =1,000/m, and L=L_min =1.44 cm. The transmission spectrum is similar to that of EIT. The FWHM bandwidth is 66.2 MHz, and the group delay in the center wavelength is 4.97 ns with group velocity reduction of 71.3 (compared to group velocity of mode A).

Fig. 3. (a) Transmission spectrum of a uniform structure. κ_ab =900/m, κ_bc =1,000/m, and L=L_min =1.44 cm. The refractive indices are n_a =1.45, n_b =1.425, and n_c =1.4. (b) Transmission spectrum in a narrower span of the same structure in (a). (c) Transmission spectrum of a uniform structure with L=6L _min . (d) Transmission spectrum of a uniform structure with L=6 L_min and periodic inversion of κ_ab . (e) Group delay of the structure in (d).

The case of uniform gratings occupying 0<z<L can be viewed as a resonator where the two backward bright modes are reflected into the forward dark mode at z=0 and conversely at z=L. The round-trip phase of the cavity includes the forward propagation of the dark mode, the backward propagation of the two bright modes, and the phase of reflectance of the two mirrors. This explains why L has to be a multiple of L_min for resonance at Δω=0. Since the transmission of mode A depends on phase matching of the three supermodes, the bandwidth is extremely narrow and the group velocity reduction is much larger than that of the dark mode in Eq. (7). At Δω=0, the field distributions of the individual modes are

A (z) = \frac{1}{1 - α^{2}} [1 - α^{2} \cos (Δ β_{0} z)]

B (z) = \frac{α}{\sqrt{1 - α^{2}}} \sin (Δ β_{0} z),

C (z) = \frac{j α}{1 - α^{2}} [1 - \cos (Δ β_{0} z)]

(8)

assuming the input A(0)=1. Figure 4 plots the field distribution for α=0.95 and L=2L_min . Since the three supermodes are nearly parallel, they interfere constructively to extremely high intensities of individual modes A, B, C in the middle of the structure, and destructively interfere to small intensity of A at the two ends. The quality factor Q of the resonator defined as ω·(energy stored)/(power loss) can be obtained from Eq. (8) assuming n_a ≈n_b ≈n_c and is given by

Q = β_{a} L \frac{1 + 2 α^{2}}{{(1 - α^{2})}^{2}},

(9)

where L is fixed to be a multiple of L_min . If α is close to 1, say 0.999 for example, the factor (1+2α ²)/(1-α ²)² is as high as 7.5×10⁵, while the length required is 14 cm assuming κ_bc =1,000/m. By employing gain in the structure with a high quality factor we can make a laser with the minimum threshold gain occurring when L is a multiple of L_min and is inversely proportional to the quality factor.

Fig. 4. Energy distribution of a uniform structure with L=2L_min and an input A(0)=1. κ_ab =900/m, κ_bc =1,000/m, and L=2.88 cm. The refractive indices are n_a =1.45, n_b =1.425, and n_c =1.4.

For the example in Figs. 3(a) and 3(b), the phase shift across the transmission band is about π, limiting the delay-bandwidth product less than 0.5, like a single resonator. Cascading N resonators, namely increasing L to N·L_min , will result in more delay but reduced bandwidth, since the phase mismatch of the three supermodes is approximately proportional to (Δω)L. In fact, as shown in Fig. 3(c), there are 2N-1 peaks in the transmission spectrum, each with a phase shift of π across, limiting the delay-bandwidth product still less than 0.5. To break this limit, we propose to invert the sign of α(=κ_ab /κ_bc ) every L_min along the waveguide, as illustrated in Fig. 5(a). By doing this periodic inversion, it can be shown that the first-order term of the reflectance at z=0 as a function of Δω is cancelled out when N is even. Figures 3(d) and 3(e) show the transmission spectrum and group delay of the modified waveguide with L=N·L_min , where the bandwidth is not reduced and the envelope of the ripples is approximately a quadratic function. The group delay at Δω=0 and the quality factor of the modified waveguide is exactly the same as the original one, while the delay-bandwidth product is unlimited and the total phase shift across the bandwidth is Nπ.

Fig. 5. (a) κ_ab and κ_bc in a uniform structure with periodic inversion. (b) κ_ab and κ_bc in an adiabatic structure. L_ad and L_uni are the lengths of the adiabatic and uniform region, respectively.

4. Adiabatic GIT waveguide

The second structure limiting the excitation to the dark mode alone is achieved by employing adiabatic transition [14

E. Peral and A. Yariv, “Supermodes of grating-coupled multimode waveguides and application to mode conversion between copropagating modes mediated by backward Bragg scattering,” J. Lightwave Technol. 17(5), 942–947 (1999). [CrossRef]

,15

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, New York and London, 1974).

]. This is accomplished by changing the structure (waveguide + gratings) sufficiently slowly along the direction of propagation (z) so that the propagating wave remains in the same local eigenmode. As shown in Fig. 5(b), the waveguide consists of two adiabatic regions at the two ends and a uniform region in the middle. κ_bc is set to be a constant throughout the structure, while κ_ab , and thus α, are functions of z. The input mode A entering the structure encounters only grating κ_bc and is thus the dark mode (Eq. (5)) where it will remain till exiting the structure. As it propagates, the grating κ_ab is turned on spatially adiabatically. In the middle of the structure, the dark mode which now has a mode C component possesses a small group velocity. At the output, κ_ab is adiabatically decreased to 0, and the propagating wave is transformed back, always staying dark, to mode A at the exit.

The adiabatic condition ensuring that a wave remains in the same (dark) mode is defined as the situation where the fraction of the energy converted to other modes never exceeds an arbitrarily chosen small number ε. The adiabatic condition of the GIT waveguide and an optimal function α(z) can be derived in a similar way as in quantum mechanics [16

L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955).

] and an adiabatic mode converter [17

X. Sun, H.-C. Liu, and A. Yariv, “Adiabaticity criterion and the shortest adiabatic mode transformer in a coupled-waveguide system,” Opt. Lett. 34(3), 280–282 (2009). [CrossRef]

], with careful treatment of the non-orthogonal basis. The three eigenvectors in Eqs. (5) and (6) should be renormalized such that the total forward energy flow is 1 or -1, given by

v_{dark} = \frac{1}{\sqrt{1 - α^{2}}} [\begin{array}{c} 1 \\ 0 \\ j α \end{array}], v_{bright - 1, 2} = \frac{1}{\sqrt{2 (1 - α^{2})}} [{\sqrt{1 - α^{2}}}_{1}^{- j α}] .

(10)

When decomposing a vector x =[A B C]^T into the three supermodes, the amplitude of each supermode can be determined by the product with the associated left eigenvectors of the coupling matrix K since u ^T _iv_j =δ_ij , where u^T _i and v_i are respectively the i-th left and right eigenvectors associated with the eigenvalue λ_i of the coupling matrix K . The fields x can be written as the superposition of the supermodes v_i ,

x (z) = \sum_{i = d ark, bright} a_{i} (z) v_{i} (z) \exp [- j \int_{0}^{z} β_{i} (z') d z'] .

(11)

Substituting Eq. (11) into the coupled equations

\frac{d}{d z} x (z) = K (z) x (z)

(Eq. (4)) leads to

\sum_{i = dark, bright} [(\partial_{z} a_{i}) v_{i} + a_{i} (\partial_{z} v_{i})] \exp [- j \int_{0}^{z} β_{i} (z') d z'] = 0 .

(12)

For adiabatic transition of the Dark mode ( v_dark ), a_dark ≈1 and a _bright-1,2≈0. Multiplying u _bright-1 ^T from the left side results in

{d a}_{bright - 1} / d z = u_{bright - 1}^{T} (\partial_{z} v_{dark}) \exp [j \int_{0}^{z} β_{bright - 1} (z') d z'],

(13)

since β_dark =0. u _bright-1 ^T (∂_z v_dark ) can be shown to be

\frac{j}{\sqrt{2} (1 - α^{2})} \frac{\partial α}{\partial z},

, and with the assumption that it is slowly varying compared to the exponential term, Eq. (13) can be solved as

a_{bright - 1} (z) = \frac{- \sqrt{2 j}}{κ_{bc} {(1 - α^{2})}^{3 ⁄ 2}} \frac{\partial α}{\partial z} \sin (\frac{β_{brigth - 1}}{2} z) \exp [j \frac{β_{bright - 1}}{2} z]

(14)

Likewise for a _bright-2(z). The adiabatic condition where only the Dark mode exists requires |a _bright-1(z)|²+|a _bright-2(z)|²≤ε, which gives

\frac{d α}{d z} \leq \frac{κ_{bc}}{4} {(1 - α^{2})}^{\frac{3}{2}} \sqrt{ε} .

(15)

The closer α is to 1, the smaller dα/dz is required to achieve adiabaticity. By solving Eq. (15) with equality, we obtain an optimal function α(z) given by

α (z) = \sin^{- 1} (\tan (a z)),

(16)

where a is a scaling constant. It takes infinite length to transform α to 1 to reach zero group velocity. Figure 6 shows the transmission spectrum and group delay of an adiabatic GIT structure, where α of the uniform region is 0.9 and the total length is 10 cm. The transmission spectrum resembles a rectangular function, with a bandwidth of 4.48 GHz, equal to that of the dark mode of the uniform region. The group delay mainly depends on the group velocity of the dark mode of the uniform region and is 2.92 ns at the center wavelength with group velocity reduction of 6.04. The delay-bandwidth product, which is 13.1 in this case, can be made arbitrarily large by increasing the length of the uniform region without reducing the bandwidth.

Fig. 6. (a) Transmission and (b) group delay of an adiabatic structure. L_ad =3 cm, L_uni =4 cm, κ_bc =4,000/m, and α _max=0.9. The refractive indices are n_a =1.45, n_b =1.425, and n_c =1.4.

5. Discussion and conclusion

We have analyzed the EIT-analog waveguide and proposed two types of structures to control the propagation of supermodes. This GIT waveguide can be implemented using a few-mode fiber or a few-mode on-chip waveguide. One short-period grating and another long-period grating are imposed on the waveguide. For a waveguide with geometric symmetry, the modes are either symmetric or anti-symmetric, so are the gratings that couple the modes. Therefore, the two gratings can be independent of each other if they have opposite symmetry. The symmetry of a grating can be controlled by tilting the grating in fibers or designing the transverse profile of a waveguide grating. The phase indices of the three modes should be carefully chosen, in case any unwanted mode is coupled by one of the gratings, since there are three forward and three backward modes in total. Moreover, to reduce loss, coupling into the radiation modes should be prevented.

In conclusion, the new energy storage mechanism resulting from the internal bouncing of waves between two Bragg gratings gives rise to a new class of propagating modes which includes a dark mode with “slow light” characteristics. These modes can potentially form the basis of a new class of narrow band filters, high Q resonators and lasers. The adiabatic GIT structures which excite the dark mode alone possess high delay-bandwidth product.

Acknowledgements

The authors acknowledge the support of Taiwan Merit Scholarship TMS-094-2-A-039.

References and links

1.	S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997). [CrossRef]
2.	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]
3.	C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001). [CrossRef]
4.	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]
5.	L. Maleki, A. B. Matsko, A. A. Savchenkov, and V. S. Ilchenko, “Tunable delay line with interacting whispering-gallery-mode resonators,” Opt. Lett. 29(6), 626–628 (2004). [CrossRef]
6.	M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93(23), 233903 (2004). [CrossRef]
7.	A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, “Induced transparency and absorption in coupled whispering-gallery microresonators,” Phys. Rev. A 71(4), 043804 (2005). [CrossRef]
8.	Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. H. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96(12), 123901 (2006). [CrossRef]
9.	Q. F. Xu, J. Shakya, and M. Lipson, “Direct measurement of tunable optical delays on chip analogue to electromagnetically induced transparency,” Opt. Express 14(14), 6463–6468 (2006). [CrossRef]
10.	K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98(21), 213904 (2007). [CrossRef]
11.	L. Yosef Mario and M. K. Chin, “Optical buffer with higher delay-bandwidth product in a two-ring system,” Opt. Express 16(3), 1796–1807 (2008). [CrossRef]
12.	Y. F. Xiao, B. K. Min, X. Jiang, C. H. Dong, and L. Yang, “Coupling Whispering-Gallery-Mode Microcavities With Modal Coupling Mechanism,” IEEE J. Quantum Electron. 44(11), 1065–1070 (2008). [CrossRef]
13.	A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University Express, New York, 2007).
14.	E. Peral and A. Yariv, “Supermodes of grating-coupled multimode waveguides and application to mode conversion between copropagating modes mediated by backward Bragg scattering,” J. Lightwave Technol. 17(5), 942–947 (1999). [CrossRef]
15.	D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, New York and London, 1974).
16.	L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955).
17.	X. Sun, H.-C. Liu, and A. Yariv, “Adiabaticity criterion and the shortest adiabatic mode transformer in a coupled-waveguide system,” Opt. Lett. 34(3), 280–282 (2009). [CrossRef]

OCIS Codes
(050.2770) Diffraction and gratings : Gratings
(230.5750) Optical devices : Resonators
(230.7370) Optical devices : Waveguides

ToC Category:
Optical Devices

History
Original Manuscript: May 18, 2009
Revised Manuscript: June 21, 2009
Manuscript Accepted: June 21, 2009
Published: June 26, 2009

Citation
Hsi-Chun Liu and Amnon Yariv, "Grating induced transparency (GIT) and the dark mode in optical waveguides," Opt. Express 17, 11710-11718 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-11710

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References

S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50(7), 36-42 (1997). [CrossRef]
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