Abstract

We investigate the electromagnetically induced transparency (EIT) and nonlinear pulse propagation in a Λ-type three-level atomic gas filled in a slot waveguide, in which electric field is strongly confined inside the slot of the waveguide due to the discontinuity of dielectric constant. We find that EIT effect can be greatly enhanced due to the reduction of optical-field mode volume contributed by waveguide geometry. Comparing with the atomic gases in free space, the EIT transparency window in the slot waveguide system can be much wider and deeper, and the Kerr nonlinearity of probe laser field can be much stronger. We also prove that using slot waveguide ultraslow optical solitons can be produced efficiently with extremely low generation power.

© 2013 OSA

1. Introduction

Over the past two decades, quantum interference phenomena has attracted much attention due to their fundamental interest and promising applications for optical and quantum information processing. One of such phenomena is electromagnetically induced transparency (EIT), which can be used to substantially enhance the efficiency of nonlinear optical processes in addition to a large suppression of optical absorption [1]. Another noticeable effect of EIT is drastic reduction of group velocity of optical pulses, which has important applications such as slow light [2, 3], quantum memory [46], quantum phase gates [7, 8], and slow light solitons [9, 10].

However, up to now most works on EIT and optical pulse propagations have been performed with atomic gases in bulk samples [1]. Since optical pulses in such systems are unguided plane waves, interaction strength between optical pulses and quantum emitters is limited and thus EIT effect is weak. It is desirable to use optical waveguides where optical fields are guided and optical energy is concentrated in small spatial regions. Such structures can be used to not only for enhancing EIT effect, but also for raising the efficiency of nonlinear optical processes based on gases phase media such as atoms or molecules. In recent years, there have been several works on EIT and related studies in waveguide structures where atomic or molecular gases are filled in low-index regions, including hollow-core photonic crystal fibers [1117], nanofibers [18], and so on.

Guiding light in low-refractive-index materials (e.g air) is thought to be prohibited in conventional waveguides based on total internal reflection. Usually, multiple dielectric layers [11] or hollow-core photonic crystal fibers [1217] based on external reflections are adopted. However, in order to have high reflections, such structures have relatively large dimensions and are wavelength sensitive. In 2004, Almeida et al. [19] proposed a novel structure called slot waveguide, which consists of a nanometer-size slot filled with a low-index material and embedded in high-index materials. In such structure, light is also guided by total internal reflection but it can be tightly confined and hence largely enhanced in the slot region [20, 21]. Recently, There have been a large amount of research activities on guiding and confining light by using slot waveguides [2232].

In this article, we propose a scheme to enhance quantum interference effect by using a Λtype three-level atomic gas filled in a slot waveguide, in which electric field is strongly confined inside the slot of the waveguide due to the discontinuity of dielectric constant. We find that the EIT effect can be greatly enhanced due to reduction of optical-field mode volume contributed by waveguide geometry. Comparing with atomic gases in free space, the EIT transparency window in the slot-waveguide system is much wider and deeper, and Kerr nonlinearity of probe laser field is much stronger. We also demonstrate that using the slot waveguide ultraslow optical solitons can be produced more efficiently and their generation power is extremely low.

The rest of the article is organized as follows. Sec. 2 describes our theoretical model. Sec. 3 studies the linear propagation of probe field and analyzes its EIT characters. Sec. 4 discusses nonlinear pulse propagation in the slot waveguide system. Finally, the last section summarizes the main results obtained in this work.

2. Model

The slot waveguide we adopt is similar to that suggested in Ref. [19], which is shown in the left part of Fig. 1. It consists of a very thin slot (with width 2a in z-direction) of low-index material (with index nS) embedded between two thick rectangular regions (with width ba on both sides) of high-index material (with index nH), both surrounded by a low-index cladding (with index nC). Sizes in the x- and y-directions of each rectangular region are much larger than a and b. It has been shown [19] that such slot waveguide has the ability for concentrating electromagnetic (EM) field of transverse magnetic (TM) modes basically within the slot region, and hence is very attractive for enhancing radiation-matter interaction. The expressions of guided TM eigenmodes of eigenfrequencies ωm(k) are given in Appendix A. Here k||=(kx2+ky2)1/2 with kx and ky being respectively the wavenumbers in x- and y-directions. The physical reason of the confinement and enhancement of TM modes is quite simple. For interfaces with high-index contrast, Maxwell’s equations require a continuity of normal component of electric displacement vector, which gives that electric field in the slot region is (nH / nS)2 times higher than that in the high-index region.

 

Fig. 1 Left: Schematic of slot waveguide structure. The slot width (index nS) is 2a, the width of the high-index silicon slabs (index nH) is 2b − 2a. The index of the cladding material is nC. Right: Level diagram of the three-level atomic system. Ground state |1〉 couples to the exited |2〉 and |3〉 with the control field Ωc and the probe field Ωp. Δ2 and Δ3 are the detunings of control and probe fields, respectively. Γ3132) is the spontaneous emission decay rate from |3〉 to |1〉 (|2〉). Γ12 and Γ21 are incoherent population exchange rates. Below the level diagram is the coordinate system chosen for theoretical calculations. The slot region is |z| ≤ a, and the slabs are in the region a < |z| < b.

Download Full Size | PPT Slide | PDF

Since kx, ky can take any continuous values, and m takes non-zero integers (i.e. m = 1, 2, 3,···), the guided TM eigenmodes propagate in the xy-plane but is confined in the slot region of the waveguide. For simplicity, we study the lowest-order (i.e. m = 1) TM guided mode, and assume kx = 0, ky = k without loss of generality. Then from the Appendix A we have k = (0, k, 0), k̂ = ey, k = k, and

ETM(r,t)=kh¯ω1(k)2ε0V1u1,k(z)a1(k)ei[kyω1(k)t]+c.c.,
u1,k(z)=c2N1ω1(k)n2(z)[kH1,k(z)ez+idH1,k(z)dzey].
Here ω1(k)=(κH12+k2)1/2c/nH=(k2γS12)1/2c/nS, V1 is the mode volume, ey (ez) is the unit vector along the y (z) direction, u1,k(z) is the mode function satisfying dz|u1,k(z)|2=V1, ε(z) ≡ n2(z) is dielectric function with n(z) = nS (|z| < a), nH (a < |z| < b), and nC (|z| > b) (see Fig. 1). The concrete expressions of V1 and N1 (normalized constant) can be found in the Appendix A (i.e. Eq. (27)) for m = 1.

For convenience, we take k as a function of ω1. Replacing ω1 by ω, the electric-field expression of the lowest-order TM guided mode reads ETM(r,t)=ωω(W1V1)12u1,ω(z)ei[(ω)yωt)]+c.c., with u1,ω(z){c/[N1ωn2(z)]}{[kH1,ω(z)ez+i[dH1,ω(z)/dz]ey}. Here k(ω)=[nH2ω2/c2κH12]1/2=[γS12+nS2ω2/c2]1/2 and ω=h¯ω2ε0W1a1(ω), with W1 being the mode volume without the slot (its expression is given in the Appendix B (i.e. Eq. (30)) for m = 1), which is taken to be a reference mode volume for the discussions in the following.

Our aim is to investigate the resonant interaction between the TM-mode of EM field and quantum emitters that are embedded in the slot of the waveguide. For simplicity, we assume the media in the slot and the cladding regions are air (i.e. nS = nC = 1), and the quantum emitters are gaseous atoms with a Λ-type three-level configuration (see the right part of Fig. 1), which is filled in the slot region. Atomic energy levels are two ground states |1〉 and |2〉 and one exited state |3〉, which couple with a weak, pulsed probe field with center angular frequency ωp = kp/c and half Rabi frequency Ωp (i.e. |1〉 → |3〉 transition) and a strong, continuous-wave control field of center angular frequency ωc = kc/c and half Rabi frequency Ωp (i.e. |2〉 → |3〉 transition). Γ3132) is the spontaneous emission decay rate from |3〉 to |1〉 (|2〉). Γ2112) is the incoherent population exchange rate from |1〉 to |2〉 (|2〉 to |1〉), introduced to reflect the transient relaxation process of the atoms entering and leaving interaction region. We assume that both the probe and control fields belong to the lowest-order TM guided mode given in Eq. (1), which has the form

ETM(r,t)=l=p,c1(W1V1)12u1,l(z)exp{i[k(ωl)yωlt]}+c.c..

The Hamiltonian of the system in interaction picture reads

^=h¯j3Δj|jj|h¯[ζp*(z)Ωp*|13|+ζc*(z)Ωc*|23|+h.c.],
where Δ3ωp − (ω3ω1) and Δ2ωpωc − (ω2ω1) are respectively the one- and two-photon detunings, ζp(z)(W1V1)12e31u1,p(z) and ζc(z)(W1V1)12e32u1,c(z) are respectively the mode functions of the probe and control fields, Ωpp31p/h̄ and Ωcp32c/h̄ are respectively their corresponding half Rabi frequencies, with ejl the unit vector of the electric dipole matrix element pjl, i.e. pjl = ejlpjl.

Taking Doppler effect into account, the equation of motion of σjl, i.e. the density matrix elements in the interaction picture, are given by

itσ11+iΓ21σ11iΓ12σ22iΓ13σ33+ζp*(z)Ωp*σ31ζp(z)Ωpσ31*=0,
itσ22iΓ21σ11+iΓ12σ22iΓ23σ33+ζc*(z)Ωc*σ32ζc(z)Ωcσ32*=0,
itσ33+i(Γ13+Γ23)σ33ζp*(z)Ωp*σ31+ζp(z)Ωpσ31*ζc*(z)Ωc*σ32+ζc(z)Ωcσ32*=0,
(it+d21)σ21ζp(z)Ωpσ32*+ζc*(z)Ωc*σ31=0,
(it+d31)σ31ζp(z)Ωp(σ33σ11)+ζc(z)Ωcσ21=0,
(it+d32)σ32ζc(z)Ωc(σ33σ22)+ζp(z)Ωpσ21*=0,
where d21 = −(kpkc)v + Δ2 + 21, d31 = −kpv + Δ3 + 31 and d32 = −kcv + (Δ3 − Δ2) + 32 (with v the atom velocity), Γj = ∑j>i Γij and γij=(Γi+Γj)/2+γijcol are population and coherence decay rates, respectively. Here Γij represents the spontaneous emission decay rate from state |j〉 to state |i〉, γijcol are dephasing rates, and Γ21 is the incoherent population exchange from state |1〉 to state |2〉. Note that the atom-photon interactions in waveguide geometries may introduce some undesirable effects, such as the atomic collisions with waveguide walls and the adhesion to the walls [17]. We assume these interfacial effects can be weakened by using similar experimental techniques as did in Refs. [13, 16, 17], like coating the inner walls with some materials (i.e. paraffin) or using light-induced atomic desorption. Note that the controlling field Rabi frequency Ωc in our system is strong enough, so that the change of the decay rate due to the confinement of atoms plays no significant role.

The electric polarization intensity of the system reads

P=𝒩advf(v)[p13σ31ei(kpyωpt)+p23σ32ei(kcyωct)+c.c.],
where 𝒩a is atomic concentration, f(v)=1/(πvT)exp[(v/vT)2] is Maxwell velocity distribution, vT=2kBT/M is the most probable speed at temperature T, with M the atomic mass and kB the Boltzmann constant. Because the integration over the Maxwell distribution is not easy to analyze, as did in Refs. [33, 34] we replace it by the modified Lorentzian velocity distribution f(v)=vT/[π(v2+vT2)].

The motion of the electric field is controlled by Maxwell equation, which under the slowly varying envelope approximation reduces to

i(y+1cn2(z)nefft)Ωp+c2ωpneff2Ωpx2+κ13dvfvσ31(v,z)=0,
where κ13 = p|p13|2/(2ε0h̄cneff) and neff = ck(ωp)/ωp is the effective refraction index. The quantity Q(z)dzζ*(z)Q(z)/dz|ζ(z)|2 for any function of Q(z).

3. EIT characters

3.1. Base state

When the probe field is absent (i.e. Ωp = 0), the Maxwell-Bloch (MB) Eqs. (4), (6) have the steady-state solution

σ11(0)=Γ12(Γ13+Γ23)X1+(Γ12+Γ13)|ζ(z)Ωc|2X2,
σ22(0)=Γ21(Γ13+Γ23)X1+Γ21|ζ(z)Ωc|2X2,
σ33(0)=Γ21|ζ(z)Ωc|2X2,
σ32(0)=ζ(z)Ωcd32Γ21(Γ13+Γ23)X1X2,
and σ21(0)=σ31(0)=0, where X1={[kcv(Δ3Δ2)]2+γ322}/2γ32 and X2 = (Γ12 + Γ21)(Γ13 + Γ23)X1 + (Γ12 + 2Γ21 + Γ13)|ζ(zc|2. Note that we have taken ζc(z) ≈ ζp(z) ≡ ζ(z) because ωpωc.

From (7) we see that, due to the incoherent population exchange, there are population occupation in all three levels. However, the population in states |2〉 and |3〉 are small because generally Γ21 and Γ12 are small. When Γ21 = Γ12 = 0, one has σ11 = 1 and σjl = 0 (j, l ≠ 1).

3.2. EIT characters

We now study the solution of linear excitations of the system, which can be done by linearizing the MB Eqs. (4) and (6) around the base state (7). Taking σjj=σjj(0)(j = 1, 2, 3), σ32=σ32(0), and assuming Ωp and σj1 are small quantities proportional to exp[i(K(ω)yωt)], we obtain the linear dispersion relation

K(ω)=1dz|ζ(z)|2dz|ζ(z)|2{ωcn2(z)neff+κ13dvf(v)ζ(z)Ωcσ32*(0)+(ω+d21)(2σ11(0)+σ22(0)1)|ζ(z)Ωc|2(ω+d21)(ω+d31)}.

Different from the case in free space [34], here we must calculate two-fold integration. The first one is the second term in the brace, which is a statistical average on atomic velocity v. Such integration can be calculated by the use of residue theorem [34]. Taking kpv as a complex number, we find two poles in the lower half complex plane, given by kpv = −ikpvT and kpv=i{γ322+2γ32(Γ12+2Γ21+Γ13)|ζ(z)Ωc|2/[(Γ12+Γ21)Γ3]}1/2iη. For calculating the integration, we take a contour consisting of the lower half complex plane and real axis. The use of residue theorem gives

K(ω)=dz|ζ(z)|2(ωcn2(z)neff+𝒦1(z)+𝒦2(z))/dz|ζ(z)|2,
where 𝒦1 is the contribution from the first pole point kpv = −ikpvT :
𝒦1(z)=πκ13{|ζ(z)Ωc|2Γ21Γ3(iΔωDiγ32)+(ω+iγ21)[Γ12Γ3(ΔωD2+γ322)+2γ32(Γ12Γ21+Γ13)|ζ(z)Ωc|2]}/{(ΔωD2+η2)(Γ12+Γ21)Γ3[|ζ(z)Ωc|2(ω+iγ21)(ω+iΔωD+iγ31)]},
and 𝒦2 is the contribution from the second pole point kpv = −
𝒦2(z)=πκ13ΔωD{|ζ(z)Ωc|2Γ21Γ3(iηiγ32)+(ω+iγ21)[Γ12Γ3(η2+γ322)+2γ32(Γ12Γ21+Γ13)|ζ(z)Ωc|2]}/{η(ΔωD2η2)(Γ12+Γ21)Γ3[|ζ(z)Ωc|2(ω+iγ21)(ω+iη+iγ31)]}.
The integration on z in Eq. (9) is a spatial average due to the EM field confinement by the waveguide geometry, which must be done numerically.

The expression of the imaginary part of K(ω) at ω = 0, i.e. Im(K0), is given by

Im(K0)=1dz|ζ(z)|2dz|ζ(z)|2dvf(v)κ13γ21|ζ(z)Ωc|2+γ21γ31×(1Γ21Γ3X1+3Γ21|ζ(z)Ωc|2(Γ21+Γ12)Γ3X1+(2Γ21+Γ12+Γ13)|ζ(z)Ωc|2).
From Eq. (9) to Eq. (12) we obtain the following conclusions:
  1. The linear dispersion relation K(ω) depends strongly on the slot width 2a due to the factor (W1/V1)1/2 appeared in the mode function ζ(z). Shown in Fig. 2(a) is the probe-field absorption spectrum Im(K) as a function of ω for different slot width. The red solid, black dashed and blue dashed-dotted lines are for 2a = 50, 30 and 10 nm, respectively. We see that: (i) For three different slot widths, an EIT transparency window (i.e. the dip near ω = 0) is opened. (ii) The width of the EIT transparency window becomes larger as the slot width 2a decreases, which means that quantum interference effect is enhanced when the slot width decreases. The physical reason of the enhancement of EIT effect is due to the reduction of EM-field mode volume, which results in (W1/V1)1/2 ≫ 1 and hence the giant enhancement of the interaction between light and atoms.
  2. The minimum of the absorption, i.e. Im(K0), depends not only on Ωc but also on the slot width. Fig. 2(b) shows the profile of Im(K0) as a function of Rabi frequency |Ωc| for different slot width 2a, where the red solid, black dashed and blue dashed-dotted lines are for 2a = 50,30 and 10 nm, respectively. One sees that: (i) For a given Ωc, Im(K0) for smaller slot width is obviously much smaller than that for larger slot width. As the slot width decreases, the EIT transparency window can be not only widened but also deepened dramatically. (ii) The EIT transparency window with a smaller slot width can be obtained with a much smaller Ωc than that with a larger slot width, which means that the confinement provided by the waveguide geometry can be used to get an EIT more easily than that without the confinement. Note that when plotting Fig. 2 we have used a practical example with the D1 line transition of 87Rb atoms, by selecting |5S1/2, F = 1〉, |5S1/2, F = 2〉, and |5P1/2, F = 1〉 as the atomic states |1〉, |2〉, and |3〉, respectively. The system parameters used are κ13 = 1.0 × 109 cm−1s−1, Ωc = 1.0 × 108 s−1, Γ31 = Γ32 = 1.0 × 107 s−1, and Γ21 = 10−4Γ31.
  3. The incoherent population exchange (i.e. nonzero Γ21) plays no significant role on the probe-field absorption when Ωc is very large. However, it has non-negligible influence when Ωc is not too large, reflected in the second term (i.e. the term related to Γ21) in the bracket of Eq. (12) which contributes a obvious reduction to the absorption of the probe field. In fact, by the incoherent population exchange the atoms undergo an active Raman gain process from |2〉 → |3〉 → |1〉. Shown in Fig. 3(a) is Im(K) as function of frequency ω for Ωc = 1.0 × 106 s−1 with Γ21 = 0 (red solid line), Γ21 = 0.5Γ12 (black dashed line) and Γ21 = Γ12 (blue dashed-dotted line), respectively. We see that the absorption for Γ21 = Γ12 is much smaller than that for Γ21 = 0. So a incoherent population exchange can be used to widen and deepen the EIT transparency window. Fig. 3(b) shows the profile of Im(K0) as a function of |Ωc| with Γ21 = 0 (red solid line), Γ21 = 0.5Γ12 (black dashed line) and Γ21 = Γ12 (blue dashed-dotted line), respectively. We see that for large |Ωc|, there is no obvious difference in Im(K0) for different Γ21; but for small |Ωc| the effect caused by Γ21 can be observed clearly.

 

Fig. 2 (a): Im(K) as a function of frequency ω for different slot width 2a. The red solid, black dashed, and blue dashed-dotted lines are for the slot width 2a = 50, 30 and 10 nm, respectively. (b): Im(K0) as a function of |Ωc| for different slot width 2a. The red solid, black dashed, and blue dashed-dotted lines are for the slot width 2a = 50, 30, 10 nm, respectively.

Download Full Size | PPT Slide | PDF

 

Fig. 3 (a): Im(K) as a function of ω. (b): Im(K0) as a function of |Ωc| with Γ21 = 0 (red solid line), Γ21 = 0.5Γ12 (black dashed line) and Γ21 = Γ12 (blue dashed-dotted line), respectively.

Download Full Size | PPT Slide | PDF

4. Kerr nonlinearity and ultraslow optical solitons

4.1. Kerr nonlinearity of the system

From the MB Eqs. (4) and (6), we obtain the probe field susceptibility

χp=dvf(v)𝒩a|p13|2ε0h¯σ31(v,z)Ωpχp(1)+χpp(3)|p|2,
where p = h̄Ωp/p31, χp(1) and χpp(3) are respectively first-order (linear) and third-order (Kerr) susceptibilities, with the expressions given by
χp(1)=𝒩a|p13|2ε0h¯dz|ζ(z)|2dvf(v)×d21d32*(σ33(0)σ11(0))|ζ(z)Ωc|2(σ33(0)σ22(0))d32*(d21d31|ζ(z)Ωc|2)/dz|ζ(z)|2,
χpp(3)=𝒩a|p13|4ε0h¯31dz|ζ(z)|2dz|ζ(z)|2dvf(v)×|ζ(z)|2[id32ζ(z)ΩcZ1Z4(2d21|d32|2d32|ζ(z)Ωc|2)Z2(d21|d32|22d32|ζ(z)Ωc|2)Z3]/[iZ1|d32|2(|ζ(z)Ωc|2d21d31)],
where Z1 = (Γ12 + Γ21)(Γ13 + Γ23)|d32|2 + 2γ3212 + 2Γ21 + Γ13)|ζ(zc|2, Z2=[(Γ12+Γ23)|d32|2+4γ32|ζ(z)Ωc|2](Z5c.c.)+(Γ12Γ13)[(d32*ζ*(z)Ωc*Z4*)c.c.], Z3=(Γ21+Γ13)[(d32*ζ*(z)Ωc*Z4*)c.c.][(Γ21+Γ23)|d32|2+2γ32|ζ(z)Ωc|2](Z5c.c.), Z4=[d31σ32*(0)ζ*(z)Ωc*(σ11(0)σ33(0))]/(|ζ(z)Ωc|2d21d31) and Z5=[ζ(z)Ωcσ32(0)+d21*(σ11(0)σ33(0))]/(|ζ(z)Ωc|2d21*d31*).

The Kerr effect can be enhanced due to the confinement effect induced by the waveguide geometry. When the slot width 2a decreases, the confinement of the light field increases because the factor (W1/V1) in the expression |ζ(z)|2 increases. Shown in Fig. 4 is the real part of the third-order susceptibility, i.e. Re(χpp(3)), as a function of detunning Δ3 for different slot width 2a, where the red solid, black dashed and blue dashed-dotted lines are for 2a = 50,25 and 5 nm, respectively. Parameters are the same as those used in Fig. 2 with a small air (εS = 1) slot embedded between the silicon (εH = 14) slabs. We see that the Kerr effect for large confinement (2a = 5 nm) is much larger than that for small confinement (2a = 50 nm). Hence the effect of self-phase modulation becomes stronger as the slot size 2a goes smaller, which indicates that in the slot waveguide the efficiency of producing ultraslow optical solitons may be higher than that in free space.

 

Fig. 4 Third-order susceptibility Re(χpp(3)) as a function of detunning Δ3 for different slot width 2a. The red solid, black dashed and blue dashed-dotted lines are for 2a = 50,25 and 5 nm, respectively.

Download Full Size | PPT Slide | PDF

4.2. Asymptotic expansion and nonlinear envelope equation

We now turn to study possible optical solitons in the system, which is especially interesting for the present slot waveguide geometry because the light power density in such system is increased and diffraction is suppressed in the confined direction, and thus optical solitons are easy to produce than in free space [9, 10]. Such study is also of practical interest in optical information processing and transmission in quantum hybrid systems when shape-preserving probe pulses with low light power are needed.

To this end, we employ the method of multiple scales to solve the MB equations for nonlinear propagation problems developed in Ref. [10]. Taking the asymptotic expansion σij=l=0ε(l)σij(l), Ωp=l=1ε(l)Ωp(l), where σij(0) is the base state solution given by Eq. (7) and ε is a dimensionless small parameter characterizing the amplitude of the probe field. To obtain a divergence-free expansion, all quantities on the right hand side of the expansion are considered as functions of the multi-scale variables yβ = εβy (β = 0, 1, 2) and tβ = εβt (β = 0,1). Substituting the expansion into the MB Eqs. (4) and (6), we obtain a series of equations for σij(l) and Ωp(l), which can be solved order by order.

At the first order (l = 1), we obtain the linear solution

Ωp(1)=Feiθ,
σ31(1)=(ω+d21)(2σ11(0)+σ22(0)1)+ζ(z)Ωcσ32*(0)|ζ2(z)Ωc|2(ω+d21)(ω+d31)ζ(z)Feiθ,
σ21(1)=(ω+d31)σ32*(0)+ζc*(z)Ωc*(2σ11(0)+σ22(0)1)|ζ2(z)Ωc|2(ω+d21)(ω+d31)ζ(z)Feiθ,
with other σij(1)=0. Here θ = K(ω)y0ωt0, with F being a yet to be determined envelope function depending on the slow variables (t1, y1, y2) and K(ω) being the linear dispersion relation given by Eq. (8).

At the second order (l = 2), the condition of the solution in this order is divergence-free requires i[∂F/∂y1 + (1/vg)∂F/∂t1] = 0, where vg = ∂K/∂ω is the group velocity of the envelope function F. The explicit expressions of the second-order solution are omitted here for saving space.

At the third order (l = 3), we obtain the closed equation for F:

iFy2+c2ωpneff2Fx12K222Ft12W|F|2Fe2α¯y2=0,
where K22K/∂ω2, and
W=κ13dz|ζ(z)|4dvf(v)ζ(z)Ωca32*(2)+(ω+d21)(2a11(2)+a22(2))|ζ2(z)Ωc|2(ω+d21)(ω+d31)/dz|ζ(z)|2,
and α = Im(K) = ε2ᾱ. The explicit expressions of a11(2), a22(2), a32(2) have been given in Appendix C.

Returning to the original variables, Eq. (16) becomes

i(y+α)U+c2ωpneff2Ux2K222Uτ2W|U|2U=0,
where τ = ty/vg and U = εFeαy. Equation (18) is of the form of nonlinear Schrödinger (NLS) equation, but has complex coefficients and hence is not integrable generally. If a nonlinear localized pulse is produced, it may be highly unstable during propagation. However, if a realistic set of system parameters under some conditions can be found so that the imaginary part of the coefficients can be much smaller than their real part, it is possible to obtain a shape-preserving soliton solution that can propagate for a rather long distance without significant distortion. In fact, such parameter set can indeed be found near the EIT transparency window (see below), so the imaginary parts of the coefficients are very small. In this way Eq. (18) can be written into the dimensionless form
ius+2uσ2+2u|u|2=id0u+d12uξ2,
where s = −z/(2LD), σ = τ/τ0, ξ = x/R and u = U/U0. LD=τ02/K˜2 is the characteristic dispersion length, R is the beam radius in x-direction and U0=1/τ0K˜2/W˜ is typical Rabi frequency of the probe field, with K̃2 and W̃ denoting respectively the real parts of K2 and W. In Eq. (19), d0 = LD/LA and d1 = LD/Ldiff are two dimensionless coefficients, with LA = 1/2α the characteristic absorption length and Ldiff=ωpneffRx2 the characteristic diffraction length, respectively. Under the condition d0, d1 ≪ 1, Eq. (19) reduces to an integrable NLS equation, which allows multi-soliton solutions. A single soliton solution reads
u=2βsech[2β(σσ0+4δs)]exp[2iδσ4i(δ2β2)siϕ0],
where β, δ, σ0 and ϕ0 are real free parameters that determine the amplitude (also width), propagating velocity, initial position, and initial phase of the soliton, respectively. Taking β = 1/2, δ = σ0 = ϕ0 = 0, we have u = 2β sechexp(is); or in terms of Rabi frequency
Ωp=1τ0K˜2W˜sech[1τ0(tyv˜g)]exp[iK˜0y+iy2LD],
with K̃0 = Re(K0). Equation (21) describes a bright soliton traveling with velocity ṽg = Re(vg).

We now consider a practical example for the formation of the optical soliton given above. We choose 87Rb D1-line transition, with system parameters given by κ13 = 1.0 × 109 cm−1s−1, Δ2 = 2.5 × 105 s−1, Δ3 = 5.9 × 107 s−1 and slot width 2a = 10 nm. In this case, the coefficients in Eq. (21) are K2 = (1.59 + 0.14i) × 10−14 cm−1s−2 and W = (5.1 + 0.36i) × 10−15 cm−1s−2. We see that the imaginary parts of these coefficients are indeed much smaller than their corresponding real parts. The physical reason of so small imaginary parts is due to the quantum interference effect induced by the control field, by which the role of population and coherence decay rates for the propagation of the soliton is largely suppressed. When taking τ0 = 1.5 × 10−7 s, Rx = 0.05 cm we have the characteristic lengths LD = 1.4 cm, LA = 38.4 cm and Ldiff = 107 cm, which ensure the validity of neglecting absorption and diffraction of the probe pulse when propagating a distance not much larger than the dispersion length, i.e. d0 ≪ 1 and d0 ≪ 1 is satisfied. With these parameters we obtain the group velocity Vg = 1.6 × 10−5c. Consequently, the optical soliton obtained travels with an ultraslow propagating velocity in the system.

The input power for generating the ultraslow optical soliton may be estimated by calculating Poyntings vector. The average flux of energy over carrier-wave period is P̄/S0 = (P̄max/S0)sech2[(tz/Ṽg)/τ0] with the peak power P¯max=2ε0cnpS0(h/p31)2K˜2/(W˜τ02). Here, np = nneff + cK̃0/ωp is the refractive index and S0 is the cross section area of the probe beam. With the values of coefficients given above, we obtain P̄max = 1.19 μW. Thus, very low input power is needed for generating the ultraslow optical soliton in the slot waveguide system.

In order to make a further confirmation of the soliton solutions and check their stability, a numerical simulation is carried out. Shown in Fig. 5(a) is the three-dimensional plot for the wave shape |Ωp/U0|2 as a function of z/LD and t/τ0. The initial condition of the simulation is given by Ωp(0, σ) = U0sech(t/τ0). We see that the amplitude of the soliton undergoes only a slight decrease and its width undergoes slight increase due to the influence of the small imaginary parts of the coefficients. The properties of collision between two ultraslow optical solitons are also investigated numerically by taking Ωp(0, σ) = U0sech(t/τ0 − 5)+U0sech(t/τ0 + 5) as the initial condition without any approximation. As time goes on, they collide, pass through, and depart from each other, as shown in Fig. 5(b). The two solitons recover their initial waveforms after the collision.

 

Fig. 5 (a) The three-dimensional plot of the wave shape |Ωp/U0|2 as a function of z/LD and t/τ0. The solution is numerically obtained from Eq. (14) with full complex coefficients taken into account. The values of parameters are given in the text. (b): The interaction between two identical bright solitons.

Download Full Size | PPT Slide | PDF

5. Conclusion

We have investigated the EIT and nonlinear pulse propagation in a Λ-type three-level atomic gas filled in a slot waveguide, in which electric field is strongly confined inside the slot of the waveguide due to the discontinuity of dielectric constant. We have found that the EIT effect can be largely enhanced due to reduction of optical-field mode volume contributed by the waveguide geometry. In comparison with the atomic gases in free space, the EIT transparency window in the slot waveguide system are much wider and deeper, and the Kerr nonlinearity of the probe laser field are much stronger. We have also proved that by using the slot waveguide ultraslow optical solitons via EIT can be produced efficiently with extremely low generation power. The present work opens an avenue to the study EIT-related quantum coherence in nano-sized systems and the results presented may have promising applications for optical information processing and transmission.

Appendix

A. TM-modes of EM field and mode volume in the slot waveguide

For the slot waveguide, the EM field can be divided into transverse electric (TE) and transverse magnetic (TM) parts, i.e. E = ETE +ETM and H = HTE +HTM. The confinement and enhancement of EM field near the slot region is contributed by the TM part. By solving Maxwell’s equations in the absence of atoms, we can obtain the TM eigenmode solutions with the coordinate system chosen in Fig. 1 as

Hm,k||TM(r,t)=(k^||×ez)Hm,k||(z)ei(k||rωmt)+c.c.,
Em,k||TM(r,t)=iωm(k||)ε0ε(z)[ik||Hm,k||(z)ez+dHm,k||(z)dzk^||]ei(k||rωmt)+c.c.,
where k = (kx, ky, 0) (kx, ky are arbitrary real numbers) is the wavevector in the xy plane, = k/|k| and ez are respectively the unit vectors in the k- and z-directions, ωn(k) is the eigenfrequency with m = 1, 2, 3···, and ε(z) ≡ n2(z) is dielectric function with n(z) (refractive index) taking the value nS for |z| < a, nH for a < |z| < b, and nC for |z| > b (see Fig. 1).

The function Hm,k(z) in Eq. (22) satisfies the equation

d2dz2Hm,k||(z)+[(ωc)2n2(z)k||2]Hm,k||(z)=0,
with the boundary conditions Hm,k(z), dHm,k(z)/dz being continuous at the interfaces z = ±a, ±b. For guided modes, an additional condition m,k(z) → 0 for z → ±∞ is also required. Then one obtains [19]
Hm,k||TM(z)={cosh(γSmz),|z|<aCmcos[κHm(|z|a)]+Dmsin[κHm(|z|a)],a<|z|<bEmexp[γSm(|z|b)],|z|>b
where Cm = cosh(γSma), Dm=[nH2γSm/(nS2κHm)]sinh(γSma), Em={cosh(γSma)cos[κHm(ba)]+[nH2γSm/(nS2κHm)]sinh(γSa)sin[κHm(ba)]}, with κHm=[nH2ωm2(k||)/c2k||2]1/2 and γSm=[k||2nS2ωm2(k||)/c2]1/2. The eigenvalue ωm(k) is determined by the equation
tan[κHm(ba)arctan(γSmnH2κHmnS2)]=γSnnH2κHmnS2tanh(γSma).
In fact, the eigenfrequency ωm depends only on k||=(kx2+ky2)1/2, i.e. ωm = ωm(k). Obviously, the guided eigenmodes given above propagate in the xy-plane but confined basically in the slot region.

Using the formula U=12d3r(εE2+μH2) with U being EM-field energy and ε and μ (= μ0) being respectively the permittivity and permeability, we obtain the second-quantization form of the TM part

ETM(r,t)=kx,kym=1h¯ωm2ε0Vmum,k||(z)a^m(k)ei(k||rωmt)+h.c.,
HTM(r,t)=kx,kym=1h¯ωm2ε0Vm(k^||×ez)ε0cNmHm,k||(z)a^m(k)ei(k||rωmt)+h.c.,
where um,k||(z)={c/[Nmωmn2(z)]}[ezk||Hm,k||(z)+ik^||dHm,k||(z)/dz] is the mode function with dz|um,k||(z)|2=Vm, âm(k) and a^m(k) are creation and annihilation operators of TM photons. In our present study, we assume that the photon numbers in both control and probe fields are much larger than one, so âm(k) and a^m(k) are taken as dimensionless numbers with âm(k) = an(k) and a^m(k)=am*(k).

The quantity Vm appeared in Eq. (26) is the (effective) mode volume given by

Vm=S{12γSm[sinh(2γSma)+2Em2]+a2+(Cm2+Dm2)(ba)+12κHm(Cm2Dm2)sin[2κHm(ba)]}/Nm,
Nm=ωm2/(cP)2,
where S is the transverse area of the waveguide in the xy-plane, and P = (k||2+γS2)[sinh(2γSma)+2Em2]/(2γSnS4) + nH2ωm2(Cm2+Dm2)(ba)/c2 + CmDmωm2cos[2κH(ba)]/(4c2κHmnH2) + ωm2(Cm2Dm2)sinh[2κH(ba)]/(4c2κHmnH2).

B. TM-modes and mode volume of EM field without the slot

For conventional slab waveguide (i.e. the waveguide shown in Fig. 1 but with the slot width 2a = 0), the TM-modes of the EM field have the same form of Eq. (22), but here n(z) takes the values nH for |z| < b, and nC for |z| > b, we have

Hm,k||TM(z)={sinϕmexp(z+b2bψmcosϕm),z<bcos(ψmzbsinϕm),b<z<bsinϕmexp(bz2bψmcosϕm),z>b
where ϕm=(nH2ωm2(k||)/c2k||2)1/2L and ψm=(2bωm/c)(nH2nC2)1/2. The eigenvalue ωm(k) is determined by the equation ψm sinϕm = − 2ϕm (m = 1, 2, 3, ···). The second-quantization form of the EM field reads
ETM(r,t)=kx,kym=1h¯ωm2ε0Wmum,k||(z)a^n(k)ei(k||rωmt)+h.c.,
HTM(r,t)=kx,kym=1h¯ωm2ε0Wm(k^||×ez)ε0cMmHm,k||(z)a^m(k)ei(k||rωmt)+h.c.,
where um,k||(z)={c/[Mmωmn2(z)]}[ezk||Hm,k||(z)+ik^||dHm,k||(z)/dz] is the mode function with |um,k||(z)|2dz=Wma^m(k) and a^m(k) are creation and annihilation operators of TM photons, Wm is the mode volume given by
Wm=S{b[2sin2ϕm+ψmcosϕm]/(ψmcosϕm)+bsin(2ψmsinϕm)/(2ψmsinϕm)}/Mm,
Mm=ωm2/(cG)2,
with G=[2k||2bsin2ϕm]/(ψmcosϕm)+sin2ϕm[ψmcosϕm+ψm2/2+ψm2sin(2ψmsinϕm)]/(2b).

C. Expressions of aij(2) appearing in Eq. (17)

a11(2)=i{[i(Γ12+Γ23)2|ζ2(z)Ωc|2(1d321d32*)][(ω+d21*)(2σ11(0)+σ22(0)1)+ζ*(z)Ωc*σ32(0)|ζ2(z)Ωc|2(ω+d21*)(ω+d31*)c.c.]+i(Γ12Γ13)[ζ*(z)Ωc*d32(ω+d31*)σ32(0)+ζ(z)Ωc(2σ11(0)+σ22(0)1)|ζ2(z)Ωc|2(ω+d21*)(ω+d31*)+c.c.]}/[i(Γ12+Γ21)(Γ13+Γ23)(2Γ21+Γ12+Γ13)|ζ2(z)Ωc|2(1d321d32*)],
a22(2)=iΓ12Γ13[(ω+d21*)(2σ11(0)+σ22(0)1)+ζ*(z)Ωc*σ32(0)|ζ2(z)Ωc|2(ω+d21*)(ω+d31*)c.c.i(Γ21+Γ13)a11(2)],
a32(2)=1d32[(ω+d31*)σ32(0)+ζ(z)Ωc(2σ11(0)+σ22(0)1)|ζ2(z)Ωc|2(ω+d21*)(ω+d31*)ζ(z)Ωc(a11(2)+2a22(2))].

Acknowledgments

The authors thanks L. Deng and M. Xiao for useful discussions. This work was supported by NSF-China under Grant numbers 10874043 and 11174080.

References and links

1. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005). [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, 594–598 (1999). [CrossRef]  

3. M. M. Kash, V.A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M.O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999). [CrossRef]  

4. A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photon. 3, 706–714 (2009). [CrossRef]  

5. H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics 5, 628–632 (2011). [CrossRef]  

6. H. N. Dai, H. Zhang, S.-J. Yang, T.-M. Zhao, J. Rui, Y.-J. Deng, L. Li, N.-L. Liu, S. Chen, X.-H. Bao, X.-M. Jin, B. Zhao, and J.-W. Pan, “Holographic storage of biphoton entanglement,” Phys. Rev. Lett. 108, 210501 (2012). [CrossRef]   [PubMed]  

7. C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett. 90, 197902 (2003). [CrossRef]   [PubMed]  

8. C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phys. Rev. A 74, 012319 (2006). [CrossRef]  

9. Y. Wu and L. Deng, “Ultraslow optical solitons in a cold four-state medium,” Phys. Rev. Lett. 93, 143904 (2004). [CrossRef]   [PubMed]  

10. G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E. 72, 016617 (2005). [CrossRef]  

11. S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. Engeness, M. Soljacic, S. Jacobs, J. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748–779 (2001). [CrossRef]   [PubMed]  

12. S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, “Resonant optical interactions with molecules confined in photonic band-gap fibers,” Phys. Rev. Lett. 94, 093902 (2005). [CrossRef]   [PubMed]  

13. S. Ghosh, A. R. Bhagwat, C. K. Renshaw, S. Goh, and A. L. Gaeta, “Low-light-level optical interactions with rubidium vapor in a photonic band-gap fiber,” Phys. Rev. Lett. 97, 023603 (2006). [CrossRef]   [PubMed]  

14. P. S. Light, F. Benabid, F. Couny, M. Maric, and A. N. Luiten, “Electromagnetically induced transparency in Rb-filled coated hollow-core photonic crystal fiber,” Opt. Lett. 32, 1323–1325 (2007). [CrossRef]   [PubMed]  

15. F. Benabid, P. Light, F. Couny, and P. Russell, “Electromagnetically-induced transparency grid in acetylene-filled hollow-core PCF,” Opt. Express 13, 5694–5703 (2005). [CrossRef]   [PubMed]  

16. M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009). [CrossRef]   [PubMed]  

17. A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, “Spectroscopy of Rb atoms in hollow-core fibers,” Phys. Rev. A 81, 053825 (2010). [CrossRef]  

18. F. L. Kien and K. Hakuta, “Slowing down of a guided light field along a nanofiber in a cold atomic gas,” Phys. Rev. A. 79, 013818 (2009). [CrossRef]  

19. V. R. Almeida, Q. Xu, C. A. Barrios, R. R. Panepucci, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1209–1211 (2004). [CrossRef]   [PubMed]  

20. Q. Xu, V. R. Almeida, R. Panepucci, and M. Lipson, “Experimental demonstration of guiding and confining light in nanometer-size low-refractive-index material,” Opt. Lett. 29, 1626–1628 (2004). [CrossRef]   [PubMed]  

21. M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. L. Savio, A. Irrera, and F. Priolo, “Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides,” Appl. Phys. Lett. 89, 241114 (2006). [CrossRef]  

22. K. Foubert, L. Lalouat, B. Cluzel, E. Picard, D. Peyrade, F. de Fornel, and E. Hadji, “An air-slotted nanoresonator relying on coupled high Q small V Fabry-Perot nanocavities,” Appl. Phys. Lett. 94, 251111 (2009). [CrossRef]  

23. T. Yamamoto, M. Notomi, H. Taniyama, E. Kuramochi, Y. Yoshikawa, Y. Torii, and T. Kuga, “Design of a high-Q air-slot cavity based on a width-modulated line-defect in a photonic crystal slab,” Opt. Express 16, 13809–13817 (2008). [CrossRef]   [PubMed]  

24. Y. Li, J. Zheng, J. Gao, J. Shu, M. S. Aras, and C. W. Wong, “Design of dispersive optomechanical coupling and cooling in ultrahigh-Q/V slot-type photonic crystal cavities,” Opt. Express 18, 23844–23856 (2010). [CrossRef]   [PubMed]  

25. M. P. Hiscocks, C. Su, B. C. Gibson, A. D. Greentree, L. C. L. Hollenberg, and F. Ladouceur, “Slot-waveguide cavities for optical quantum information applications,” Opt. Express 17, 7295–7303 (2009). [CrossRef]   [PubMed]  

26. H. Ryu, J. Kim, Y. M. Jhon, S. Lee, and N. Park, “Effect of index contrasts in the wide spectral-range control of slot waveguide dispersion,” Opt. Express 20, 13189–13194 (2012) [CrossRef]   [PubMed]  

27. C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5990 (2007). [CrossRef]   [PubMed]  

28. P. Muellner, M. Wellenzohn, and R. Hainberger, “Nonlinearity of optimized silicon photonic slot waveguides,” Opt. Express 17, 9282–9287 (2009). [CrossRef]   [PubMed]  

29. Q. Quan, I. Bulu, and M. Lončar, “Broadband waveguide QED system on a chip,” Phys. Rev. A. 80, 011810(R) (2009). [CrossRef]  

30. L. Zhang, Y. Yue, Y. X. Li, J. Wang, R. G. Beausoleil, and A. E. Willner, “Flat and low dispersion in highly nonlinear slot waveguides,” Opt. Express 18, 13187–13193 (2010). [CrossRef]   [PubMed]  

31. Y. Yue, L. Zhang, J. Wang, R. G. Beausoleil, and A. E. Willner, “Highly efficient nonlinearity reduction in silicon-on-insulator waveguides using vertical slots,” Opt. Express 18, 22061–22066 (2010). [CrossRef]   [PubMed]  

32. R. Guo, B. Wang, X. Wang, L. Wang, L. Jiang, and Z. Zhou, “Optical amplification in Er/Yb silicate slot waveguide,” Opt. Lett. 37, 1427–1429 (2012). [CrossRef]   [PubMed]  

33. H. Lee, Y. Rostovtsev, C. J. Bednar, and A. Javan, “From laser-induced line narrowing to electromagnetically induced transparency: closed system analysis,” Appl. Phys. B 76, 33 (2003). [CrossRef]  

34. L. Li and G. Huang, “Linear and nonlinear light propagations in a Doppler-broadened medium via electromagnetically induced transparency,” Phys. Rev. A 82, 023809 (2010). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77, 633–673 (2005).
    [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, 594–598 (1999).
    [CrossRef]
  3. M. M. Kash, V.A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M.O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett.82, 5229–5232 (1999).
    [CrossRef]
  4. A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photon.3, 706–714 (2009).
    [CrossRef]
  5. H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
    [CrossRef]
  6. H. N. Dai, H. Zhang, S.-J. Yang, T.-M. Zhao, J. Rui, Y.-J. Deng, L. Li, N.-L. Liu, S. Chen, X.-H. Bao, X.-M. Jin, B. Zhao, and J.-W. Pan, “Holographic storage of biphoton entanglement,” Phys. Rev. Lett.108, 210501 (2012).
    [CrossRef] [PubMed]
  7. C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett.90, 197902 (2003).
    [CrossRef] [PubMed]
  8. C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phys. Rev. A74, 012319 (2006).
    [CrossRef]
  9. Y. Wu and L. Deng, “Ultraslow optical solitons in a cold four-state medium,” Phys. Rev. Lett.93, 143904 (2004).
    [CrossRef] [PubMed]
  10. G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E.72, 016617 (2005).
    [CrossRef]
  11. S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. Engeness, M. Soljacic, S. Jacobs, J. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express9, 748–779 (2001).
    [CrossRef] [PubMed]
  12. S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, “Resonant optical interactions with molecules confined in photonic band-gap fibers,” Phys. Rev. Lett.94, 093902 (2005).
    [CrossRef] [PubMed]
  13. S. Ghosh, A. R. Bhagwat, C. K. Renshaw, S. Goh, and A. L. Gaeta, “Low-light-level optical interactions with rubidium vapor in a photonic band-gap fiber,” Phys. Rev. Lett.97, 023603 (2006).
    [CrossRef] [PubMed]
  14. P. S. Light, F. Benabid, F. Couny, M. Maric, and A. N. Luiten, “Electromagnetically induced transparency in Rb-filled coated hollow-core photonic crystal fiber,” Opt. Lett.32, 1323–1325 (2007).
    [CrossRef] [PubMed]
  15. F. Benabid, P. Light, F. Couny, and P. Russell, “Electromagnetically-induced transparency grid in acetylene-filled hollow-core PCF,” Opt. Express13, 5694–5703 (2005).
    [CrossRef] [PubMed]
  16. M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett.102, 203902 (2009).
    [CrossRef] [PubMed]
  17. A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, “Spectroscopy of Rb atoms in hollow-core fibers,” Phys. Rev. A81, 053825 (2010).
    [CrossRef]
  18. F. L. Kien and K. Hakuta, “Slowing down of a guided light field along a nanofiber in a cold atomic gas,” Phys. Rev. A.79, 013818 (2009).
    [CrossRef]
  19. V. R. Almeida, Q. Xu, C. A. Barrios, R. R. Panepucci, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett.29, 1209–1211 (2004).
    [CrossRef] [PubMed]
  20. Q. Xu, V. R. Almeida, R. Panepucci, and M. Lipson, “Experimental demonstration of guiding and confining light in nanometer-size low-refractive-index material,” Opt. Lett.29, 1626–1628 (2004).
    [CrossRef] [PubMed]
  21. M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. L. Savio, A. Irrera, and F. Priolo, “Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides,” Appl. Phys. Lett.89, 241114 (2006).
    [CrossRef]
  22. K. Foubert, L. Lalouat, B. Cluzel, E. Picard, D. Peyrade, F. de Fornel, and E. Hadji, “An air-slotted nanoresonator relying on coupled high Q small V Fabry-Perot nanocavities,” Appl. Phys. Lett.94, 251111 (2009).
    [CrossRef]
  23. T. Yamamoto, M. Notomi, H. Taniyama, E. Kuramochi, Y. Yoshikawa, Y. Torii, and T. Kuga, “Design of a high-Q air-slot cavity based on a width-modulated line-defect in a photonic crystal slab,” Opt. Express16, 13809–13817 (2008).
    [CrossRef] [PubMed]
  24. Y. Li, J. Zheng, J. Gao, J. Shu, M. S. Aras, and C. W. Wong, “Design of dispersive optomechanical coupling and cooling in ultrahigh-Q/V slot-type photonic crystal cavities,” Opt. Express18, 23844–23856 (2010).
    [CrossRef] [PubMed]
  25. M. P. Hiscocks, C. Su, B. C. Gibson, A. D. Greentree, L. C. L. Hollenberg, and F. Ladouceur, “Slot-waveguide cavities for optical quantum information applications,” Opt. Express17, 7295–7303 (2009).
    [CrossRef] [PubMed]
  26. H. Ryu, J. Kim, Y. M. Jhon, S. Lee, and N. Park, “Effect of index contrasts in the wide spectral-range control of slot waveguide dispersion,” Opt. Express20, 13189–13194 (2012)
    [CrossRef] [PubMed]
  27. C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express15, 5976–5990 (2007).
    [CrossRef] [PubMed]
  28. P. Muellner, M. Wellenzohn, and R. Hainberger, “Nonlinearity of optimized silicon photonic slot waveguides,” Opt. Express17, 9282–9287 (2009).
    [CrossRef] [PubMed]
  29. Q. Quan, I. Bulu, and M. Lončar, “Broadband waveguide QED system on a chip,” Phys. Rev. A.80, 011810(R) (2009).
    [CrossRef]
  30. L. Zhang, Y. Yue, Y. X. Li, J. Wang, R. G. Beausoleil, and A. E. Willner, “Flat and low dispersion in highly nonlinear slot waveguides,” Opt. Express18, 13187–13193 (2010).
    [CrossRef] [PubMed]
  31. Y. Yue, L. Zhang, J. Wang, R. G. Beausoleil, and A. E. Willner, “Highly efficient nonlinearity reduction in silicon-on-insulator waveguides using vertical slots,” Opt. Express18, 22061–22066 (2010).
    [CrossRef] [PubMed]
  32. R. Guo, B. Wang, X. Wang, L. Wang, L. Jiang, and Z. Zhou, “Optical amplification in Er/Yb silicate slot waveguide,” Opt. Lett.37, 1427–1429 (2012).
    [CrossRef] [PubMed]
  33. H. Lee, Y. Rostovtsev, C. J. Bednar, and A. Javan, “From laser-induced line narrowing to electromagnetically induced transparency: closed system analysis,” Appl. Phys. B76, 33 (2003).
    [CrossRef]
  34. L. Li and G. Huang, “Linear and nonlinear light propagations in a Doppler-broadened medium via electromagnetically induced transparency,” Phys. Rev. A82, 023809 (2010).
    [CrossRef]

2012 (3)

2011 (1)

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

2010 (5)

2009 (7)

M. P. Hiscocks, C. Su, B. C. Gibson, A. D. Greentree, L. C. L. Hollenberg, and F. Ladouceur, “Slot-waveguide cavities for optical quantum information applications,” Opt. Express17, 7295–7303 (2009).
[CrossRef] [PubMed]

K. Foubert, L. Lalouat, B. Cluzel, E. Picard, D. Peyrade, F. de Fornel, and E. Hadji, “An air-slotted nanoresonator relying on coupled high Q small V Fabry-Perot nanocavities,” Appl. Phys. Lett.94, 251111 (2009).
[CrossRef]

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett.102, 203902 (2009).
[CrossRef] [PubMed]

P. Muellner, M. Wellenzohn, and R. Hainberger, “Nonlinearity of optimized silicon photonic slot waveguides,” Opt. Express17, 9282–9287 (2009).
[CrossRef] [PubMed]

Q. Quan, I. Bulu, and M. Lončar, “Broadband waveguide QED system on a chip,” Phys. Rev. A.80, 011810(R) (2009).
[CrossRef]

F. L. Kien and K. Hakuta, “Slowing down of a guided light field along a nanofiber in a cold atomic gas,” Phys. Rev. A.79, 013818 (2009).
[CrossRef]

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photon.3, 706–714 (2009).
[CrossRef]

2008 (1)

2007 (2)

2006 (3)

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. L. Savio, A. Irrera, and F. Priolo, “Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides,” Appl. Phys. Lett.89, 241114 (2006).
[CrossRef]

C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phys. Rev. A74, 012319 (2006).
[CrossRef]

S. Ghosh, A. R. Bhagwat, C. K. Renshaw, S. Goh, and A. L. Gaeta, “Low-light-level optical interactions with rubidium vapor in a photonic band-gap fiber,” Phys. Rev. Lett.97, 023603 (2006).
[CrossRef] [PubMed]

2005 (4)

S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, “Resonant optical interactions with molecules confined in photonic band-gap fibers,” Phys. Rev. Lett.94, 093902 (2005).
[CrossRef] [PubMed]

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E.72, 016617 (2005).
[CrossRef]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77, 633–673 (2005).
[CrossRef]

F. Benabid, P. Light, F. Couny, and P. Russell, “Electromagnetically-induced transparency grid in acetylene-filled hollow-core PCF,” Opt. Express13, 5694–5703 (2005).
[CrossRef] [PubMed]

2004 (3)

2003 (2)

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett.90, 197902 (2003).
[CrossRef] [PubMed]

H. Lee, Y. Rostovtsev, C. J. Bednar, and A. Javan, “From laser-induced line narrowing to electromagnetically induced transparency: closed system analysis,” Appl. Phys. B76, 33 (2003).
[CrossRef]

2001 (1)

1999 (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, 594–598 (1999).
[CrossRef]

M. M. Kash, V.A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M.O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett.82, 5229–5232 (1999).
[CrossRef]

Almeida, V. R.

Andreani, L. C.

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. L. Savio, A. Irrera, and F. Priolo, “Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides,” Appl. Phys. Lett.89, 241114 (2006).
[CrossRef]

Aras, M. S.

Artoni, M.

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett.90, 197902 (2003).
[CrossRef] [PubMed]

Bajcsy, M.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett.102, 203902 (2009).
[CrossRef] [PubMed]

Balic, V.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett.102, 203902 (2009).
[CrossRef] [PubMed]

Bao, X.-H.

H. N. Dai, H. Zhang, S.-J. Yang, T.-M. Zhao, J. Rui, Y.-J. Deng, L. Li, N.-L. Liu, S. Chen, X.-H. Bao, X.-M. Jin, B. Zhao, and J.-W. Pan, “Holographic storage of biphoton entanglement,” Phys. Rev. Lett.108, 210501 (2012).
[CrossRef] [PubMed]

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

Barrios, C. A.

Beausoleil, R. G.

Bednar, C. J.

H. Lee, Y. Rostovtsev, C. J. Bednar, and A. Javan, “From laser-induced line narrowing to electromagnetically induced transparency: closed system analysis,” Appl. Phys. B76, 33 (2003).
[CrossRef]

Behroozi, C. H.

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, 594–598 (1999).
[CrossRef]

Benabid, F.

Bhagwat, A. R.

A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, “Spectroscopy of Rb atoms in hollow-core fibers,” Phys. Rev. A81, 053825 (2010).
[CrossRef]

S. Ghosh, A. R. Bhagwat, C. K. Renshaw, S. Goh, and A. L. Gaeta, “Low-light-level optical interactions with rubidium vapor in a photonic band-gap fiber,” Phys. Rev. Lett.97, 023603 (2006).
[CrossRef] [PubMed]

Bulu, I.

Q. Quan, I. Bulu, and M. Lončar, “Broadband waveguide QED system on a chip,” Phys. Rev. A.80, 011810(R) (2009).
[CrossRef]

Canino, A.

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. L. Savio, A. Irrera, and F. Priolo, “Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides,” Appl. Phys. Lett.89, 241114 (2006).
[CrossRef]

Cataliotti, F.

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett.90, 197902 (2003).
[CrossRef] [PubMed]

Chen, S.

H. N. Dai, H. Zhang, S.-J. Yang, T.-M. Zhao, J. Rui, Y.-J. Deng, L. Li, N.-L. Liu, S. Chen, X.-H. Bao, X.-M. Jin, B. Zhao, and J.-W. Pan, “Holographic storage of biphoton entanglement,” Phys. Rev. Lett.108, 210501 (2012).
[CrossRef] [PubMed]

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

Chen, Z.-B.

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

Cluzel, B.

K. Foubert, L. Lalouat, B. Cluzel, E. Picard, D. Peyrade, F. de Fornel, and E. Hadji, “An air-slotted nanoresonator relying on coupled high Q small V Fabry-Perot nanocavities,” Appl. Phys. Lett.94, 251111 (2009).
[CrossRef]

Couny, F.

Dai, H. N.

H. N. Dai, H. Zhang, S.-J. Yang, T.-M. Zhao, J. Rui, Y.-J. Deng, L. Li, N.-L. Liu, S. Chen, X.-H. Bao, X.-M. Jin, B. Zhao, and J.-W. Pan, “Holographic storage of biphoton entanglement,” Phys. Rev. Lett.108, 210501 (2012).
[CrossRef] [PubMed]

Dai, H.-N.

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

de Fornel, F.

K. Foubert, L. Lalouat, B. Cluzel, E. Picard, D. Peyrade, F. de Fornel, and E. Hadji, “An air-slotted nanoresonator relying on coupled high Q small V Fabry-Perot nanocavities,” Appl. Phys. Lett.94, 251111 (2009).
[CrossRef]

Deng, L.

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E.72, 016617 (2005).
[CrossRef]

Y. Wu and L. Deng, “Ultraslow optical solitons in a cold four-state medium,” Phys. Rev. Lett.93, 143904 (2004).
[CrossRef] [PubMed]

Deng, Y.

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

Deng, Y.-J.

H. N. Dai, H. Zhang, S.-J. Yang, T.-M. Zhao, J. Rui, Y.-J. Deng, L. Li, N.-L. Liu, S. Chen, X.-H. Bao, X.-M. Jin, B. Zhao, and J.-W. Pan, “Holographic storage of biphoton entanglement,” Phys. Rev. Lett.108, 210501 (2012).
[CrossRef] [PubMed]

Dutton, Z.

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, 594–598 (1999).
[CrossRef]

Engeness, T.

Fink, Y.

Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77, 633–673 (2005).
[CrossRef]

Foubert, K.

K. Foubert, L. Lalouat, B. Cluzel, E. Picard, D. Peyrade, F. de Fornel, and E. Hadji, “An air-slotted nanoresonator relying on coupled high Q small V Fabry-Perot nanocavities,” Appl. Phys. Lett.94, 251111 (2009).
[CrossRef]

Freude, W.

Fry, E. S.

M. M. Kash, V.A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M.O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett.82, 5229–5232 (1999).
[CrossRef]

Gaeta, A. L.

A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, “Spectroscopy of Rb atoms in hollow-core fibers,” Phys. Rev. A81, 053825 (2010).
[CrossRef]

S. Ghosh, A. R. Bhagwat, C. K. Renshaw, S. Goh, and A. L. Gaeta, “Low-light-level optical interactions with rubidium vapor in a photonic band-gap fiber,” Phys. Rev. Lett.97, 023603 (2006).
[CrossRef] [PubMed]

S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, “Resonant optical interactions with molecules confined in photonic band-gap fibers,” Phys. Rev. Lett.94, 093902 (2005).
[CrossRef] [PubMed]

Galli, M.

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. L. Savio, A. Irrera, and F. Priolo, “Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides,” Appl. Phys. Lett.89, 241114 (2006).
[CrossRef]

Gao, J.

Gerace, D.

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. L. Savio, A. Irrera, and F. Priolo, “Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides,” Appl. Phys. Lett.89, 241114 (2006).
[CrossRef]

Ghosh, S.

S. Ghosh, A. R. Bhagwat, C. K. Renshaw, S. Goh, and A. L. Gaeta, “Low-light-level optical interactions with rubidium vapor in a photonic band-gap fiber,” Phys. Rev. Lett.97, 023603 (2006).
[CrossRef] [PubMed]

S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, “Resonant optical interactions with molecules confined in photonic band-gap fibers,” Phys. Rev. Lett.94, 093902 (2005).
[CrossRef] [PubMed]

Gibson, B. C.

Goh, S.

S. Ghosh, A. R. Bhagwat, C. K. Renshaw, S. Goh, and A. L. Gaeta, “Low-light-level optical interactions with rubidium vapor in a photonic band-gap fiber,” Phys. Rev. Lett.97, 023603 (2006).
[CrossRef] [PubMed]

Greentree, A. D.

Guo, R.

Hadji, E.

K. Foubert, L. Lalouat, B. Cluzel, E. Picard, D. Peyrade, F. de Fornel, and E. Hadji, “An air-slotted nanoresonator relying on coupled high Q small V Fabry-Perot nanocavities,” Appl. Phys. Lett.94, 251111 (2009).
[CrossRef]

Hafezi, M.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett.102, 203902 (2009).
[CrossRef] [PubMed]

Hainberger, R.

Hakuta, K.

F. L. Kien and K. Hakuta, “Slowing down of a guided light field along a nanofiber in a cold atomic gas,” Phys. Rev. A.79, 013818 (2009).
[CrossRef]

Hang, C.

C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phys. Rev. A74, 012319 (2006).
[CrossRef]

Harris, S. E.

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, 594–598 (1999).
[CrossRef]

Hau, L. V.

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, 594–598 (1999).
[CrossRef]

He, Y.

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

Hiscocks, M. P.

Hofferberth, S.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett.102, 203902 (2009).
[CrossRef] [PubMed]

Hollberg, L.

M. M. Kash, V.A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M.O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett.82, 5229–5232 (1999).
[CrossRef]

Hollenberg, L. C. L.

Huang, G.

L. Li and G. Huang, “Linear and nonlinear light propagations in a Doppler-broadened medium via electromagnetically induced transparency,” Phys. Rev. A82, 023809 (2010).
[CrossRef]

C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phys. Rev. A74, 012319 (2006).
[CrossRef]

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E.72, 016617 (2005).
[CrossRef]

Ibanescu, M.

Imamoglu, A.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77, 633–673 (2005).
[CrossRef]

Irrera, A.

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. L. Savio, A. Irrera, and F. Priolo, “Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides,” Appl. Phys. Lett.89, 241114 (2006).
[CrossRef]

Jacobs, S.

Jacome, L.

Javan, A.

H. Lee, Y. Rostovtsev, C. J. Bednar, and A. Javan, “From laser-induced line narrowing to electromagnetically induced transparency: closed system analysis,” Appl. Phys. B76, 33 (2003).
[CrossRef]

Jhon, Y. M.

Jiang, L.

Jiang, X.

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

Jin, X.-M.

H. N. Dai, H. Zhang, S.-J. Yang, T.-M. Zhao, J. Rui, Y.-J. Deng, L. Li, N.-L. Liu, S. Chen, X.-H. Bao, X.-M. Jin, B. Zhao, and J.-W. Pan, “Holographic storage of biphoton entanglement,” Phys. Rev. Lett.108, 210501 (2012).
[CrossRef] [PubMed]

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

Joannopoulos, J.

Johnson, S.

Kash, M. M.

M. M. Kash, V.A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M.O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett.82, 5229–5232 (1999).
[CrossRef]

Kien, F. L.

F. L. Kien and K. Hakuta, “Slowing down of a guided light field along a nanofiber in a cold atomic gas,” Phys. Rev. A.79, 013818 (2009).
[CrossRef]

Kim, J.

Koos, C.

Kuga, T.

Kuramochi, E.

Ladouceur, F.

Lalouat, L.

K. Foubert, L. Lalouat, B. Cluzel, E. Picard, D. Peyrade, F. de Fornel, and E. Hadji, “An air-slotted nanoresonator relying on coupled high Q small V Fabry-Perot nanocavities,” Appl. Phys. Lett.94, 251111 (2009).
[CrossRef]

Lee, H.

H. Lee, Y. Rostovtsev, C. J. Bednar, and A. Javan, “From laser-induced line narrowing to electromagnetically induced transparency: closed system analysis,” Appl. Phys. B76, 33 (2003).
[CrossRef]

Lee, S.

Leuthold, J.

Li, L.

H. N. Dai, H. Zhang, S.-J. Yang, T.-M. Zhao, J. Rui, Y.-J. Deng, L. Li, N.-L. Liu, S. Chen, X.-H. Bao, X.-M. Jin, B. Zhao, and J.-W. Pan, “Holographic storage of biphoton entanglement,” Phys. Rev. Lett.108, 210501 (2012).
[CrossRef] [PubMed]

L. Li and G. Huang, “Linear and nonlinear light propagations in a Doppler-broadened medium via electromagnetically induced transparency,” Phys. Rev. A82, 023809 (2010).
[CrossRef]

Li, Y.

Y. Li, J. Zheng, J. Gao, J. Shu, M. S. Aras, and C. W. Wong, “Design of dispersive optomechanical coupling and cooling in ultrahigh-Q/V slot-type photonic crystal cavities,” Opt. Express18, 23844–23856 (2010).
[CrossRef] [PubMed]

C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phys. Rev. A74, 012319 (2006).
[CrossRef]

Li, Y. X.

Light, P.

Light, P. S.

Lipson, M.

Liscidini, M.

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. L. Savio, A. Irrera, and F. Priolo, “Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides,” Appl. Phys. Lett.89, 241114 (2006).
[CrossRef]

Liu, N.-L.

H. N. Dai, H. Zhang, S.-J. Yang, T.-M. Zhao, J. Rui, Y.-J. Deng, L. Li, N.-L. Liu, S. Chen, X.-H. Bao, X.-M. Jin, B. Zhao, and J.-W. Pan, “Holographic storage of biphoton entanglement,” Phys. Rev. Lett.108, 210501 (2012).
[CrossRef] [PubMed]

Loncar, M.

Q. Quan, I. Bulu, and M. Lončar, “Broadband waveguide QED system on a chip,” Phys. Rev. A.80, 011810(R) (2009).
[CrossRef]

Londero, P.

A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, “Spectroscopy of Rb atoms in hollow-core fibers,” Phys. Rev. A81, 053825 (2010).
[CrossRef]

Luiten, A. N.

Lukin, M. D.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett.102, 203902 (2009).
[CrossRef] [PubMed]

M. M. Kash, V.A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M.O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett.82, 5229–5232 (1999).
[CrossRef]

Lvovsky, A. I.

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photon.3, 706–714 (2009).
[CrossRef]

Ma, L.

C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phys. Rev. A74, 012319 (2006).
[CrossRef]

Marangos, J. P.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77, 633–673 (2005).
[CrossRef]

Maric, M.

Miritello, M.

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. L. Savio, A. Irrera, and F. Priolo, “Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides,” Appl. Phys. Lett.89, 241114 (2006).
[CrossRef]

Muellner, P.

Notomi, M.

Ottaviani, C.

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett.90, 197902 (2003).
[CrossRef] [PubMed]

Ouzounov, D. G.

S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, “Resonant optical interactions with molecules confined in photonic band-gap fibers,” Phys. Rev. Lett.94, 093902 (2005).
[CrossRef] [PubMed]

Pan, G.-S.

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

Pan, J.-W.

H. N. Dai, H. Zhang, S.-J. Yang, T.-M. Zhao, J. Rui, Y.-J. Deng, L. Li, N.-L. Liu, S. Chen, X.-H. Bao, X.-M. Jin, B. Zhao, and J.-W. Pan, “Holographic storage of biphoton entanglement,” Phys. Rev. Lett.108, 210501 (2012).
[CrossRef] [PubMed]

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

Panepucci, R.

Panepucci, R. R.

Park, N.

Patrini, M.

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. L. Savio, A. Irrera, and F. Priolo, “Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides,” Appl. Phys. Lett.89, 241114 (2006).
[CrossRef]

Payne, M. G.

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E.72, 016617 (2005).
[CrossRef]

Peyrade, D.

K. Foubert, L. Lalouat, B. Cluzel, E. Picard, D. Peyrade, F. de Fornel, and E. Hadji, “An air-slotted nanoresonator relying on coupled high Q small V Fabry-Perot nanocavities,” Appl. Phys. Lett.94, 251111 (2009).
[CrossRef]

Peyronel, T.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett.102, 203902 (2009).
[CrossRef] [PubMed]

Picard, E.

K. Foubert, L. Lalouat, B. Cluzel, E. Picard, D. Peyrade, F. de Fornel, and E. Hadji, “An air-slotted nanoresonator relying on coupled high Q small V Fabry-Perot nanocavities,” Appl. Phys. Lett.94, 251111 (2009).
[CrossRef]

Politi, A.

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. L. Savio, A. Irrera, and F. Priolo, “Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides,” Appl. Phys. Lett.89, 241114 (2006).
[CrossRef]

Poulton, C.

Priolo, F.

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. L. Savio, A. Irrera, and F. Priolo, “Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides,” Appl. Phys. Lett.89, 241114 (2006).
[CrossRef]

Quan, Q.

Q. Quan, I. Bulu, and M. Lončar, “Broadband waveguide QED system on a chip,” Phys. Rev. A.80, 011810(R) (2009).
[CrossRef]

Renshaw, C. K.

S. Ghosh, A. R. Bhagwat, C. K. Renshaw, S. Goh, and A. L. Gaeta, “Low-light-level optical interactions with rubidium vapor in a photonic band-gap fiber,” Phys. Rev. Lett.97, 023603 (2006).
[CrossRef] [PubMed]

Rostovtsev, Y.

H. Lee, Y. Rostovtsev, C. J. Bednar, and A. Javan, “From laser-induced line narrowing to electromagnetically induced transparency: closed system analysis,” Appl. Phys. B76, 33 (2003).
[CrossRef]

M. M. Kash, V.A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M.O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett.82, 5229–5232 (1999).
[CrossRef]

Rui, J.

H. N. Dai, H. Zhang, S.-J. Yang, T.-M. Zhao, J. Rui, Y.-J. Deng, L. Li, N.-L. Liu, S. Chen, X.-H. Bao, X.-M. Jin, B. Zhao, and J.-W. Pan, “Holographic storage of biphoton entanglement,” Phys. Rev. Lett.108, 210501 (2012).
[CrossRef] [PubMed]

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

Russell, P.

Ryu, H.

Sanders, B. C.

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photon.3, 706–714 (2009).
[CrossRef]

Sautenkov, V.A.

M. M. Kash, V.A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M.O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett.82, 5229–5232 (1999).
[CrossRef]

Savio, R. L.

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. L. Savio, A. Irrera, and F. Priolo, “Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides,” Appl. Phys. Lett.89, 241114 (2006).
[CrossRef]

Scully, M.O.

M. M. Kash, V.A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M.O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett.82, 5229–5232 (1999).
[CrossRef]

Sharping, J. E.

S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, “Resonant optical interactions with molecules confined in photonic band-gap fibers,” Phys. Rev. Lett.94, 093902 (2005).
[CrossRef] [PubMed]

Shu, J.

Skorobogatiy, M.

Slepkov, A. D.

A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, “Spectroscopy of Rb atoms in hollow-core fibers,” Phys. Rev. A81, 053825 (2010).
[CrossRef]

Soljacic, M.

Su, C.

Taniyama, H.

Tittel, W.

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photon.3, 706–714 (2009).
[CrossRef]

Tombesi, P.

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett.90, 197902 (2003).
[CrossRef] [PubMed]

Torii, Y.

Venkataraman, V.

A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, “Spectroscopy of Rb atoms in hollow-core fibers,” Phys. Rev. A81, 053825 (2010).
[CrossRef]

Vitali, D.

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett.90, 197902 (2003).
[CrossRef] [PubMed]

Vuletic, V.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett.102, 203902 (2009).
[CrossRef] [PubMed]

Wang, B.

Wang, J.

Wang, L.

Wang, X.

Weisberg, O.

Welch, G. R.

M. M. Kash, V.A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M.O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett.82, 5229–5232 (1999).
[CrossRef]

Wellenzohn, M.

Willner, A. E.

Wong, C. W.

Wu, Y.

Y. Wu and L. Deng, “Ultraslow optical solitons in a cold four-state medium,” Phys. Rev. Lett.93, 143904 (2004).
[CrossRef] [PubMed]

Xu, Q.

Yamamoto, T.

Yang, F.

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

Yang, J.

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

Yang, S.-J.

H. N. Dai, H. Zhang, S.-J. Yang, T.-M. Zhao, J. Rui, Y.-J. Deng, L. Li, N.-L. Liu, S. Chen, X.-H. Bao, X.-M. Jin, B. Zhao, and J.-W. Pan, “Holographic storage of biphoton entanglement,” Phys. Rev. Lett.108, 210501 (2012).
[CrossRef] [PubMed]

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

Yoshikawa, Y.

Yuan, Z.-S.

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

Yue, Y.

Zhang, H.

H. N. Dai, H. Zhang, S.-J. Yang, T.-M. Zhao, J. Rui, Y.-J. Deng, L. Li, N.-L. Liu, S. Chen, X.-H. Bao, X.-M. Jin, B. Zhao, and J.-W. Pan, “Holographic storage of biphoton entanglement,” Phys. Rev. Lett.108, 210501 (2012).
[CrossRef] [PubMed]

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

Zhang, L.

Zhao, B.

H. N. Dai, H. Zhang, S.-J. Yang, T.-M. Zhao, J. Rui, Y.-J. Deng, L. Li, N.-L. Liu, S. Chen, X.-H. Bao, X.-M. Jin, B. Zhao, and J.-W. Pan, “Holographic storage of biphoton entanglement,” Phys. Rev. Lett.108, 210501 (2012).
[CrossRef] [PubMed]

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

Zhao, T.-M.

H. N. Dai, H. Zhang, S.-J. Yang, T.-M. Zhao, J. Rui, Y.-J. Deng, L. Li, N.-L. Liu, S. Chen, X.-H. Bao, X.-M. Jin, B. Zhao, and J.-W. Pan, “Holographic storage of biphoton entanglement,” Phys. Rev. Lett.108, 210501 (2012).
[CrossRef] [PubMed]

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

Zheng, J.

Zhou, Z.

Zibrov, A. S.

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett.102, 203902 (2009).
[CrossRef] [PubMed]

M. M. Kash, V.A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M.O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett.82, 5229–5232 (1999).
[CrossRef]

Appl. Phys. B (1)

H. Lee, Y. Rostovtsev, C. J. Bednar, and A. Javan, “From laser-induced line narrowing to electromagnetically induced transparency: closed system analysis,” Appl. Phys. B76, 33 (2003).
[CrossRef]

Appl. Phys. Lett. (2)

M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L. C. Andreani, A. Canino, M. Miritello, R. L. Savio, A. Irrera, and F. Priolo, “Direct evidence of light confinement and emission enhancement in active silicon-on-insulator slot waveguides,” Appl. Phys. Lett.89, 241114 (2006).
[CrossRef]

K. Foubert, L. Lalouat, B. Cluzel, E. Picard, D. Peyrade, F. de Fornel, and E. Hadji, “An air-slotted nanoresonator relying on coupled high Q small V Fabry-Perot nanocavities,” Appl. Phys. Lett.94, 251111 (2009).
[CrossRef]

Nat. Photon. (1)

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photon.3, 706–714 (2009).
[CrossRef]

Nat. Photonics (1)

H. Zhang, X.-M. Jin, J. Yang, H.-N. Dai, S.-J. Yang, T.-M. Zhao, J. Rui, Y. He, X. Jiang, F. Yang, G.-S. Pan, Z.-S. Yuan, Y. Deng, Z.-B. Chen, X.-H. Bao, S. Chen, B. Zhao, and J.-W. Pan, “Preparation and storage of frequency-uncorrelated entangled photons from cavity-enhanced spontaneous parametric downconversion,” Nat. Photonics5, 628–632 (2011).
[CrossRef]

Nature. (1)

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, 594–598 (1999).
[CrossRef]

Opt. Express (10)

S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. Engeness, M. Soljacic, S. Jacobs, J. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express9, 748–779 (2001).
[CrossRef] [PubMed]

F. Benabid, P. Light, F. Couny, and P. Russell, “Electromagnetically-induced transparency grid in acetylene-filled hollow-core PCF,” Opt. Express13, 5694–5703 (2005).
[CrossRef] [PubMed]

T. Yamamoto, M. Notomi, H. Taniyama, E. Kuramochi, Y. Yoshikawa, Y. Torii, and T. Kuga, “Design of a high-Q air-slot cavity based on a width-modulated line-defect in a photonic crystal slab,” Opt. Express16, 13809–13817 (2008).
[CrossRef] [PubMed]

Y. Li, J. Zheng, J. Gao, J. Shu, M. S. Aras, and C. W. Wong, “Design of dispersive optomechanical coupling and cooling in ultrahigh-Q/V slot-type photonic crystal cavities,” Opt. Express18, 23844–23856 (2010).
[CrossRef] [PubMed]

M. P. Hiscocks, C. Su, B. C. Gibson, A. D. Greentree, L. C. L. Hollenberg, and F. Ladouceur, “Slot-waveguide cavities for optical quantum information applications,” Opt. Express17, 7295–7303 (2009).
[CrossRef] [PubMed]

H. Ryu, J. Kim, Y. M. Jhon, S. Lee, and N. Park, “Effect of index contrasts in the wide spectral-range control of slot waveguide dispersion,” Opt. Express20, 13189–13194 (2012)
[CrossRef] [PubMed]

C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express15, 5976–5990 (2007).
[CrossRef] [PubMed]

P. Muellner, M. Wellenzohn, and R. Hainberger, “Nonlinearity of optimized silicon photonic slot waveguides,” Opt. Express17, 9282–9287 (2009).
[CrossRef] [PubMed]

L. Zhang, Y. Yue, Y. X. Li, J. Wang, R. G. Beausoleil, and A. E. Willner, “Flat and low dispersion in highly nonlinear slot waveguides,” Opt. Express18, 13187–13193 (2010).
[CrossRef] [PubMed]

Y. Yue, L. Zhang, J. Wang, R. G. Beausoleil, and A. E. Willner, “Highly efficient nonlinearity reduction in silicon-on-insulator waveguides using vertical slots,” Opt. Express18, 22061–22066 (2010).
[CrossRef] [PubMed]

Opt. Lett. (4)

Phys. Rev. A (3)

L. Li and G. Huang, “Linear and nonlinear light propagations in a Doppler-broadened medium via electromagnetically induced transparency,” Phys. Rev. A82, 023809 (2010).
[CrossRef]

A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, “Spectroscopy of Rb atoms in hollow-core fibers,” Phys. Rev. A81, 053825 (2010).
[CrossRef]

C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phys. Rev. A74, 012319 (2006).
[CrossRef]

Phys. Rev. A. (2)

F. L. Kien and K. Hakuta, “Slowing down of a guided light field along a nanofiber in a cold atomic gas,” Phys. Rev. A.79, 013818 (2009).
[CrossRef]

Q. Quan, I. Bulu, and M. Lončar, “Broadband waveguide QED system on a chip,” Phys. Rev. A.80, 011810(R) (2009).
[CrossRef]

Phys. Rev. E. (1)

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E.72, 016617 (2005).
[CrossRef]

Phys. Rev. Lett. (7)

S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, “Resonant optical interactions with molecules confined in photonic band-gap fibers,” Phys. Rev. Lett.94, 093902 (2005).
[CrossRef] [PubMed]

S. Ghosh, A. R. Bhagwat, C. K. Renshaw, S. Goh, and A. L. Gaeta, “Low-light-level optical interactions with rubidium vapor in a photonic band-gap fiber,” Phys. Rev. Lett.97, 023603 (2006).
[CrossRef] [PubMed]

M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett.102, 203902 (2009).
[CrossRef] [PubMed]

Y. Wu and L. Deng, “Ultraslow optical solitons in a cold four-state medium,” Phys. Rev. Lett.93, 143904 (2004).
[CrossRef] [PubMed]

M. M. Kash, V.A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M.O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett.82, 5229–5232 (1999).
[CrossRef]

H. N. Dai, H. Zhang, S.-J. Yang, T.-M. Zhao, J. Rui, Y.-J. Deng, L. Li, N.-L. Liu, S. Chen, X.-H. Bao, X.-M. Jin, B. Zhao, and J.-W. Pan, “Holographic storage of biphoton entanglement,” Phys. Rev. Lett.108, 210501 (2012).
[CrossRef] [PubMed]

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett.90, 197902 (2003).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77, 633–673 (2005).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Left: Schematic of slot waveguide structure. The slot width (index nS) is 2a, the width of the high-index silicon slabs (index nH) is 2b − 2a. The index of the cladding material is nC. Right: Level diagram of the three-level atomic system. Ground state |1〉 couples to the exited |2〉 and |3〉 with the control field Ωc and the probe field Ωp. Δ2 and Δ3 are the detunings of control and probe fields, respectively. Γ3132) is the spontaneous emission decay rate from |3〉 to |1〉 (|2〉). Γ12 and Γ21 are incoherent population exchange rates. Below the level diagram is the coordinate system chosen for theoretical calculations. The slot region is |z| ≤ a, and the slabs are in the region a < |z| < b.

Fig. 2
Fig. 2

(a): Im(K) as a function of frequency ω for different slot width 2a. The red solid, black dashed, and blue dashed-dotted lines are for the slot width 2a = 50, 30 and 10 nm, respectively. (b): Im(K0) as a function of |Ωc| for different slot width 2a. The red solid, black dashed, and blue dashed-dotted lines are for the slot width 2a = 50, 30, 10 nm, respectively.

Fig. 3
Fig. 3

(a): Im(K) as a function of ω. (b): Im(K0) as a function of |Ωc| with Γ21 = 0 (red solid line), Γ21 = 0.5Γ12 (black dashed line) and Γ21 = Γ12 (blue dashed-dotted line), respectively.

Fig. 4
Fig. 4

Third-order susceptibility Re ( χ p p ( 3 ) ) as a function of detunning Δ3 for different slot width 2a. The red solid, black dashed and blue dashed-dotted lines are for 2a = 50,25 and 5 nm, respectively.

Fig. 5
Fig. 5

(a) The three-dimensional plot of the wave shape |Ωp/U0|2 as a function of z/LD and t/τ0. The solution is numerically obtained from Eq. (14) with full complex coefficients taken into account. The values of parameters are given in the text. (b): The interaction between two identical bright solitons.

Equations (50)

Equations on this page are rendered with MathJax. Learn more.

E TM ( r , t ) = k h ¯ ω 1 ( k ) 2 ε 0 V 1 u 1 , k ( z ) a 1 ( k ) e i [ k y ω 1 ( k ) t ] + c . c . ,
u 1 , k ( z ) = c 2 N 1 ω 1 ( k ) n 2 ( z ) [ k H 1 , k ( z ) e z + i d H 1 , k ( z ) d z e y ] .
E TM ( r , t ) = l = p , c 1 ( W 1 V 1 ) 1 2 u 1 , l ( z ) exp { i [ k ( ω l ) y ω l t ] } + c . c ..
^ = h ¯ j 3 Δ j | j j | h ¯ [ ζ p * ( z ) Ω p * | 1 3 | + ζ c * ( z ) Ω c * | 2 3 | + h . c . ] ,
i t σ 11 + i Γ 21 σ 11 i Γ 12 σ 22 i Γ 13 σ 33 + ζ p * ( z ) Ω p * σ 31 ζ p ( z ) Ω p σ 31 * = 0 ,
i t σ 22 i Γ 21 σ 11 + i Γ 12 σ 22 i Γ 23 σ 33 + ζ c * ( z ) Ω c * σ 32 ζ c ( z ) Ω c σ 32 * = 0 ,
i t σ 33 + i ( Γ 13 + Γ 23 ) σ 33 ζ p * ( z ) Ω p * σ 31 + ζ p ( z ) Ω p σ 31 * ζ c * ( z ) Ω c * σ 32 + ζ c ( z ) Ω c σ 32 * = 0 ,
( i t + d 21 ) σ 21 ζ p ( z ) Ω p σ 32 * + ζ c * ( z ) Ω c * σ 31 = 0 ,
( i t + d 31 ) σ 31 ζ p ( z ) Ω p ( σ 33 σ 11 ) + ζ c ( z ) Ω c σ 21 = 0 ,
( i t + d 32 ) σ 32 ζ c ( z ) Ω c ( σ 33 σ 22 ) + ζ p ( z ) Ω p σ 21 * = 0 ,
P = 𝒩 a d v f ( v ) [ p 13 σ 31 e i ( k p y ω p t ) + p 23 σ 32 e i ( k c y ω c t ) + c . c . ] ,
i ( y + 1 c n 2 ( z ) n eff t ) Ω p + c 2 ω p n eff 2 Ω p x 2 + κ 13 d v f v σ 31 ( v , z ) = 0 ,
σ 11 ( 0 ) = Γ 12 ( Γ 13 + Γ 23 ) X 1 + ( Γ 12 + Γ 13 ) | ζ ( z ) Ω c | 2 X 2 ,
σ 22 ( 0 ) = Γ 21 ( Γ 13 + Γ 23 ) X 1 + Γ 21 | ζ ( z ) Ω c | 2 X 2 ,
σ 33 ( 0 ) = Γ 21 | ζ ( z ) Ω c | 2 X 2 ,
σ 32 ( 0 ) = ζ ( z ) Ω c d 32 Γ 21 ( Γ 13 + Γ 23 ) X 1 X 2 ,
K ( ω ) = 1 d z | ζ ( z ) | 2 d z | ζ ( z ) | 2 { ω c n 2 ( z ) n eff + κ 13 d v f ( v ) ζ ( z ) Ω c σ 32 * ( 0 ) + ( ω + d 21 ) ( 2 σ 11 ( 0 ) + σ 22 ( 0 ) 1 ) | ζ ( z ) Ω c | 2 ( ω + d 21 ) ( ω + d 31 ) } .
K ( ω ) = d z | ζ ( z ) | 2 ( ω c n 2 ( z ) n eff + 𝒦 1 ( z ) + 𝒦 2 ( z ) ) / d z | ζ ( z ) | 2 ,
𝒦 1 ( z ) = π κ 13 { | ζ ( z ) Ω c | 2 Γ 21 Γ 3 ( i Δ ω D i γ 32 ) + ( ω + i γ 21 ) [ Γ 12 Γ 3 ( Δ ω D 2 + γ 32 2 ) + 2 γ 32 ( Γ 12 Γ 21 + Γ 13 ) | ζ ( z ) Ω c | 2 ] } / { ( Δ ω D 2 + η 2 ) ( Γ 12 + Γ 21 ) Γ 3 [ | ζ ( z ) Ω c | 2 ( ω + i γ 21 ) ( ω + i Δ ω D + i γ 31 ) ] } ,
𝒦 2 ( z ) = π κ 13 Δ ω D { | ζ ( z ) Ω c | 2 Γ 21 Γ 3 ( i η i γ 32 ) + ( ω + i γ 21 ) [ Γ 12 Γ 3 ( η 2 + γ 32 2 ) + 2 γ 32 ( Γ 12 Γ 21 + Γ 13 ) | ζ ( z ) Ω c | 2 ] } / { η ( Δ ω D 2 η 2 ) ( Γ 12 + Γ 21 ) Γ 3 [ | ζ ( z ) Ω c | 2 ( ω + i γ 21 ) ( ω + i η + i γ 31 ) ] } .
Im ( K 0 ) = 1 d z | ζ ( z ) | 2 d z | ζ ( z ) | 2 d v f ( v ) κ 13 γ 21 | ζ ( z ) Ω c | 2 + γ 21 γ 31 × ( 1 Γ 21 Γ 3 X 1 + 3 Γ 21 | ζ ( z ) Ω c | 2 ( Γ 21 + Γ 12 ) Γ 3 X 1 + ( 2 Γ 21 + Γ 12 + Γ 13 ) | ζ ( z ) Ω c | 2 ) .
χ p = d v f ( v ) 𝒩 a | p 13 | 2 ε 0 h ¯ σ 31 ( v , z ) Ω p χ p ( 1 ) + χ p p ( 3 ) | p | 2 ,
χ p ( 1 ) = 𝒩 a | p 13 | 2 ε 0 h ¯ d z | ζ ( z ) | 2 d v f ( v ) × d 21 d 32 * ( σ 33 ( 0 ) σ 11 ( 0 ) ) | ζ ( z ) Ω c | 2 ( σ 33 ( 0 ) σ 22 ( 0 ) ) d 32 * ( d 21 d 31 | ζ ( z ) Ω c | 2 ) / d z | ζ ( z ) | 2 ,
χ p p ( 3 ) = 𝒩 a | p 13 | 4 ε 0 h ¯ 3 1 d z | ζ ( z ) | 2 d z | ζ ( z ) | 2 d v f ( v ) × | ζ ( z ) | 2 [ i d 32 ζ ( z ) Ω c Z 1 Z 4 ( 2 d 21 | d 32 | 2 d 32 | ζ ( z ) Ω c | 2 ) Z 2 ( d 21 | d 32 | 2 2 d 32 | ζ ( z ) Ω c | 2 ) Z 3 ] / [ i Z 1 | d 32 | 2 ( | ζ ( z ) Ω c | 2 d 21 d 31 ) ] ,
Ω p ( 1 ) = Fe i θ ,
σ 31 ( 1 ) = ( ω + d 21 ) ( 2 σ 11 ( 0 ) + σ 22 ( 0 ) 1 ) + ζ ( z ) Ω c σ 32 * ( 0 ) | ζ 2 ( z ) Ω c | 2 ( ω + d 21 ) ( ω + d 31 ) ζ ( z ) Fe i θ ,
σ 21 ( 1 ) = ( ω + d 31 ) σ 32 * ( 0 ) + ζ c * ( z ) Ω c * ( 2 σ 11 ( 0 ) + σ 22 ( 0 ) 1 ) | ζ 2 ( z ) Ω c | 2 ( ω + d 21 ) ( ω + d 31 ) ζ ( z ) Fe i θ ,
i F y 2 + c 2 ω p n eff 2 F x 1 2 K 2 2 2 F t 1 2 W | F | 2 Fe 2 α ¯ y 2 = 0 ,
W = κ 13 d z | ζ ( z ) | 4 d v f ( v ) ζ ( z ) Ω c a 32 * ( 2 ) + ( ω + d 21 ) ( 2 a 11 ( 2 ) + a 22 ( 2 ) ) | ζ 2 ( z ) Ω c | 2 ( ω + d 21 ) ( ω + d 31 ) / d z | ζ ( z ) | 2 ,
i ( y + α ) U + c 2 ω p n eff 2 U x 2 K 2 2 2 U τ 2 W | U | 2 U = 0 ,
i u s + 2 u σ 2 + 2 u | u | 2 = i d 0 u + d 1 2 u ξ 2 ,
u = 2 β sech [ 2 β ( σ σ 0 + 4 δ s ) ] exp [ 2 i δ σ 4 i ( δ 2 β 2 ) s i ϕ 0 ] ,
Ω p = 1 τ 0 K ˜ 2 W ˜ sech [ 1 τ 0 ( t y v ˜ g ) ] exp [ i K ˜ 0 y + i y 2 L D ] ,
H m , k | | TM ( r , t ) = ( k ^ | | × e z ) H m , k | | ( z ) e i ( k | | r ω m t ) + c . c . ,
E m , k | | TM ( r , t ) = i ω m ( k | | ) ε 0 ε ( z ) [ i k | | H m , k | | ( z ) e z + d H m , k | | ( z ) d z k ^ | | ] e i ( k | | r ω m t ) + c . c . ,
d 2 d z 2 H m , k | | ( z ) + [ ( ω c ) 2 n 2 ( z ) k | | 2 ] H m , k | | ( z ) = 0 ,
H m , k | | TM ( z ) = { cosh ( γ S m z ) , | z | < a C m cos [ κ H m ( | z | a ) ] + D m sin [ κ H m ( | z | a ) ] , a < | z | < b E m exp [ γ S m ( | z | b ) ] , | z | > b
tan [ κ H m ( b a ) arctan ( γ S m n H 2 κ H m n S 2 ) ] = γ S n n H 2 κ H m n S 2 tanh ( γ S m a ) .
E TM ( r , t ) = k x , k y m = 1 h ¯ ω m 2 ε 0 V m u m , k | | ( z ) a ^ m ( k ) e i ( k | | r ω m t ) + h . c . ,
H TM ( r , t ) = k x , k y m = 1 h ¯ ω m 2 ε 0 V m ( k ^ | | × e z ) ε 0 c N m H m , k | | ( z ) a ^ m ( k ) e i ( k | | r ω m t ) + h . c . ,
V m = S { 1 2 γ S m [ sinh ( 2 γ S m a ) + 2 E m 2 ] + a 2 + ( C m 2 + D m 2 ) ( b a ) + 1 2 κ H m ( C m 2 D m 2 ) sin [ 2 κ H m ( b a ) ] } / N m ,
N m = ω m 2 / ( c P ) 2 ,
H m , k | | TM ( z ) = { sin ϕ m exp ( z + b 2 b ψ m cos ϕ m ) , z < b cos ( ψ m z b sin ϕ m ) , b < z < b sin ϕ m exp ( b z 2 b ψ m cos ϕ m ) , z > b
E TM ( r , t ) = k x , k y m = 1 h ¯ ω m 2 ε 0 W m u m , k | | ( z ) a ^ n ( k ) e i ( k | | r ω m t ) + h . c . ,
H TM ( r , t ) = k x , k y m = 1 h ¯ ω m 2 ε 0 W m ( k ^ | | × e z ) ε 0 c M m H m , k | | ( z ) a ^ m ( k ) e i ( k | | r ω m t ) + h . c . ,
W m = S { b [ 2 sin 2 ϕ m + ψ m cos ϕ m ] / ( ψ m cos ϕ m ) + b sin ( 2 ψ m sin ϕ m ) / ( 2 ψ m sin ϕ m ) } / M m ,
M m = ω m 2 / ( c G ) 2 ,
a 11 ( 2 ) = i { [ i ( Γ 12 + Γ 23 ) 2 | ζ 2 ( z ) Ω c | 2 ( 1 d 32 1 d 32 * ) ] [ ( ω + d 21 * ) ( 2 σ 11 ( 0 ) + σ 22 ( 0 ) 1 ) + ζ * ( z ) Ω c * σ 32 ( 0 ) | ζ 2 ( z ) Ω c | 2 ( ω + d 21 * ) ( ω + d 31 * ) c . c . ] + i ( Γ 12 Γ 13 ) [ ζ * ( z ) Ω c * d 32 ( ω + d 31 * ) σ 32 ( 0 ) + ζ ( z ) Ω c ( 2 σ 11 ( 0 ) + σ 22 ( 0 ) 1 ) | ζ 2 ( z ) Ω c | 2 ( ω + d 21 * ) ( ω + d 31 * ) + c . c . ] } / [ i ( Γ 12 + Γ 21 ) ( Γ 13 + Γ 23 ) ( 2 Γ 21 + Γ 12 + Γ 13 ) | ζ 2 ( z ) Ω c | 2 ( 1 d 32 1 d 32 * ) ] ,
a 22 ( 2 ) = i Γ 12 Γ 13 [ ( ω + d 21 * ) ( 2 σ 11 ( 0 ) + σ 22 ( 0 ) 1 ) + ζ * ( z ) Ω c * σ 32 ( 0 ) | ζ 2 ( z ) Ω c | 2 ( ω + d 21 * ) ( ω + d 31 * ) c . c . i ( Γ 21 + Γ 13 ) a 11 ( 2 ) ] ,
a 32 ( 2 ) = 1 d 32 [ ( ω + d 31 * ) σ 32 ( 0 ) + ζ ( z ) Ω c ( 2 σ 11 ( 0 ) + σ 22 ( 0 ) 1 ) | ζ 2 ( z ) Ω c | 2 ( ω + d 21 * ) ( ω + d 31 * ) ζ ( z ) Ω c ( a 11 ( 2 ) + 2 a 22 ( 2 ) ) ] .

Metrics