# Electromagnetically induced transparency for guided light in an atomic array outside an optical nanofiber

###### Abstract

We study the propagation of guided light along an array of three-level atoms in the vicinity of an optical nanofiber under the condition of electromagnetically induced transparency. We examine two schemes of atomic levels and field polarizations where the guided probe field is quasilinearly polarized along the major or minor principal axis, which is parallel or perpendicular, respectively, to the radial direction of the atomic position. Our numerical calculations indicate that 200 cesium atoms in a linear array with a length of 100 m at a distance of 200 nm from the surface of a nanofiber with a radius of 250 nm can slow down the speed of guided probe light by a factor of about (the corresponding group delay is about 1.17 s). In the neighborhood of the Bragg resonance, a significant fraction of the guided probe light can be reflected back with a negative group delay. The reflectivity and the group delay of the reflected field do not depend on the propagation direction of the probe field. However, when the input guided light is quasilinearly polarized along the major principal axis, the transmittivity and the group delay of the transmitted field substantially depend on the propagation direction of the probe field. Under the Bragg resonance condition, an array of atoms prepared in an appropriate internal state can transmit guided light polarized along the major principal in one specific direction even in the limit of infinitely large atom numbers. The directionality of transmission of guided light through the array of atoms is a consequence of the existence of a longitudinal component of the guided light field as well as the ellipticity of both the field polarization and the atomic dipole vector.

###### pacs:

42.50.Gy, 42.50.Nn, 42.81.Dp, 42.81.Gs## I Introduction

Optical properties of materials can be dramatically modified by quantum interference between the excitation pathways Harris review (); Scully review (). Intensive attention has been devoted to the manipulation and control of the propagation of light through coherently driven optical media, especially in the connection with the possibility of enormous slowing down, storage, and retrieval of optical pulses Harris review (); Scully review (); slow light review (); most recent review (); Scully (); Agarwal book (); vapor (); ultracold (); polariton (). The interest to this topic is related to the applications for optical delay lines, optical data storage, optical memories, quantum computing, and sensitive measurements. Through the technique of electromagnetically induced transparency (EIT) Harris review (); Scully review (); slow light review (); most recent review (); Scully (); Agarwal book (), ultralow group velocities of light have been obtained in hot vapor () and cold ultracold () atomic gases. This technique allows one to render the material highly transparent and still retain the strong dispersion required for the generation of slow light. In addition, the transmitted pulse can experience strong nonlinear effects due to the constructive interference in the third-order susceptibility . This leads to new techniques in nonlinear optics at the few-photon level, which may find important applications to quantum information processing.

The EIT technique has been extended to media embedded in a variety of waveguide systems, such as rectangular waveguides Shen (), hollow-core photonic-crystal fibers Andre (); Ghosh (); Bajcsy09 (), coupled resonator optical waveguides Neff (), quantum well waveguides Li (), waveguide-cavity systems Bermel (), and optical nanofibers Patnaik (); propag (); fibercavity (); Spillane (); Laurat15 (); Sayrin15a (). EIT-based photon switches in waveguides Bajcsy09 (); Ginzburg () have been examined. The generation and waveguiding of solitons in an EIT medium have been studied Hong (). Waveguiding of ultraslow light in an atomic Bose-Einstein condensate Tarhan () has been investigated.

Nanofibers are optical fibers that are tapered to a diameter comparable to or smaller than the wavelength of light Mazur's Nature (); Birks (); taper (). Slowing down of guided light in an optical nanofiber embedded in an EIT medium has been investigated Patnaik (); propag (); fibercavity (); Spillane (); Laurat15 (); Sayrin15a (). The first observation of EIT at very low power levels of guided pump and probe light was reported in Ref. Spillane (). Very recently, coherent storage of guided light has been experimentally demonstrated Laurat15 (); Sayrin15a (). In Ref. Patnaik (), the propagation of light is described in terms of the averages of the local refractive index, the local absorption coefficient, and the local group delay in the fiber cross-section plane. The studies in Refs. propag (); fibercavity () are based on a more rigorous formalism that takes into account the inhomogeneous density distribution of the atomic gas and the inhomogeneous mode-profile function of the guided field in the fiber transverse plane. The atomic medium considered in Patnaik (); propag (); fibercavity (); Spillane (); Laurat15 () is continuous. Meanwhile, recent experiments with atom-waveguide interfaces Dawkins11 (); Reitz13 (); Reitz14 (); Mitsch14a (); Mitsch14b (); Sayrin15b () used linear arrays of atoms prepared in a nanofiber-based optical dipole trap Vetsch10 (); Goban12 (); Polzik14 (). It has recently been demonstrated experimentally that spin-orbit coupling of guided light can lead to directional spontaneous emission Mitsch14b (); Petersen14 () and optical diodes Sayrin15b (). When the array period is near to the Bragg resonance, the discreteness and periodicity of the array may lead to significant effects, such as nearly perfect atomic mirrors, photonic band gaps, long-range interaction, and self-ordering Deutsch95 (); Birkl95 (); Henkel03 (); Artoni05 (); Petrosyan07 (); Schilke11 (); Schilke12 (); Chang07 (); Chang11 (); Chang12 (); Ritsch14a (); Ritsch14b (); AtomArray (). In the prior work on atoms trapped in a one-dimensional optical lattice under the EIT condition Petrosyan07 (); Schilke12 (), scalar light fields in free space were considered. Scattering of a scalar light field from an array of three-level atoms with two degenerate lower levels in a waveguide has also been studied Tsoi09 ().

In a nanofiber, the guided field penetrates an appreciable distance into the surrounding medium and appears as an evanescent wave carrying a significant fraction of the propagation power and having a complex polarization pattern fiber books (); fibermode (). Since the nanofiber is thin, the guided modes of the nanofiber are the fundamental HE modes fiber books (); fibermode (). These modes are hybrid modes. The field in such a mode has longitudinal electric and magnetic components. The local polarization of the mode varies in the fiber cross-section plane fibermode () and depends on the propagation direction Mitsch14b (); Petersen14 (). Therefore, the use of the scalar field formalism to treat the interaction of a nanofiber-guided field with an atom is not always appropriate. When the local polarization of the guided field is elliptical and the dipole matrix-element vector of the atom is a complex vector, direction-dependent effects in the atom-field interaction may occur Mitsch14b (); Petersen14 (); Sayrin15b (); AtomArray (); Fam14 ().

In view of the recent results and insights, it is necessary to develop a systematic theory for the propagation of guided light under the EIT condition in an atomic array taking into account the vector nature of the guided field and the discreteness and periodicity of the array.

In this paper, we study EIT in a one-dimensional periodic array of three-level atoms trapped along an optical nanofiber. We examine two schemes of atomic levels and field polarizations where the guided probe field is quasilinearly polarized along the major principal axis or the minor principal axis , which lie in the fiber cross-section plane and are parallel or perpendicular, respectively, to the radial direction of the atomic position. We take into account the vector nature of the guided field and the discreteness and periodicity of the atomic array. We study the transmittivity and reflectivity of guided light and the time evolution of the transmitted and reflected fields.

The paper is organized as follows. In Sec. II we describe two schemes of atomic levels and field polarizations for nanofiber-based EIT, and present the coupled-mode propagation equations. In Sec. III we study EIT in the homogeneous-medium approximation and the phase-matching approximation. In Sec. IV we investigate EIT in a discrete array of atoms with the help of the transfer matrix formalism. Our conclusions are given in Sec. V.

## Ii Nanofiber-based EIT schemes and coupled-mode propagation equations

We consider a linear periodic array of -type three-level atoms trapped outside an optical nanofiber (see Fig. 1). The nanofiber has a cylindrical silica core, with radius and refractive index , surrounded by vacuum, with refractive index . We use the Cartesian coordinate system and the associated cylindrical coordinate system , with being the fiber axis. We assume that the array of atoms is parallel to the fiber axis and lies in the plane. The positions of the atoms in the array are characterized by the Cartesian coordinates , , and . Here, the index labels the atoms, with being the number of atoms in the array, and the parameter is the period of the array. The axes and , which lie in the fiber cross-section plane and are parallel and perpendicular, respectively, to the radial direction of the atomic position, are called the major and minor principal axes, respectively. Although our theory is general and applicable, in principle, to arbitrary multilevel atoms, we assume cesium atoms throughout this paper. For simplicity, we neglect the effect of the surface-induced potential on the atomic energy levels. This approximation is reasonable when the atoms are not close to the fiber surface. We also neglect the effect of the far-detuned trapping light fields.

### ii.1 Quasilinearly polarized nanofiber-guided modes

We represent the electric component of a nanofiber-guided light field as , where is the angular frequency and is the slowly varying envelope of the positive-frequency part, with and being the field amplitude and the polarization vector, respectively. In general, the amplitude is a complex scalar and the polarization vector is a complex unit vector. The guided light field can be decomposed into a superposition of quasilinearly polarized modes fiber books (). These guided modes can be labeled by the index , where or (or simply or ) stands for the positive () or negative () propagation direction, respectively, and or stands for the major polarization axis. In the cylindrical coordinates, the transverse-plane profile functions of the positive-frequency parts of the electric components of the modes are given by AtomArray (); Fam14 (); fiber books (); fibermode ()

(1) |

Here, the notations , , and stand for the unit basis vectors of the cylindrical coordinate system, where and are the unit basis vectors of the Cartesian coordinate system for the fiber cross-section plane . The notations , , and stand for the cylindrical components of the profile function of the forward counterclockwise polarized guided mode and are given in Refs. fiber books (); fibermode (); cesium decay (); AtomArray (); Fam14 (). Equations (1) show that the - and -polarized guided modes have, in general, not only transverse but also longitudinal components. The local polarizations of these modes vary in the fiber cross-section plane, and are generally not strictly linear fiber books (); fibermode (). It is interesting to note from Eqs. (1) that the signs of the longitudinal components and for the - and -polarized modes, respectively, depend on the propagation direction . Thus, the difference between the mode profile functions for the forward () and backward () guided fields is expressed by the change in sign of the longitudinal components. This change may affect the magnitude of the coupling between the atom and the field and, consequently, may lead to directional spontaneous emission and directional scattering Fam14 (); AtomArray (); Mitsch14b (); Petersen14 (); Sayrin15b ().

Since the radial direction of the atomic position in our study is parallel to the axis , the polar angle for the atomic position is . Therefore, when we evaluate the mode functions (1) at the positions of the atoms, we find

(2a) | |||||

(2b) |

In deriving the above equations we have used the properties , , and AtomArray (); Fam14 (); fiber books (); fibermode ().

Equation (2a) indicates that, on the axis, the -polarized guided mode has two components, the transverse component (aligned along the axis) and the longitudinal component (aligned along the axis). The difference in phase between these components is . It depends of the mode propagation direction . The polarization vector of the -polarized guided mode on the axis is

(3) |

It is clear that the local polarization of the -polarized guided mode on the axis is elliptical in the plane. The ellipticity vector is given by , where

(4) |

is the ellipticity parameter. The circulation direction of the above elliptical polarization depends on the mode propagation direction . The polarization vector (3) is a linear superposition of the circular polarization basis vectors and with the coefficients

(5) |

The corresponding weight factors are given in terms of the polarization ellipticity parameter as . When , we have , , and for . In this case, the local polarization of the -polarized guided mode at the position of the atom on the axis is almost circular. As an example, we consider the case where the nanofiber radius is nm and the atom-to-surface distance is nm. These parameters correspond to the Vienna atom trap experiment Vetsch10 (). We find the ratio , which leads to , , and for Mitsch14b (); Petersen14 ().

Equation (2b) indicates that, on the axis, the -polarized guided mode has a single component , which is aligned along the axis. Thus, the local polarization of the -polarized guided mode at the position of the atomic array is exactly linear along the axis. This local polarization does not depends on the mode propagation direction .

### ii.2 Atomic levels and EIT schemes

We assume that the atoms have a single upper level of energy and two lower levels and of energies and , respectively. The atoms are initially prepared in the lower level . The levels and are coupled by a weak probe field of frequency . The levels and are coupled by a strong control field of frequency . The transition between the lower levels and is electric-dipole forbidden. We assume that the probe field is guided by the nanofiber and propagates along the fiber axis in the direction , where or corresponds to the positive direction or the negative direction , respectively. The detuning of the probe field with respect to the atomic transition is denoted by

(6) |

where . The control field is an external plane-wave field propagating perpendicularly to the fiber axis . The two-photon (Raman) transition between the lower levels and may be off resonance, and the corresponding two-photon detuning is denoted by

(7) |

In our general analytical calculations, the single-photon detuning and the two-photon detuning can be different from each other. However, in our numerical calculations, we will study only the case where , that is, the case where the control field is on exact resonance with the atomic transition .

To be specific, we use the transitions between the Zeeman sublevels of the line of atomic cesium in our calculations. In order to specify the internal atomic states, we use the minor principal axis as the quantization axis . The purpose of this choice is that it allows us to identify appropriate atomic states and which are coupled to each other by just one type of polarization of guided probe light. Indeed, as shown in the previous subsection, at an arbitrary position on the axis , the local polarization of -polarized guided light is elliptical in the plane and the local polarization of -polarized guided light is exactly linear along the axis. Therefore, an atomic transition between two Zeeman sublevels specified with respect to the quantization axis can interact with either - or -polarized guided light but not with - or -polarized guided light, respectively.

We consider two schemes that are illustrated in parts (a) and (b) of Fig. 1. In both schemes, we use the excited-state sublevel as the upper level . Furthermore, we use the ground-state sublevels and as the levels and , respectively, in scheme (a) of Fig. 1, and as the levels and , respectively, in scheme (b) of Fig. 1. The effects of other Zeeman sublevels are removed by applying an external magnetic field. We emphasize that the atomic states in the above schemes are specified by using the minor principal axis as the quantization axis. In addition, the quantum numbers of the lower states and are interchangeable between the two different schemes (see Fig. 1).

In scheme (a) of Fig. 1, the guided probe field is quasilinearly polarized along the direction. Meanwhile, the control field is a plane wave propagating along the axis and linearly polarized along the axis in accordance with the type of the atomic transition . In scheme (b) of Fig. 1, the guided probe field is quasilinearly polarized along the direction. Meanwhile, the control field is a plane wave propagating along the axis and counterclockwise or clockwise circularly polarized in accordance with the or type of the atomic transition . In what follows, schemes (a) and (b) of Fig. 1 are called the - and -polarization schemes, respectively. In both schemes, we neglect the reflection of the control field from the fiber surface. Since the control field propagation direction and the atomic array axis are perpendicular and parallel, respectively, to the fiber axis , the effect of the reflection of the control field can be easily accounted for by modifying the magnitude of at the position of the atomic array.

We introduce the notation for the magnetic quantum number of the atomic level , where . In the analytical calculations, we use and for the -polarization scheme, and for the -polarization scheme, and for both schemes.

We note that, in both schemes (a) and (b) of Fig. 1, the probe transition is not coupled to the guided modes with the polarization that is orthogonal to the polarization of the incident guided probe field. Here, we have introduced the notation for and for . For the control field , we can use a guided field instead of an external plane-wave field. Indeed, the control transition can be coupled by an additional guided field that is quasilinearly polarized along the direction in scheme (a) of Fig. 1 or along the direction in scheme (b) of the figure.

Due to the interaction between the atoms and the guided probe field , a guided reflected field with the frequency , the propagation direction , and the polarization may be generated. In order to describe the reflection, we need to include both propagation directions and into the analysis. We introduce the notation for the positive frequency component of the electric part of the field in the guided mode with the frequency , the polarization , and the propagation direction . The electric field vector is related to the photon flux amplitude via the formula

(8) |

Here, is the group velocity of the guided field, where is the longitudinal propagation constant. The notation stands for the normalized profile function for the guided mode with the frequency , the propagation direction , and the polarization fiber books (); cesium decay (); AtomArray (); Fam14 (). According to propag (), the amplitudes of the photon fluxes of the guided fields are governed, in the framework of the slowly varying envelope approximation, by the propagation equations

(9) |

Here, is the longitudinal propagation constant for the forward and backward guided fields,

(10) |

is the one-dimensional atom-number density, and are the coupling coefficients for the guided fields and , respectively, and with are the elements of the density matrix the atom in the interaction picture. The coupling coefficient is defined as

(11) |

Here, is the dipole matrix element for the atomic transition . We emphasize that the photon flux amplitudes are independent of and .

It is clear from Eq. (11) that the atom-field coupling coefficient depends on the local polarization of the guided probe field at the position of the atom. This coefficient also depends on the orientation and magnitude of the dipole matrix element . We emphasize again that, in our study, the internal atomic states and the atomic transitions are specified by using the minor principal axis as the quantization axis. Moreover, in order to obtain nonzero atom-field coupling coefficients, different transitions of atomic cesium, with different dipole matrix elements , are used in the different polarization schemes (see Fig. 1).

In the case of the -polarization scheme, the coupling coefficient is given as

(12) |

Here, and are respectively the radial and axial components of the mode profile function of the forward counterclockwise quasicircularly polarized guided modes fiber books (); cesium decay (); AtomArray (); Fam14 ().

In the case of the -polarization scheme, the coupling coefficient is given as

(13) |

Here, is the azimuthal component of the mode profile function of the forward counterclockwise quasicircularly polarized guided modes fiber books (); cesium decay (); AtomArray (); Fam14 ().

Note that , where is the rate of spontaneous emission of the atomic transition into the guided mode with the propagation direction and the polarization . Since the probe transition is not coupled to the guided modes with the polarization in the cases of schemes (a) and (b) of Fig. 1, we have the relation , where is the rate of spontaneous emission of the atomic transition into the guided modes with the propagation direction . Hence, we obtain .

The Rabi frequency caused by the guided field is given by

(14) |

The Rabi frequency caused by the control field is , where is the dipole matrix element for the atomic transition . We assume that . In the adiabatic approximation, the expression for to first order in is found to be

(15) |

where Harris review (); Scully review (); slow light review (); most recent review (); Scully (); Agarwal book ()

(16) |

Here, is the decay rate of the atomic probe transition coherence , and is the decay rate of the lower-level coherence .

## Iii Continuous-medium approximations

In this section, we approximate the generalized-function representation (10) of the one-dimensional atom-number density by two different continuous-function representations and present the corresponding analytical and numerical results. In Sec. III.1, we replace by a constant and solve the corresponding coupled-mode propagation equations. In Sec. III.2, we expand into a Fourier series and neglect the terms that do not correspond to the phase-matching condition in the coupled-mode propagation equations.

### iii.1 Homogeneous-medium approximation

We consider the case where the lattice constant is not close to any integer multiple of the in-fiber half-wavelength of the probe field, that is, the atomic array is far off the Bragg resonance. In this case, the effect of the interference between the beams reflected from different atoms in the array is not significant and, therefore, we can neglect the discreteness and periodicity of the atomic array. This approximation means that we can use the one-dimensional atom-number distribution

(19) |

which is continuous and constant in the axial coordinate . With the use of this approximation, expression (18) for the coefficients reduces to

(20) |

where the coefficients

(21) |

are independent of . Note that . In the case of the -polarization scheme, we have and . In the case of the -polarization scheme, we have . It is convenient to introduce the notations and .

We can easily solve Eqs. (II.2) with the constant coefficients given by Eq. (20). We assume that and are the left- and right-edge positions of the atomic medium, respectively. In the case where the incident probe field is , the boundary condition is . In this case, the reflection and transmission coefficients are and , respectively. In the case where the incident probe field is , the boundary condition is . In this case, the reflection and transmission coefficients are and , respectively. With the help of the relation , we can show that . The expressions for and are found to be

(22) |

where

(23) |

We note that, in the case of the -polarization scheme, we have , which leads to .

Since , , and , we have . Hence, we find . With this approximation, Eqs. (III.1) reduce to

(24) |

Since , the reflectivity of the array is . The transmission of the probe field is , where

(25) |

is the absorption coefficient for the probe field in the case where the reflection is negligible. The corresponding phase shift coefficient for the probe field is

(26) |

The optical depth per atom is

(27) |

The phase shift per atom is

(28) |

We now describe time dependence of the guided probe field , where . In the case where the period of the atomic array is far from the Bragg resonance, the reflection is, as shown analytically above and illustrated numerically in Figs. 4(b) and 5(b) below, negligible. Then, in the frequency domain, the Fourier-transformed amplitude of the probe field is governed by the propagation equation

(29) |

Here, we have introduced the notation , where is given by Eq. (21) with the substitution . We neglect the dispersion of the fiber-mode group velocity, that is, we take , where is the central frequency of the input guided probe field. We expand up to the second order of as

(30) |

where

(31) |

We substitute Eq. (30) into Eq. (29) and perform the inverse Fourier transformation. Then, we obtain

(32) | |||||

In general, , , and are complex parameters. The imaginary and real parts of the parameter , namely, the coefficients and , are, as already discussed above, the absorption and phase shift coefficients, respectively, for the guided probe light field. We emphasize that the coefficients , , and are the propagation characteristics for the photon flux amplitude .

We analyze the case of exact one- and two-photon resonances, that is, the case where . It is clear from the expression for in Eqs. (III.1) that, when , the absorption coefficient and the phase shift coefficient are small. These features are the signatures of EIT Harris review (); Scully review (); slow light review (); most recent review (); Scully (); Agarwal book (). We note that the width of the corresponding transparency window is given by . When the dephasing rate is negligible, the expression for in Eqs. (III.1) yields , leading to Harris review (); Scully review (); slow light review (); most recent review (); Scully (); Agarwal book ()

(33) |

When the input probe pulse is long enough, the group velocity of the probe field is determined by the equation . When the dephasing rate is negligible, the expression for in Eqs. (III.1) yields , leading to Harris review (); Scully review (); slow light review (); most recent review (); Scully (); Agarwal book ()

(34) |

where the atom-number density in the framework of the homogeneous-medium approximation is given by Eq. (19).

We calculate numerically the optical depth per atom , the phase shift per atom , the group-velocity reduction factor , the transmittivity , and the reflectivity as functions of the detuning of the guided probe field. In the numerical calculations presented in this paper, we use, as already stated, the outermost Zeeman sublevels of the hyperfine levels , , and of the line of atomic cesium, with the free-space wavelength nm. In the calculations for the -polarization scheme, we use the levels , , and . In the calculations for the -polarization scheme, we use the levels , , and . The rate of decay of the atomic optical-transition coherence is calculated by using the results of Ref. cesium decay (). The obtained value is MHz. The corresponding value of the linewidth of the upper level is MHz. This value is slightly larger than the literature value MHz for the atomic natural linewidth Steck02 (); coolingbook (). The obtained increase of the atomic linewidth is caused by the presence of the nanofiber. The lower-level decoherence rate is assumed to be kHz. This value is comparable to the experimental value of about kHz, measured in the Vienna experiment Mitsch14a (). The control field is at exact resonance with the atomic transition . The fiber radius is nm and the distance from the atoms to the fiber surface is nm Vetsch10 ().

We plot in Figs. 2 and 4 the results of the calculations for scheme (a) of Fig. 1, where the guided probe field is polarized. We show in Figs. 3 and 5 the results for scheme (b) of Fig. 1, where the guided probe field is polarized. For the calculations of Figs. 2–5, we use the one-dimensional atom-number density , where nm. The chosen value of is one-half of the in-fiber wavelength of the red-detuned standing-wave guided light field used in the Vienna atom trap experiment Vetsch10 (). Such a value of the array period is far from the Bragg resonance (the Bragg resonant array period is , where and nm for the probe field with the free-space atomic resonance wavelength nm).

Figures 2 and 3 show the familiar features of EIT: in the vicinity of the exact resonance, the optical depth achieves a low minimum, the magnitude of the phase shift is small but the slope of the phase shift is steep, and consequently the reduction of the group velocity is large Harris review (); Scully review (); slow light review (); most recent review (); Scully (); Agarwal book (); vapor (); ultracold (); polariton (). These features occur in a region of frequency which is narrower than the atomic natural linewidth and is called the EIT window.

Figures 4(a) and 5(a) show that the transmittivity has a narrow peak at . We observe from Figs. 4(b) and 5(b) that the reflectivity is very small and is slightly asymmetric with respect to . The asymmetry results from the fact that the relative phase is not an integer multiple of for the parameters used. Due to this fact, the magnitude of the reflectivity depends on the sign of the detuning .

When we compare the left and right columns of Fig. 2, where the probe field polarization is and the propagation direction is for the left column and for the right column, and compare these two columns with Fig. 3, where and , we see that the optical depth per atom , the phase shift per atom , the group-velocity reduction factor are different in the three cases. Similarly, when we compare the solid red and dashed blue curves of Fig. 4(a), where the probe field polarization is and the propagation direction is for the solid red curve and for the dashed blue curve, and compare these two curves with the curve of Fig. 5(a), where and , we observe that the transmittivity is also different in the three cases. The differences between the results for the three cases arise from the fact that the atom-field coupling characterized by the coefficient depends on the local polarization of the guided probe field at the position of the atom [see Eq. (11)], that is, on the mode polarization and on the propagation direction in the case of [see Eqs. (12) and (13)]. Another reason is that different atomic transitions , which have different complex dipole matrix elements , are used in the - and -polarization schemes (see Fig. 1).

Thus, in the case of the -polarization scheme, the optical depth , the phase shift , the group velocity , and the transmittivity substantially depend on the propagation direction . The directionality of transmission of guided light through the array of atoms is a consequence of the existence of a longitudinal component of the guided light field as well as the ellipticity of both the field polarization and the atomic dipole vector AtomArray (); Fam14 (). We note that directional spontaneous emission into an optical nanofiber has been recently demonstrated experimentally for trapped atoms Mitsch14b () and for nanoparticles Petersen14 (). An optical diode based on the chirality of guided photons has also been reported Sayrin15b ().

We observe from Figs. 2(a), 2(b), and 4(a) for the -polarization scheme and Figs. 3(a) and 5(a) for the -polarization scheme that, at zero detuning, the absorption is not completely suppressed and, hence, the induced transparency is not 100%. This is a consequence of the fact that a nonzero lower-level decoherence rate is included in our calculations. Without this decoherence, the induced transparency is 100% at zero detuning. Comparison between Figs. 2(a), 2(b), and 3(a) and between Figs. 4(a) and 5(a) shows that, for a given lower-level decoherence, the residual absorption depends on the probe field polarization and on the propagation direction in the case of . The reason is twofold: (1) the atom-field coupling depends on the local polarization of the field and (2) the different atomic transitions are used in the different polarization schemes.

### iii.2 Phase-matching approximation

We now take into account the periodicity of the atomic array but still consider the array as a continuous one-dimensional medium. For this purpose, we rewrite expression (10) for the one-dimensional atom-number distribution in the Fourier series form for . Then, Eq. (18) for the coupling coefficients becomes