Slow Light in Various Media: a Tutorial

Slow light in various media: a tutorial

Jacob B. Khurgin

Department of Electrical and Computer Engineering, Johns Hopkins University, Balti- more, Maryland 21218, USA ([email protected])

Received October 26, 2009; revised February 4, 2010; accepted February 8, 2010; published March 23, 2010 (Doc. ID 119072)

I consider the physical basics of slow light propagation in atomic media, photonic structures, and optical ﬁbers. I show similarities and differences between all of the above media and develop set of criteria that are then used to compare different media. Special attention is given to dispersion of group velocity and loss, which are shown to limit the bandwidth and delay capacity of all the slow light schemes. © 2010 Optical Society of America OCIS codes: 260.0260, 030.1670, 060.5530, 350.5500, 260.2030.

1. Introduction...... 288 2. Background: Light Propagation in the Vicinity of an Atomic Resonance...... 289 3. Double Atomic Resonance Reduces Dispersion...... 295 4. Tunable Double Resonance and Electromagnetically Induced Transparency...... 299 5. Bandwidth Limitations in Atomic Schemes...... 303 6. Photonic Resonances: Bragg Gratings...... 305 7. Double-Resonant Photonic Structures...... 307 8. Coupled Resonator Structures and Their Comparison with Atomic Delay Lines...... 308 9. Slow Light in Optical Fibers...... 311 10. Conclusions...... 313 References...... 313

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 287 1943-8206/10/020287-32/$15.00 © OSA Slow light in various media: a tutorial

Jacob B. Khurgin

1. Introduction

The subject of slow light has caused keen interest in the past decade, mostly because of the groundbreaking successes of achieving very slow propagation velocity [1] and stopping the light entirely [2]. Following these pioneering results, a large amount of work has been performed in recent years, with slow light propagation attained in diverse media such as metal vapors with and without electromagnetically induced transparency (EIT), rare-earth-doped materials, Raman and Brillouin fiber optical amplifiers, photonic crystal waveguides, and microresonators. As the field has matured a number of interesting linear and nonlinear optical devices have been proposed, including optical buffers, switches, and interferometers. With all this variety of sources, media, and applications, the slow light science is a truly multidisciplinary field whose understanding requires significant background knowledge from those who want to fa- miliarize themselves with the field and, possibly, to contribute to it. Furthermore, since potential applications of slow light field are so often in fields far from optical physics, there is a need for a basic treatise of the field accessible to the engineering community. At this time, in our view, this demand had not been adequately filled. Of course, there exists a body of solid review work on slow light, starting with 2002 reviews by Boyd and Gauthier [3] and Milonni [4], who also published a book in 2005 including chapters on slow, fast, and left-handed light [5]. The first comprehen- sive description of the state-of-the-art slow light research has been given in our recent book [6], which contains contributions from 18 groups that have been ac- tively involved in the slow light field and have all made significant contributions in recent years. But all of the existing work is geared toward the scientists with extensive knowledge of optical physics, while the basics are not covered in sufficient detail. This tutorial is an attempt to provide the most basic treatment of the principles of most slow light schemes and present the order-of-magnitude as- sessment of their performance for various applications. It should be mentioned that the while the term “slow light” has appeared only recently, the history of light propagating with reduced group velocity goes back to the 19th century when the classical theory of dispersion of electromagnetic waves was first formulated in works of Lorentz [7] and others. Slow wave propagation was also observed and widely used in the microwave range as early as the 1940s [8]. The first experimental observation of slow light in the nonlinear regime was made in 1967 by McCall and Hahn when they studied the effect of self-induced transparency in ruby [9]. Soon afterward Grischkowsky and others observed slow light in the linear regime [10]. Yet it was only with the discovery

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 288 of EIT that slow light science was given a strong impetus—and this can be ex- plained by the simple fact that EIT is a wonderful method of obtaining strong optical resonance without extensive loss and dispersion. It will become clear later in this tutorial why resonances with low loss and low dispersion are so critical to slow light and why they are so difficult to find in nature or to create artifi- cially. Indeed, with all the seeming diversity of slow light schemes they can be all characterized by a single common feature—the existence of a sharp single resonance or multiple resonances. The resonance can be defined by a simple atomic transition, by a Bragg grating, by a microresonator or other resonant photonic structure, or by an external laser as in the schemes involving various nonlinear processes—resonant scattering, spectral hole burning, or four-wave mixing. But no matter what is the physical origin of the resonance, it can be characterized by a relatively short list of parameters that can immediately be used to estimate the performance of a given slow light scheme. In this tutorial I emphasize the com- monality of all slow light approaches as well as their distinctive features.

2. Background: Light Propagation in the Vicinity of an Atomic Resonance

Let us consider a simple atomic resonance characterized by the resonant fre- ␻ quency 12 as shown in Fig. 1(a). Using the familiar harmonic oscillator model, the

Figure 1

(a) Two-level atomic resonance. (b) Spectrum of absorption coefﬁcient. (c) Spectrum of refractive index. (d) Dispersion diagram and group velocity in the vicinity of a resonance. (e) Spectrum of slow down factor. (f) Spectrum of group-velocity dispersion (GVD).

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 289 dielectric constant in the vicinity of the resonance has a familiar expression:

⍀2 p ⑀͑␻͒ = ⑀¯ + , ͑1͒ ␻2 ␻2 ␻␥ 12 − − j 12 where ⑀¯ =n¯ 2 is the nonresonant part, or “background” part, of the dielectric con- ␥ stant, and the damping constant 12 is the decay rate of the polarization. The ex- ⍀ pression in the numerator p is the “plasma frequency,” which can be found as

2 Nae ⍀2 = f , ͑2͒ p ⑀ 12 0m0 ⑀ where e is an electron charge, 0 is the dielectric permittivity of vacuum, m0 is a free electron mass, Na is the concentration of active atoms, and f12 is the oscillator strength of a transition,

␻ 2 2m0 12d12 f = . ͑3͒ 12 ប

Here ed12 is the dipole matrix element of the transition and is the only quantum me- chanical variable that will be encountered in this tutorial. What is important for better understanding of the rest of the tutorial is that the minimum value of oscillator strength is zero, while the maximum value of oscillator strength is unity. Strictly speaking, the transitions with zero oscillator strength are called “forbidden,” while all other transitions are called “allowed,” but it is also commonplace to refer to all the transitions with a very small oscillator strength as “forbidden” ones. Also important ␥ to us is the general trend of decay rates 12 to increase with both the oscillator strength of the transition and the concentration of active atoms. According to Eq. (1) this fact limits the absolute value of the dielectric constant in the optical range to less than 10—hence, the phase velocity of light can never be reduced by more than a factor of a few, and it is the group velocity of light that is being modiﬁed in all the slow light schemes. Furthermore, in all practical slow light schemes the concentration of active atoms (or ions) responsible for the resonant reduction of the speed of light is fairly ⍀2 ␻␥ small, such that p / 12 1, and when we introduce the complex refractive index as n˜͑␻͒=⑀1/2͑␻͒, we can immediately use a series expansion of it around the back- ͉␻ ground value, and for the frequencies relatively close to the resonance 12 −␻͉␻ we obtain

⍀2 1/2 ⍀2 p 1 p n˜͑␻͒ = ͩn¯ 2 + ͪ Ϸ n¯ + ␻2 ␻2 ␻␥ ␻2 ␻2 ␻␥ 12 − − j 12 2n¯ 12 − − j 12 ⍀2͑␻ ␻͒ ⍀2␥ 1 p 12 − j p 12 Ϸ n¯ + + . ͑4͒ ␻ ͑␻ ␻͒2 ␥2 ␻ ͑␻ ␻͒2 ␥2 4n¯ 12 − + 12/4 8n¯ 12 − + 12/4 From real and imaginary parts of relation (4) one immediately obtains the ex- pressions for the absorption coefﬁcient shown in Fig. 1(b),

␻ ⍀2␥ 2 1 p 12 ␣͑␻͒ = Im͑n˜͒Ϸ , ͑5͒ ͑␻ ␻͒2 ␥2 c 4n¯c 12 − + 12/4 and the refractive index [Fig. 1(c)]

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 290 ⍀2͑␻ ␻͒ ␻ ␻ 1 p 12 − c 12 − n͑␻͒ =Re͑n˜͒Ϸn¯ + = n¯ + ␣͑␻͒ . ͑6͒ ␻ ͑␻ ␻͒2 ␥2 ␻ ␥ 4n¯ 12 − + 12/4 12 The absorption spectrum presents a Lorentzian peak with a full width at half- ␥ maximum (FWHM) of exactly 12. In the vicinity of the resonance the refractive index ﬁrst increases, then rapidly decreases to the point where n͑␻͒Ͻn¯, and then increases once again to the value of the background index.The last relation in Eq. (6) is a particular form of the more general Kramer–Kronig relation between the real and the imaginary parts of dielectric constant,

1 ϱ Im͑⑀͑␻Ј͒͒ Re͑⑀͑␻͒͒ =1+ P͵ , ␲ −ϱ ␻Ј − ␻

1 ϱ Re͑⑀͑␻Ј͒͒ −1 Im͑⑀͑␻͒͒ =− P͵ , ͑7͒ ␲ −ϱ ␻Ј − ␻ where P indicates the principal value of the integral. These relations are extremely important because they show that no modiﬁcation of refractive index is possible without also affecting the absorption. Let us now consider a plane wave of frequency ␻ propagating in the medium de- ﬁned by relation (6),

͑ ͒ ͑ ␻ ͒ ͑ ͒ E z,t = E0 exp jkz − j t , 8 with the wave vector

k͑␻͒ = n͑␻͒ · ␻/c. ͑9͒ The dispersion relation between the wave vector and frequency is shown in Fig. 1(d). The velocity with which the phase of the plane wave propagates,

͑␻͒ ␻ ͑␻͒ ͑ ͒ vp = /k = c/n , 10 deviates somewhat from its background value c/n¯ near the resonance, but, as mentioned above, these changes do not lead to the slow light phenomenon. In fact, the phase velocity is meaningful only for a pure monochromatic light, i.e., for the harmonic wave with no features, which carries no information. The information propagates in the form of a wave packet comprising more than a single frequency. In this work we shall consider a Gaussian wave packet with the FWHM equal to ⌬t:

͑ 2 ⌬ 2͒ ␻ ͑ ͒ −2 ln 2 t / t j 0t ͑ ͒ E t = E0e e . 11 The power spectrum of the Gaussian wave packet is also Gaussian:

͓͑␻ ␻ ͒2 ⌬␻2 ͔ ␻ ͉ ͑␻͉͒2 2 −4 ln 2 − 0 / 1/2 j 0t ͑ ͒ E = E0e e , 12 ⌬␻ where the spectral FWHM 1/2 is ⌬␻ ⌬ ͑ ͒ 1/2 =4ln2/ t. 13 The envelope of the wave packet propagates with the group velocity deﬁned as the slope of the dispersion curve of Fig. 1(d):

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 291 ͒ ͑ ץ ␻ץ vg = / k. 14 It is more convenient to deal with the variable that is the inverse of the group velocity, n͑␻͒ץ n͑␻͒ץ k͑␻͒ 1ץ v−1͑␻͒ = = ͩn͑␻͒␻ + ␻ ͪ = v−1͑␻͒ + ␻ . ͑15͒ ␻ץ ␻ pץ ␻ cץ g Clearly the group velocity can be signiﬁcantly smaller or larger than the phase velocity, depending on the sign of the derivative of the index in Eq. (15).To gauge this chance, let us introduce the slow down factor—a relative reduction of group velocity due to resonance: n͑␻͒ץ c n͑␻͒ ␻ S͑␻͒ = = + . ͑16͒ ␻ץ ␻͒͑ n¯vg n¯ n¯ Substituting Eq. (6) into Eq. (16), we obtain

⍀2 ͑␻ ␻͒2 ␥2 ␣͑␻͒ ͑␻ ␻͒2 ␥2 p 12 − − 12/4 c 12 − − 12/4 S͑␻͒Ϸ1+ =1+ , 2 ͓͑␻ ␻͒2 ␥2 ͔2 ␥ ͑␻ ␻͒2 ␥2 4n¯ 12 − + 12/4 n¯ 12 12 − + 12/4 ͑17͒ shown in Fig. 1(e). One can now see that there are two regions in which SϾ0, i.e., the slow light regime and one region where SϽ1, which can be either the fast light regime ͑0ϽSϽ1͒ or the negative group velocity regime ͑SϽ0͒, indicating that the light becomes reﬂected. The slow down phenomenon is closely connected to the absorption, and, when detuning signiﬁcantly exceeds the ͉␻ ␻͉␥ FWHM of absorption, 12 − 12, this relation becomes particularly simple:

c␣͑␻͒ S͑␻͒Ϸ1+ . ͑18͒ ␥ n¯ 12 ␥ Equation (18) can be interpreted in an interesting way—parameter Lc =c/n¯ 12 is a coherence length in the medium, the length over which the material polarization ␣ (dipole moments of individual atoms) preserve the phase, while La =1/ is the absorption length. Therefore, Eq. (18) can be rewritten as ͑␻͒Ϸ ͑␻͒ ͑ ͒ S 1+Lc/La , 19 indicating that slow light is a coherent effect associated with coherent transfer of energy between the light and medium. One can get a better intuitive picture of the physical phenomena governing slow light propagation by plotting the dispersion curve in the absence of loss (i.e., ␥ 12 =0) as the dashed curve in Fig. 1(d). This curve corresponds to a well-known coupled modes model, also known as the polariton dispersion curve in solid-state physics. The ﬁrst mode is a photon, which in the absence of an atomic transition is ␻ described by the linear dispersion curve p =ck/n¯. The second mode is the atomic ␻ polarization characterized by a resonance frequency 12. The dispersion curve of ␻ ␻ atomic polarization is a horizontal line a = 12, indicating the obvious fact that it has zero group velocity, as atoms do not move, at least not on the scale of the speed of light. In the vicinity of the resonance two modes couple into each other, and the modiﬁed dispersion curve is split into two branches separated by the gap in which the light cannot propagate. Notice that for each wave vector there are two coupled

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 292 solutions characterized by two different group velocities. The one further away from the resonance has the higher group velocity and is usually referred to as “photon- like,” while the one closer to the resonance has lower group velocity and is usually referred to as “atomlike.” One can then interpret the slow light propagation phenomenon in the following way: the energy gets constantly coupled from the electromagnetic field to the atomic polarization and back.The longer the time the energy spends in the form of atomic excitation, the slower the coupled mode propagates. Thus the slow light propagation in an atomic system can be understood as constant excitation and de-excitation of atoms in which coherence is preserved. The intuitive picture of photon–matter coupling is shown in Fig. 2, and it starts with a photon arriving at an unexcited atom [Fig. 2(a)]. Then the energy is transferred from the photon to the oscillations of the dipole associated with the resonant atomic transition [Fig. 2(b)]. Then the energy is transferred back to the electromagnetic field as a new photon appears, and this new photon is in phase with the original photon [Fig. 2(c)]. The new photon then travels to the next atom and the process repeats once again. One important manifestation of this energy transfer is related to the strength of the electric field in the electromagnetic wave propagating in a slow light regime caused by an atomic resonance. Since the group velocity is also the velocity with which the energy propagates, the local energy density of light beam with power density P is

͑ ͒ ͑ ͒ U = P/vg = Pn¯/c S, 20 and it becomes enhanced by a slow down factor in the slow light medium. But the energy density is related to electric ﬁeld as

Figure 2

Intuitive interpretation of the slow light propagation near atomic resonance: (a) photon approaches unexcited atom, (b) atomic polarization is excited, (c) energy is transferred back to the photon.

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 293 ␻⑀͒ 1͑ץ 1 U = ⑀ E2 + µ H2, ͑21͒ ␻ 4 0ץ 0 4 where the first term corresponds to electric energy and the second term to the ␩ magnetic energy. The magnetic field is related to the electric field as H=n¯ 0E, ␩ ͱ ⑀ ⍀ where 0 = µ0 / 0 =377 is the vacuum impedance. Therefore we obtain from, Eq. (21), n͑␻͒ץ n2͑␻͒ 1 1 1ץ 1 1 U = ⑀ ⑀¯E2 + ␻⑀ E2 + ⑀ ⑀¯E2 = ⑀ ⑀¯E2 + ⑀ n¯␻ E2 ␻ץ ␻ 4 0 2 0 2 0ץ 0 4 0 4 n͑␻͒ 1ץ ␻ 1 ⑀ ⑀¯ 2ͩ ͪ ⑀ ⑀¯ 2 ͑ ͒ = 0 E 1+ = 0 E S 22 ␻ 2ץ ¯n 2 It follows from equating the energy density in Eqs. (20) and (22) that E2 ␩ =2 0P/n¯. Thus the electric field does not become enhanced in the atomic slow light medium. In our intuitive picture this simply means that all the additional energy compressed into the medium is stored in the atomic polarization. This fact has important implications in the nonlinear optics. Below, when we turn our attention to the photonic slow light schemes, we shall see that the picture is dra- matically different there, as the electric field density does become enhanced sub- stantially. Also, one can see that without absorption the region of fast light ͑0 ϽSϽ1͒ is absent in Fig. 1(d), and it is important that fast light is associated with the absorption peak, while the slow light is associated only with the off-resonant absorption; hence significant delays of the signal in the slow light are quite possible, while significant advances are difficult to observe because of absorption that not only reduces the energy of the signal but also causes its reshaping, as the slower spectral components of the signal tend to be attenuated less than faster components. When it comes to practical applications of slow light, it is important to achieve large delays over significant bandwidth and in a compact device. The obstacles on the way to this goal include the aforementioned loss and the dispersion of group velocity and dispersion of absorption [11–13]. To ascertain the importance of the group-velocity dispersion (GVD), one only needs to use the Taylor ͑␻͒ ␻ series expansion of the dispersion relation k near the signal frequency 0 to obtain the expression for the group velocity:

1 v−1͑␻͒ = v−1͑␻ ͒ + ␤ ͑␻ ͒ · ͑␻ − ␻ ͒ + ␤ ͑␻ ͒ · ͑␻ − ␻ ͒2 + ... , ͑23͒ g g 0 2 0 0 2 3 0 0 ␻nץ n ͑␻͒ץ ␻͒͑ ␤ where n = k / . The delay is also a function of frequency:

1 T ͑␻͒ = v−1͑␻͒L = v−1͑␻ ͒L + ␤ ͑␻ − ␻ ͒L + ␤ ͑␻ − ␻ ͒2L + .... ͑24͒ d g g 0 2 0 2 3 0 One can estimate the limitations imposed by GVD by considering a typical return-to-zero signal consisting of the series of Gaussian pulses (11) shown in ⌬ 1 −1 Fig. 3(a), whose FWHM is equal to one half of the bit interval, t= 2 B , and the ⌬␻ bandwidth [Fig. 3(b)], according to Eq. (13),is 1/2 =8 ln 2B Different frequencies in the spectrum will then arrive at different times [Fig. 3(c)], and the pulse will broaden in time as shown in Fig. 3(d), eventually spilling into the adjacent bit intervals and thus causing intersymbol interference.The effective broadening of the pulse

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 294 Figure 3

Deleterious effect of GVD: (a) input signal, (b) spectrum of the input signal, (c) group velocity and group delay dispersion, (d) output signal with intersymbol interference. can be estimated as the difference between the delay times of spectral components at ␻ ⌬␻ the edges of the signal bandwidth, i.e., at 0 ± 1/2 /2, ⌬ ͑ ͒Ϸ␤ ⌬␻ ͑ ͒ Td L 2 1/2L. 25 Now to ensure that the signal does not spill over into the adjacent bit interval, we shall insist that the broadening should be less than one half of the bit interval, i.e., ␤ ⌬␻ ␤ Ͻ ͑ ͒ 2 1/2L = 28ln2BL 1/2B, 26 which leads to the condition relating the maximum allowable length and bit rate:

1 ͉␤ ͉B2L Ͻ . ͑27͒ 2 16 ln 2 One can use different criteria for the maximum allowable broadening, but all of them will give a result that is of the same order as Eq. (27). This condition shows ␤ the severe limitations imposed by the second-order GVD term 2, whose spectrum is plotted in Fig. 1(f). In addition to GVD, the signal bandwidth is also limited by the dispersion of absorption; the fact that different frequency components of the signal experience different attenuation causes signal distortion, but it is usually GVD that is the main factor that limits slow light scheme performance. Hence, while slow light had been observed in single-atomic resonance schemes as early as the 1960s and 1970s [12–16], it was only in 1990s when ␤ double-resonance schemes with complete cancellation of 2 had been discov- ered [17] that the slow light research really took off.

3. Double Atomic Resonance Reduces Dispersion

Although the ﬁrst practical slow light results were achieved by using the phenomenon of EIT [17,18], where the double resonance is created by strong pump beam, the main features of the double-resonant atomic schemes can be under-

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 295 stood more easily if one considers that closely spaced narrow resonances do oc- cur naturally in metal vapors, such, as, for instance, in Rb85 [19], where two D2 ␯ resonances near 780 nm separated by 32 =3 GHz have been used in the (to date) most successful slow light experiments in atomic medium. The rational for using the double resonance can be seen from Figs. 1(d) and 1(f), which show that the lowest- ␤ order GVD term 2 has opposite signs below and above resonance. Then, if one can ␤ combine two resonances as in Fig. 4(a), only the third-order GVD 3 will be a factor ␻ for signals centered at frequency 0 in the middle between two transitions where the absorption [Fig. 4(b)] is low, as is evident from the dispersion curves, Fig. 4(c) and Fig. 4(d). Indeed the refractive index of the double-resonant scheme can be obtained by simply combining two resonant terms [Eq. (6)] and neglecting the dephasing terms in the denominator:

⍀2 ⍀2 ⍀2 ␻ ␻ 1 p 1 p p 0 − n͑␻͒Ϸn¯ + + = n¯ + , ͑28͒ ␻ ␻ ␻ ␻ ␻ ␻ ␻ ␻2 8n¯ 31 − 8n¯ 21 − 4n¯ 32 ͑␻ − ␻͒2 − 0 4

where we have taken into account the fact that the oscillator strength is evenly split between two transitions. The curve [Fig. 4(d)] described by Eq. (28) is per- ␻ fectly antisymmetric around 0; thus only odd terms are present in its expansion. Note that the slow down factor in the double-resonant scheme is identical to that

Figure 4

(a) Double atomic resonance and (b) dispersion and group velocity near it. (c) Absorption spectrum. (d) Refractive index spectrum.

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 296 ␻ of the single-resonant scheme detuned by 32 /2: ␻2 32 ͑␻ − ␻͒2 + ⍀2 0 p 4 S͑␻͒Ϸ1+ . ͑29͒ 2 ␻2 2 4n¯ 32 ͫ͑␻ − ␻͒2 − ͬ 0 4

A closer look at the dispersion curve, Fig. 4(c), reveals the basic trade-off inherent in every slow light scheme—the dispersion curve is split into three branches, of which the central one, squeezed between two atomic resonances, is the one with a slow group velocity. Clearly, the group velocity, being a slop of the curve, ␯ is inversely proportional to the splitting between two resonances 32, i.e., to the maximum theoretical bandwidth of the scheme. In reality the practical bandwidth (or bit rate for digital signals) is even smaller than that and is mostly limited by the third-order dispersion [11,20].According to Eq. (24), we obtain the limitation for the ␻ relative delay between the spectral component at the central frequency 0 and the ␻ ⌬␻ spectral components at the edges of the signal bandwidth, i.e., at 0 ± 1/2 /2

⌬␻ 2 1 1/2 1 ⌬T ͑L͒Ϸ ␤ ͩ ͪ L =8͑ln 2͒2␤ B2L Ͻ B−1, ͑30͒ d 2 3 2 3 2 which leads us to the criterion

͉␤ ͉ 3 Ͻ ͑ ͒2 ͑ ͒ 3 B L 1/16 ln 2 . 31 This criterion contains a bit rate in the third power, while criterion (27) contained the square of the bit rate because it depended on second-order GVD. Another factor that limits the group delay is that the dispersion of the absorption ␻ [Fig. 4(b)] in the vicinity of 0 has the shape ⍀2␥ ⍀2␥ 1 p 21 1 p 21 ␣͑␻͒Ϸ + ͑␻ ␻ ␻ ͒2 ͑␻ ␻ ␻ ͒2 8n¯c 0 − + 32/2 8n¯c 0 − − 32/2 ␻2 32 ͑␻ − ␻͒2 + ⍀2␥ 0 ␥ p 21 4 n¯ 21 = = ͑S͑␻͒ −1͒ . ͑32͒ ␻2 2 4n¯c 32 c ͫ͑␻ − ␻͒2 − ͬ 0 4 ␻ The absorption reaches its minimum value at 0, ⍀2␥ p 12 ␣ = ␣͑␻ ͒ = ͑33͒ 0 0 ␻2 n¯c 32 and then increases as

͑␻ ␻͒2 0 − ␣͑␻͒Ϸ␣ +12␣ . ͑34͒ 0 0 ␻2 32 The transmission of the length L of the double-resonant atomic medium thus can be described as

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 297 ␣ ␣ ͓͑␻ ␻͒2 ␻2 ͓͑␻ ␻ ͒2 ⌬␻2͔ ͑␻͒ − 0L −12 0L 0 − / 32͔ −4 ln 2 − 0 / t , ͑ ͒ T = e e = T0e 35 where FWHM of the transmission window is

ln 2 1/2 ⌬␻ = ␻ ͩ ͪ . ͑36͒ t 32 ␣ 3 0L Therefore the Gaussian signal (11), shown in Fig 5(a) with its spectrum shown in Fig. 5(b), passes through the slow light medium with the transmission described by Eq. (35) and shown in Fig. 5(c). Its bandwidth narrows, as shown in Fig 5(d), according to

1 1 1 = + ͑37͒ ⌬␻2 ⌬␻2 ⌬␻2 1/2,L 1/2 t When the transmission window equals the original bandwidth of the signal, ⌬␻ 1/2 1/2 =8 ln 2B, the spectrum narrows by a factor of 2 , and the pulse becomes wider by the same amount, as shown in Fig. 5(e). This appears to be a reasonable criterion for tolerable pulse distortion, and it leads to the condition

B2 ln 2 ␣ L Ͻ . ͑38͒ 0 ␻2 32 192 It is important to note that to a certain extent small absorption can actually miti- gate the deleterious effect of the dispersion, as reduced FWHM leads to smaller third-order dispersion [Eq. (30)]. This effect was indeed observed in [19], where

Figure 5

Deleterious effect of residual absorption dispersion: (a) Input signal, (b) spectrum of the input signal, (c) spectrum of residual absorption, (d) narrowed output spectrum, (e) broadened output signal with inter symbol interference.

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 298 Figure 6

Experimental apparatus (left) used in [19] to obtain the variable pulse delays (right) at various optical depths by using double resonance in Rb vapor. group delays in excess of 100 ns with 2 ns pulses at 780 nm were obtained, which correspond to about 50 bit tunable optical buffers. The delay could be tuned by changing the concentration of Rb vapor as shown in Fig. 6.

4. Tunable Double Resonance and Electromagnetically Induced Transparency

The double atomic resonant scheme described cannot be adapted to variable bandwidth because the width of the passband cannot be changed. To change the passband width, one can consider the alternative of spectral hole burning in the inhomogeneously broadened transition [21,22]. As shown in Fig. 7(a), a strong pump pulse creates a situation in which the absorption in the frequency range ⌬␻ becomes depleted. The profile of the absorption spectrum shown in Fig. 7(b) looks remarkably like the double-resonant profile of Fig. 4. With the refractive index profile shown in Fig. 7(c) one can see that a strong reduction of group velocity can be expected near the center of the spectral hole.

Figure 7

(a) Slow light scheme based on spectral hole burning. (b) Absorption spectrum. (c) Refractive index spectrum.

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 299 By changing the spectrum of the pump, for instance, using intensity or frequency modulation, one can change ⌬␻ [23] to achieve the maximum delay without distortion for a given bit rate. In [23] delays of 2 bit intervals were achieved for a moderate bandwidth of 100 MHz but only in a 40 cm long Rb vapor delay line. Since the background absorption in the hole burning is always high, it is the dispersion of loss, represented by the second term in Eq. (34), that causes the signal distortion and is in fact an ultimate limitation in this scheme. The scheme also suffers from the large energy dissipation as the pump is absorbed. To avoid large background absorption and to achieve wide passband tunability one uses an entirely different slow light scheme based on EIT, first considered by Harris [24–26]. Without trying to explain all the intricacies of EIT, one can un- derstand the rationale of using it. Since finding two closely spaced atomic resonances is not trivial, once should consider the means for their artificial creation. Now, the atomic oscillator in the absence of external modulation has just one ␻ resonant frequency 12 in its response spectrum, just like any harmonic wave ␻ whose spectrum contains just one frequency component 0 [Fig. 8(a)]. But if the wave is amplitude modulated with some frequency ⍀, there will appear two side- ␻ ⍀ bands at frequencies 12 ± in its spectrum [Fig. 8(b)]; when the modulation depth ␻ reaches 100%, the carrier frequency 0 becomes entirely suppressed, and the spectrum shows just two sidebands separated by 2⍀ [Fig. 8(c)]. Now, according to this analogy, if one strongly modulates the strength of the atomic oscillator with some external frequency ⍀, one should expect the absorption spectrum to behave in fashion similar to the spectrum of amplitude modulated wave; i.e., it should show two absorption lines separated by 2⍀. The mate- ␻ rial should become transparent as the resonant frequency 0 increases—hence the term “EIT.” To accomplish the EIT transmission modulation there exist numerous schemes, but we shall consider only one—the most widely used three-level “⌳” scheme ␻ [1] shown in Fig. 9, in which the ground-to-excited state transition 12 is reso- ␻ nant with the frequency of the optical signal carrier 0 and has a dephasing rate of ␥ 21. In the absence of the pump the absorption spectrum (dashed curve in Fig. 9)isa normal Lorentzian line. There also exists a strong transition coupling the excited

Figure 8

Intuitive interpretation of EIT: (a) harmonic wave and its spectrum, (b) amplitude modulated harmonic wave and its spectrum with two subbands, (c) 100% amplitude modulated harmonic wave and its spectrum with carrier suppressed.

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 300 Figure 9

Principle of electromagnetic transparency in atomic ⌳ scheme.

level 2 with the third level 3, but, and this is critical, the transition between levels 1 and 3 is forbidden. When a strong resonant pump with intensity Ipump at frequency ␻ 23 is turned on, the mixing of states 2 and 3 causes modulation of the absorption of signal. As expected, the Lorentzian peak in the absorption spectrum splits into two ␻ ⍀ smaller peaks at frequencies 0 ± , where the Rabi frequency is

␲␣ 1/2 4 f Ipump ⍀ = ͩf ͪ , ͑39͒ 23 ␻ mn¯ 23

␣ 2 ␲⑀ ប where f =e /4 0 c=1/137 is a ﬁne structure constant. Thus changing the pump intensity allows one to achieve full tunability of the group velocity and to achieve very small group velocities [3,4] with a slow down factor [Eq. (29)]

⍀2 ͑␻ ␻͒2 ⍀2 p 0 − + S͑␻͒Ϸ1+ . ͑40͒ 2 ͓͑␻ ␻͒2 ⍀2͔2 4n¯ 0 − −

If one introduces the density of pump phonons inside the medium as

n¯Ipump N = , ͑41͒ pump ប␻ c 23

one can obtain a different expression for the Rabi frequency,

2 1/2 Npumpe ⍀ = ͩf ͪ , ͑42͒ 23 ⑀⑀ m¯ 0

which resembles the expression for the plasma frequency (2). The photon density in the Rabi frequency plays exactly the same role that the density of active atoms plays in plasma frequency. One then obtains a very simple expression for the slow down factor in the middle of the passband from Eq. (40),

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 301 ⍀2 p Na S͑␻ ͒Ϸ1+ =1+ . ͑43͒ 0 2 2 4n¯ ⍀ 4Npump But aside from being a convenient technique of implementing the tunable double-resonant scheme, EIT has a signiﬁcant advantage over other schemes because the residual absorption rate at the resonant frequency,

1 ⍀2 n¯␥ ␣͑␻ ͒ p ␥ ͓ ͑␻ ͒ ͔ 31 ͑ ͒ 0 = 31 = S 0 −1 , 44 n¯c 4⍀2 c is proportional to the dephasing rate of the intra-atomic excitation [30], which is ␥ ␥ not coupled to outside world. Thus typically 31 21, and the residual absorption is much weaker in the EIT than in the case of two independent resonances. This indicates that EIT is a coherent effect and that the reduction of absorption occurs because of the destructive interference of the absorption by two sidebands. Another way to underscore the coherent nature of slow light in EIT is to calculate the incre- mental time that the light spends inside the slow light medium before its intensity decreases by 1/e, i.e. over one absorption length:

͓ ͑␻ ͒ ͔ La La n¯ S 0 −1 ⌬T = − = = ␥−1, ͑45͒ d ␣͑␻ ͒ 31 vg c/n¯ 0 c which is precisely the coherence time of the intra-atomic excitation between levels 1 and 3! Thus the slow light effect has a rather simple physical interpretation. While in the simple single or double-resonant slow light scheme the energy is transferred from the electromagnetic wave to the atomic excitation and back, in the EIT scheme the process involves more steps, as shown in Fig. 10. First [Fig. 10(a)], as the signal photon propagates in the EIT medium it transfers its energy to the excitation of the atomic transition between levels 1 and 2 [Fig. 10(b)]. Because of the presence of strong pump wave coupling between levels 2 and 3, the exci-

Figure 10

Intuitive interpretation of the slow light propagation in an EIT medium: (a) photon approaches unexcited atom, (b) atomic polarization is excited at 1-to-2 transition, (c) energy is transferred to the long-lived polarization at 3-to-1 transition, (d) energy is transferred back to 1-to-2 transition and (e) back to the photon.

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 302 tation is almost instantly transferred to the long-lived excitation between levels 1 and 3 [Fig. 10(c)]. Then the process occurs in reverse, and the energy is transferred back to the transition between levels 1 and 2 [Fig. 10(d)] and ﬁnally back into the photon [Fig. 10(e)]. Then the process repeats itself. Overall, most of the time the energy is stored in the form of 1–3 excitations, and thus it propagates with a very slow group velocity. Furthermore, the actual absorption event occurs only when the excitation 1–3 loses coherence and the energy cannot get back to ␥ ␥ the photon. Naturally, it is the dephasing rate of this excitation, i.e., 31 21, that determines the residual absorption loss in Eq. (44). I stress here once again that since the energy is stored in the form of atomic excitation, one cannot expect enhancement in the strength of the optical ﬁeld.

5. Bandwidth Limitations in Atomic Schemes

The most spectacular results were achieved in slow light experiments of EIT in which the light velocity was slowed down to pedestrian speed [3] and then even stopped [4], first in metal vapors and then in a solid-state medium containing rare-earth ions [27]. These achievements are of great importance for physics when it comes to manipulating single photons [28] and coherent control [29] Also important are imaging applications of slow light, where indeed impressive results were observed by a number of groups [30,31], extrahigh-resolution interferometers [32,33], and rotation sensors [34]. There has also been significant progress on using an electrically pumped semiconductor medium to achieve slow light in an EIT configuration or using coherent population oscillations [35–37]. But overall the best results in terms of the delay–bit-rate product were achieved in metal vapors [19,24] at relatively narrow bandwidths. The delays were usually limited by the third-order dispersion. These results follow from the basic properties of the Lorentzian dispersion: according to Eq. (16) the slow- ͑␻ ␻͒−2 down factor S is proportional to 12 − —hence the total delay can be very large in the narrow frequency band near the resonance, but then it changes and become much smaller. A number of publications have been devoted to the limits of the delay–bit-rate product [9–11,38,39]. In fact it was shown in [11] and then in [39] that the most relevant figure of merit for a slow light delay line is the minimum length of the delay line L required in order to store a number of bits Nst at a given bit rate B. The required length was found in [11]tobe

͑ ͒ Nst ϳ 2 ͓⍀ ␲͔−2 ͑ ͒ LB cBNst P/2 , 46 indicating that the performance suffers at higher bit rates and also that the required length increases nonlinearly with storage capacity. The key factor in Eq. (46) is that the required length grows nonlinearly with the number of stored bits. This nonlinear dependence can be easily understood on the intuitive level as illustrated in Fig. 11. We consider a delay line of length ͑ ͒ Nst ⍀ LB and the transmission window of width 2 that successfully delays the signal of ͑Nst͒ −1 bit rate B by the delay time Td =NstB with the acceptable intersymbol interference caused by the dispersion as shown in Fig. 11(a). Now, suppose we want to increase the storage capacity by, say, a factor of two and increase length of the buffer to ͑ ͒ ͑ ͒ 2Nst Nst ϫ LB =LB 2 as shown in Fig. 11(b). Sure enough, the delay time will increase to ͑2Nst͒ −1 the required value of Td =2NstB , but the differential group delay, Eq. (30),

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 303 Figure 11

Why the length of EIT delay line growth quadratically with delay capacity. (a) A delay line with delay of Nst bits works well without excessive distortion. (b) The length and delay time are increased two-fold—strong distortion ensues. (c) The passband is increased to reduce distortion—the delay time decreases. (d) Finally, the delay time of 2Nst bits with acceptable distortion is achieved in the delay line that is four times as long as original one.

͑ ͒ ͑ ͒ ͑ ͒ ⌬ ͑ Nst ͒ ͑ ͒2␤ 2 2Nst ⌬ ͑ Nst ͒ ͑ ͒ Td 2LB =8 ln 2 3B LB =2 Td LB , 47 will also increase by a factor of two and cause unacceptable intersymbol interference. The third-order dispersion,

3 ⍀2 ␤ ͑␻ ͒ p ͑ ͒ 3 0 = , 48 2n¯c ⍀4 can be reduced twofold by increasing the Rabi frequency by a factor of 21/4, which can be conveniently achieved by reducing the pump intensity Ipump, Eq. (39), by a factor of 21/2. That, as shown in Fig. 11(c), will bring the intersymbol interference down to an acceptable value, but, unfortunately the slow down factor

⍀2 ͑␻ ͒Ϸ p ͑ ͒ S 0 1+ 49 4n¯ 2⍀2 will now decrease by about a factor of 21/2 with a commensurate decrease in the delay time. Therefore, the length will have to be further increased by a factor of 21/2, which in turn will cause additional distortion, and the process will continue iteratively until the required length converges at

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 304 ͑ ͒ ͑ ͒ ͑ ͒ 2Nst Nst ϫ ϫ 1/2 ϫ 1/4 Nst ϫ 2 ͑ ͒ LB = LB 2 2 2 ....=LB 2 50 as shown in Fig. 11(d), which proves the quadratic dependence of required length on the number of stored bits. Notice that as the length has increased by a factor of four, the slow down factor has decreased by a factor of two. If we continue increasing Nst, we will have to further decrease the slow down factor [Eq. (49)] until it asymptotically approaches the value of unity! The higher is the bit rate ⍀2 and the smaller is p, the earlier will the saturation take place. Therefore, we can introduce a critical parameter of cutoff bit rate,

͑atom͒͑ ͒ϳ͓⍀ ␲ ͔ −1/2 ͑ ͒ Bcut Nst P/4 n¯ Nst , 51 which is the maximum bandwidth at which one can use the atomic slow light scheme to achieve meaningful slow down of the Nst bit sequence. In other words, ͑atom͒͑ ͒ when the bit rate approaches Bcut Nst , the slow light delay line has to be just as −1 long as the length of a conventional delay line, i.e., c/n¯NstB Now, as mentioned before, plasma frequency in most of the slow light atomic media is limited because as the atomic concentrations increase, the transitions ⍀ ␲ broadened, reducing the oscillator strength. As a result P /2 is typically in the range of few hundreds of gigahertz. For example, in Rb87 it is 100 GHz, and in Pb205 it is about 440 GHz. For solid-state slow light media using rare-earth ions, ⍀ ␲ such as Pr:Y2SiO5, P /2 is only 40 GHz. As a result, the maximum bit rate at which meaningful slow down of light with a total delay of, say, 10 bits can be achieved is only 10 Gbit/s in Pb205 and less than that in other atomic media. Indeed most of the demonstrated and proposed slow light schemes based on atomic transitions do not show spectacular results at bit rates above 1 Gbit/s, and for this reason a different class of resonances will be considered.

6. Photonic Resonances: Bragg Gratings

As was already mentioned, in the case of atomic resonance the apparent slow down of light is caused by the resonant energy transfer to and from the excitation of atomic polarization. An entirely different resonance is the photonic resonance in which the energy is resonantly transferred between two or more modes of electromagnetic radiation. When the transfer takes place between forward and backward propagating wave slow light, the group velocity is reduced, and we are once again faced with the slow light phenomenon, albeit of an entirely different nature from the slow light in atomic media. The most common photonic resonance is the Bragg grating [Fig. 12(a)]—a structure in which the refractive index is periodically modulated with period ⌳,

n = n¯ + ␦n cos͑2␲/⌳z͒. ͑52͒

As a result a photonic bandgap opens in the vicinity of Bragg frequency [40],

␲c ␻ ͑ ͒ B = , 53 ⌳n¯ and the dispersion law becomes modiﬁed as

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 305 Figure 12

(a) Bragg grating and its index proﬁle. (b) Dispersion of Bragg grating. (c) Cas- caded Bragg grating. (d) Dispersion curve of cascaded grating.

␻ ␻ 2 ␦ 2 k − kB − B n = ͱͩ ͪ − ͩ ͪ . ͑54͒ ␻ kB B 2n¯ This dispersion law is plotted in Fig. 12(b). One can see the similarities between it and the dispersion of a single atomic resonance [Fig. 1(d)], especially the presence of the gap near the resonance frequency indicating that the light is reﬂected from the grating just as it is reﬂected from the atomic medium at resonance. Close to the gap the group velocity indeed becomes reduced, with the slow down factor being

␻ ␻ ͯ − B ͯ ␻ B S = . ͑55͒ ␻ ␻ 2 ␦ 2 − B n ͱͩ ͪ − ͩ ͪ ␻ B 2n¯ ⌬␻ ␻ ␦ Furthermore, the width of the forbidden gap is gap = B n/n¯, and hence the index contrast ␦n/n¯ can be called the “strength” of the grating. This grating strength plays a role equivalent to that played by the oscillator strength of the atomic resonance. But the physics is quite different—the slow down effect in a photonic structure is the result of the transfer of energy between the forward and backward propagating waves—no energy is transferred to the medium; hence the strength of the electric ﬁeld in photonic slow light structures becomes greatly enhanced, with important implications for nonlinear optics. One can also use simple photonic crystals [41,42] whose dispersion curves are similar to Bragg gratings, but the problem of structures with a single photonic resonance is identical to the single atomic

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 306 ␤ resonance—strong second-order dispersion 2. Clearly, in order to compensate for it, one should consider structures with more than one photonic resonance.

7. Double-Resonant Photonic Structures

Since any atomic slow light scheme requires operation near a particular narrow linewidth absorption resonance, finding such a resonance near a particular wave- length is not an easy task, and, in fact, only a very few absorption lines have been employed in practice, Rb vapors being a workhorse. Finding two closely spaced narrow lines is even more difficult, and even if two such lines can be found, the ␯ splitting between them 32 is fixed—hence the slow light delay will be optimized for one particular combination of storage capacity and bit rate. In contrast, the photonic double resonance can be easily implemented by simply ⌳ ⌳ combining two Bragg gratings with slightly different periods 1 and 2 as shown in Fig. 12(c). Such combination was first suggested for dispersion compensation [43,44] and then considered for application in electro-optic modula- tors [45]. As long as one deals with linear devices, such as delay lines, one can simply cascade two Bragg gratings sequentially, and the resulting dispersion curve will be simply the mean of the individual dispersion curves. For nonlinear and electro-optic devices one can alternate the short segments of Bragg gratings ⌳ ⌳ with periods 1 and 2. The dispersion curve of Fig. 12(d) is remarkably similar to the dispersion curve of the atomic double resonance in Fig. 4(b). Two gratings ␻ ␲ ⌳ engender two photonic bandgaps, centered at B,i = c/ in¯, of almost equal ⌬␻ ␻ ␦ Ϸ␻ ␦ ⌬␻ widths gap,i = B,i n/n¯ 0 n/n¯ with a narrow passband between them. By ⌳ ⌳ ␦ ⌬␻ choosing the periods 1 and 2 for a given index modulation n, one can design to be arbitrarily narrow or wide. This fact gives the designer true flexibility. In fact, one can show that the slow down factor in this scheme is

⌬␻ 1/2 gap S͑␻ ͒Ϸͫ1+ ͬ . ͑56͒ 0 2⌬␻ Thus, as one can see, the dispersion curve becomes squeezed between two gaps, and the group velocity decreases with the passband, but the dependence is not as strong as in the case of atomic resonance; thus the photonic structures in general should have far superior performance at large bandwidths when compared with atomic medium. A similar approach can be cascaded photonic crystal waveguides, as demonstrated in [46–50]. However, cascaded Bragg gratings also have a number of disadvantages, the first of which is a relatively small avail- able index contrast, and the second is difficulty in fabricating two gratings with a prescribed value of frequency offset. Furthermore, the cascaded geometry is applicable only to the linear devices that do not incorporate any nonlinear or electro-optic component. For this reason it is preferable to use alternating short segments of gratings with different periods. But a periodic sequence of short Bragg grating segments can be considered a new grating with periodically modulated properties—or a Moiré grating [Fig. 13(a)]. In a Moiré grating the segments are not independent but interact coherently—hence its properties are somewhat different from the cascaded grating, as was shown in [51], with the main distinction being the fact that the dispersion curve of a Moiré grating with period d is also periodic in wave-vector space with a period 2␲/d. The ability of a Moiré grating to slow down the light was first predicted in [51] and demonstrated in [52]. It was also noted that the Moiré grating is only one example of

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 307 periodically structured photonic media in which slow light can be observed. In periodically structured media the light energy density is distributed periodically, and the periodically spaced regions of high intensity can be thought of as resonators coupled to one another. Thus I shall refer to them as coupled resonator structures (CRSs). [53]

8. Coupled Resonator Structures and Their Comparison with Atomic Delay Lines

Aside from using Moiré gratings, CRSs can be fabricated by coupling Fabry– Perot resonators [Fig 13(b)], ring resonators [Fig. 13(c)][54], or so-called defect modes in the photonic crystal [Fig. 13(d)][55,56]. These and other CRS imple- mentations are discussed at length in the literature [41,57–61], and here I shall give only a short description of their properties and compare them with the EIT- like photonic structures. A periodic chain of coupled resonators is characterized by three parameters: period d, the time of a one-way pass through each resonator, and the coupling (or transmission) coefﬁcient ␬. The dispersion relation in this chain can be written as

sin ␻␶ = ␬ sin kd. ͑57͒ The dispersion curves are shown in Fig. 13(e) and consist of the series of pass- ␯ ␶ bands around resonant frequencies m =m/2 separated by the wide gaps. The presence of multiple resonances indicates that the light propagating through the CRS can be considered a superposition of more than one forward and more than one backward propagating wave. Alternatively, one can also think about the resonators as “photonic atoms” analogous to real atoms in EIT. At any rate, the dispersion curve is quite similar to the EIT dispersion curve in the sense that it becomes squeezed into a narrow passband of width

Figure 13

Photonic slow light structures based on coupled resonators. (a) Moiré grating, (b) coupled Fabry-Perot resonators, (c) coupled ring resonators, (d) coupled defect modes in photonic crystal, (e) dispersion in a typical CRS.

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 308 ⌬␯ ͑␲␶͒−1 −1͑␬͒ ͑ ͒ pass = sin . 58 The three parameters d, ␶, and ␬ are not independent of one another. First, d and ␶ are obviously related to each other. This relation can be obtained from taking a limit of Eq. (57) at ␬=1,

d ␻ c = = , ͑59͒ ␶ k n¯ which simply indicates that with 100% coupling the light simply propagates through the medium without reflections. Also related are the size of the resonator, i.e., d, and the coupling coefficient ␬: to achieve small ␬, one needs to con- fine the light tightly within the resonators, which requires a large spacing between them. If the index contrast ␦n/n¯ is large, a high degree of confinement can be achieved within a relatively small resonator; otherwise the light will leak from one resonator to another. This issue has been addressed in detail in [11], but here it is simply assumed that one uses the smallest resonator size that can be fabricated by using a technology with a given index contrast. Using the Taylor expansion of dispersion relation (58), one obtains the group velocity:

␶ n¯ v−1 = = ␬−1. ͑60͒ g d␬ c

Hence the slow down factor is simply

S = ␬−1. ͑61͒

At the same time the third-order dispersion is

␶ 2 ␤ = v−1ͩ ͪ ͑1−␬2͒, ͑62͒ 3 g ␬ indicating that one can always optimize the performance of a CRS delay line by choosing the proper coupling coefﬁcient to strike the balance between sufﬁcient delay and low distortion caused by dispersion. As a result one can achieve the following relation between the storage capacity Nst and the required delay line length:

␦ −1 ͑ ͒ n L Nst ϳ cN3/2ͩ␻ ͪ . ͑63͒ B st n¯

Comparing Eq. (63) with the results for the atomic delay line, Eq. (46), one can ﬁrst notice that the required length of the buffer (essentially the number of coupled resonators) does not increase with the bit rate. Thus the CRS should have much better performance at high bit rates compared with EIT delay lines. This result (and the fact that required length increases only as a power 3/2 of storage capacity) follows from the fact that the slow down factor is not as strong a function of passband width in CRS as in atomic media. Indeed, for weak coupling we can obtain from Eqs. (59) and (61)

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 309 ͑␲␶⌬␯ ͒−1 ͑ ͒ S = pass 64 indicating that at low bit rates the performance of the CRS optical buffers is not as spectacular as that of buffers based on atomic media, where according to Eq. (49) the slow down factor is inversely proportional to the square of the passband. At the same time the third-order dispersion in a CRS,

n¯ ␤ = ͑␲⌬␯ ͒−3, ͑65͒ 3 c␶ pass is inversely proportional to the third power of passband width and not to the fourth power of it as in atomic medium, Eq. (48). Therefore one can expect CRSs to perform much better than atomic media at high bit rates. Furthermore, the expression inside the parentheses can become commensurate with the optical frequency when large index contrast is used. This contrast can be as high as a factor of two in Si on silicon-on-insulator ring resonators [61,62]or photonic crystals [41], which means that one can ultimately store Nst bits of op- ϳ 3/2␭ tically encoded information in roughly L Nst length [63]. As far as the bandwidth limitations go, one can show that the cutoff bit rate for the photonic structures is ␦ ͑ ͒ n Nst ͑ ͒ϳͩ␯ ͪ −1/2 ͑ ͒ B Nst N , 66 cut n¯ st indicating that photonic slow light structures can perform at very high bit rates approaching terabits per second, even though the slow down factor can never ex- ceed a few dozen times. The limitations imposed by high-order dispersion can be mitigated by using a ␤ combination of CRSs with 3s of different sign as shown in [64]. Alternatively, one can use the so-called dynamic slow light scheme [65–67] in which the properties of the CRS are adiabatically tuned when the light enters it, allowing compression of the spectrum and thus reducing the deleterious effect of dispersion. Similar results can be in principle achieved by using semiconductor devices [68,69] combining Bragg gratings with quantum wells that are resonant near the Bragg frequency. It should be noted, however, that even dynamic structures have limited storage capacity because of dispersion occurring before the spectral compression, i.e., when the light enters the delay line [70,71]. To summarize, coupled resonators are similar to atoms in their ability to delay the light. The slow down factors in them are not as spectacular as in EIT slow light buffers, yet they can operate at much higher bit rates. Now, before we move on, it is crucial to stress another distinctive feature of all photonic structures— the slow down effect in them is the consequence of energy being transferred back and forth between the forward and the backward waves, or simply of light making circles in the ring resonator. The energy is never transferred from the electromagnetic wave to the medium as in the case of atomic slow light schemes. As a result the strength of the electromagnetic ﬁeld inside photonic structures does increase by a factor of S1/2, which is very advantageous for various nonlinear optical devices. Also, as was already mentioned, the losses in photonic structures are determined only by fabrication techniques and not by the fundamental coherence times; therefore, in principle photonic structures should perform with less loss than atomic ones.

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 310 9. Slow Light in Optical Fibers

In the prior Sections I have described slow light in the vicinity of atomic and photonic resonances that are always associated with loss. In the case of atomic resonance the loss is inherent and can be traced to dephasing of atomic polarization. In the case of photonic resonances the loss occurs simply because the light spends more time inside the delay line (or, in one case, it takes a longer effective path by bouncing back and forth or making circles). But it is well known that strong dispersion also takes place in the spectral vicinity of the resonant gain. The advantages of using gain are manyfold: first, one is not faced with attenuation, and, second, the gain can be changed at will by changing the pump strength and spectrum—hence the delay and the bandwidth can be made tunable. Consider a gain profile g͑␻͒ shown in Fig. 14(a) and characterized by its peak ͑␻ ͒ ␦␻ value g0 =g 0 and the FWHM linewidth 1/2. Usually the peak gain is inversely proportional to the linewidth, and therefore it makes sense to introduce the integrated optical gain that is proportional to the total pump power ␥ ϳ ␦␻ g0 1/2. Applying a Kramers–Kronig transform to the gain profile, one can obtain the refractive index spectrum shown in Fig. 14(b). There is a positive slope at the center of the gain, and this leads to a slow down factor of

͑␻ ͒ϳ ␦␻ ϳ ␥ ␦␻2 ͑ ͒ S 0 1+g0c/n¯ 1/2 1+ c/n¯ 1/2, 67

shown in Fig. 14(c). To achieve tunable gain, most often one uses stimulated Brillouin scattering (SBS) or stimulated Raman scattering (SRS). One unique advantage of these processes is that they can be observed in many media, including the ubiquitous silica optical fiber, which allows the slow light devices to be easily integrated into communication systems. In the SBS process, a high-frequency acoustic wave is induced in the material via electrostriction, for which the density of a material increases in regions of high optical intensity. The process of SBS can be described classically as a non- ␻ linear interaction between the pump (at angular frequency p) and a probe (or ͑␻ ͒ ͑⍀ ͒ Stokes) field s through the induced acoustic wave of frequency A [72]. The acoustic wave in turn modulates the refractive index of the medium and scatters pump light into the probe wave when its frequency is downshifted by the acoustic frequency. This process leads to a strong coupling among the three waves when this resonance condition is satisfied, which results in exponential

Figure 14

Slow light in an optical ampliﬁer: (a) gain, (b) index, (c) slow down factor.

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 311 amplification (absorption) of the probe wave. Efficient SBS occurs when both energy and momentum are conserved, which is satisfied when the pump and probe waves counterpropagate. This leads to a very narrow gain bandwidth typically of about 30 MHz. With the peak values of Brillouin gain of about g0 ϳ0.02 m−1 one can then expect the slow down factor to be of the order of only a few percent different from unity—hence the value of this scheme is not in the absolute delay time but in adjustable additional delay,

⌬ ϳ ␦␻ ͑ ͒ Td g0L/ 1/2. 68

Since the large value of total gain g0L can lead to Brillouin lasing, it is usually limited to about 10–15, meaning a maximum pulse delay of about 2 pulse lengths [73]. It is important that the main limitation for the achievable delay in amplifiers is the dispersion of gain rather than dispersion of group velocity [73]. While this delay of a few pulse lengths is relatively small, it is tunable and as such may be sufficient for many applications. Tunable slow light delay via SBS in an optical fiber was first demonstrated inde- pendently by Song et al. [74] and Okawachi et al. [75] In the experiment pre- formed in [74] delay could be tuned continuously by as much as 25 ns by adjust- ing the intensity of the pump field, and the technique can be applied to pulses as short as 15 ns. A fractional slow light delay of 1.3 was achieved for the 15 ns long input pulse with a pulse broadening of 1.4. Following the first demonstrations of SBS slow light in optical fibers, there has been considerable interest in exploiting the method for telecommunication applications. One line of research has focused on reducing pulse distortion by reducing the distortion caused by the gain dispersion performance [76–81]byes- sentially trying to shape the Brillouin gain with multiple pumps. In numerous experimental demonstrations with multiple closely spaced SBS gain lines gen- erated by a multifrequency pump, a significant increases in slow light pulse delay was achieved as compared with the optimum single-SBS-line delay. Another line of research has focused on broadband SBS slow light [82–86]. The width of the resonance that makes the slow light effect possible limits the minimum duration of the optical pulse that can be effectively delayed without much distortion and therefore limits the maximum data rate of the optical system. Her- raez et al. [82] were the first to increase the SBS slow light bandwidth and achieved a bandwidth of about 325 MHz by broadening the spectrum of the SBS pump field. Zhu et al. extended this work to achieve a SBS slow light bandwidth as large as 12.6 GHz, thereby supporting a data rate of more than 10 Gbits/s [83]. The latest results on expanding bandwidth are summarized in the review article [87] An alternative way to achieve tunable delays in optical fibers is via SRS, which can also be achieved in optical fibers. SRS arises from exciting vibrations in individual molecules, also known as optical phonons—as opposed to exciting sound waves (acoustic phonons) as in the SBS process. The optical phonons, un- like acoustic phonons, are localized and have very short lifetimes, measured in picoseconds or fractions of picoseconds. Furthermore, in amorphous materials, such as glass, the frequencies of optical phonons are spread over a large interval (measured in terahertz)—hence the Raman gain is much boarder than the Bril- louin gain, but is also much smaller in absolute value. Integrated gain ␥ ϳ ␦␻ g0 1/2 is similar for SBS and SRS, but in SRS it is spread out over much ␦␻ wider 1/2.

Advances in Optics and Photonics 2, 287–318 (2010) doi:10.1364/AOP.2.000287 312 Sharping et al. demonstrated an ultrafast all-optical controllable delay in a fiber Raman amplifier [88]. In this experiment a 430 fs pulse is delayed by 85% of its pulse width by using SRS in a 1 km long high-nonlinearity fiber. The ability to ac- commodate the bandwidth of pulses shorter than 1 ps in a fiber-based system makes SRS slow light useful for producing controllable delays in ultrahigh-bandwidth telecommunication systems. In addition to optical fibers, SRS slow light has also been demonstrated in a silicon-on-insulator planar waveguide [89]. Since Si is a single-crystalline material, the Raman gain is concentrated into the narrower bandwidth than in glass, but this bandwidth is still sufficient for delaying short optical pulses of 3 ps for 4 ps in a very short ͑8mm͒ waveguide. This scheme represents an important step in the development of chip-scale photonics devices for telecommunication and optical signal processing. Slow light propagation was also demonstrated in Er-doped fibers [90,91] using coherent population oscillations, but the bandwidth, related to the relaxation time in the Er ion was only of the order of kilohertz. Other methods included using the parametric gain [92] in the optical fiber as well as taking advantage of EIT in hollow optical fibers [93].

10. Conclusions

In this tutorial I have only given a very brief overview of the basics of physics involved in the phenomenon of slow light propagation. I have shown that while the underlying mechanisms can be diverse, they can all be characterized by a relatively narrow resonance, whether it is due to atomic transition or resonance in the photonic structure or is determined by a pump in optical amplifier. As a result the delay, bandwidth, and physical dimensions of all slow light media are interdependent and must be optimized for a given set of objectives that depends on the application. The applications of slow light, in both linear and nonlinear optics, are so diverse, that it is impossible to cover them all even briefly in this survey. They involve optical communications, optical signal processing, microwave photonics, precise interferometric instruments, and many others, and many new ones appear every year. The multidisciplinary field of slow light is experi- encing rapid growth, and entirely new directions, such as, for instance using metamaterials to achieve slow light [94], continue to appear. This brief survey of the current state of affairs is nothing more than a snapshot intended to provide sufficient background to the newcomers into this field and to suggest a number of sources from which deeper knowledge can be gained.

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Jacob B. Khurgin graduated with M.S. in Optics from the Institute of Fine Mechanics and Optics in St. Petersburg, Russia, in 1979. In 1980 he emigrated to the U.S. and joined Philips Laboratories of NV Philips in Briarcliff Manor, New York.There for eight years he worked on min- iature solid-state lasers, type II–VI semiconductor lasers, various display and lighting ﬁxtures, x-ray imaging, and small appliances. Simultaneously he was pursuing his graduate studies at Polytechnic Institute of NewYork,where he received a Ph.D. in Electro-physics in January 1987. In January 1988 he joined the Electrical En- gineering Department of John Hopkins University, where he is currently a Pro- fessor. His research topics over the years have included an eclectric mixture of optics of semiconductor nanostructures, nonlinear optical devices, optical communications, microwave photonics, THz technology, and condensed matter physics. Currently he is working in the areas of mid-infrared lasers and detectors, plasmonics, laser cooling, RF photonics, IR detectors, phonon engineering for high-frequency transistors, coherent optical communications, plasmonics, and slow light propagation. His publications include six book chapters, one book edited, more than 220 papers in refereed journals, and 14 patents. Prof. Khurgin is an OSA Fellow.

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