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
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  and stopping the light entirely . 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 familiarize 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  and Milonni , who also published a book in 2005 including chapters on slow, fast, and left-handed light . The first comprehensive description of the state-of-the-art slow light research has been given in our recent book , which contains contributions from 18 groups that have been actively 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 assessment 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  and others. Slow wave propagation was also observed and widely used in the microwave range as early as the 1940s . 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 . Soon afterward Grischkowsky and others observed slow light in the linear regime . Yet it was only with the discovery of EIT that slow light science was given a strong impetus—and this can be explained 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 artificially.
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 commonality 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 frequency as shown in Fig. 1(a). Using the familiar harmonic oscillator model, the dielectric constant in the vicinity of the resonance has a familiar expression: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 small, such that , and when we introduce the complex refractive index as , we can immediately use a series expansion of it around the background value, and for the frequencies relatively close to the resonance we obtain4) one immediately obtains the expressions for the absorption coefficient shown in Fig. 1(b),1(c)]6) is a particular form of the more general Kramer–Kronig relation between the real and the imaginary parts of dielectric constant,
Let us now consider a plane wave of frequency ω propagating in the medium defined by relation (6),1(d). The velocity with which the phase of the plane wave propagates,
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 :1(d):15). To gauge this chance, let us introduce the slow down factor—a relative reduction of group velocity due to resonance: 6) into Eq. (16), we obtain1(e). One can now see that there are two regions in which , i.e., the slow light regime and one region where , which can be either the fast light regime or the negative group velocity regime , 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, , this relation becomes particularly simple:18) can be interpreted in an interesting way—parameter is a coherence length in the medium, the length over which the material polarization (dipole moments of individual atoms) preserve the phase, while is the absorption length. Therefore, Eq. (18) can be rewritten as
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., ) 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 . The second mode is the atomic polarization characterized by a resonance frequency . The dispersion curve of atomic polarization is a horizontal line , 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 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 “photonlike,” 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 is21),20, 22) that . 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 dramatically different there, as the electric field density does become enhanced substantially. Also, one can see that without absorption the region of fast light 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, 12, 13]. To ascertain the importance of the group-velocity dispersion (GVD), one only needs to use the Taylor series expansion of the dispersion relation near the signal frequency to obtain the expression for the group velocity:11) shown in Fig. 3(a), whose FWHM is equal to one half of the bit interval, , and the bandwidth [Fig. 3(b)], according to Eq. (13), is 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 can be estimated as the difference between the delay times of spectral components at the edges of the signal bandwidth, i.e., at ,27). This condition shows the severe limitations imposed by the second-order GVD term , 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, 13, 14, 15, 16], it was only in 1990s when double-resonance schemes with complete cancellation of had been discovered  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 understood more easily if one considers that closely spaced narrow resonances do occur naturally in metal vapors, such, as, for instance, in , where two resonances near separated by 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), 1(f), which show that the lowest-order GVD term has opposite signs below and above resonance. Then, if one can combine two resonances as in Fig. 4(a), only the third-order GVD will be a factor for signals centered at frequency 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:4(d)] described by Eq. (28) is perfectly antisymmetric around ; thus only odd terms are present in its expansion. Note that the slow down factor in the double-resonant scheme is identical to that of the single-resonant scheme detuned by :
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 , 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 and the spectral components at the edges of the signal bandwidth, i.e., at27) 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 has the shape11), 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 to5(e). This appears to be a reasonable criterion for tolerable pulse distortion, and it leads to the condition30)]. This effect was indeed observed in , where group delays in excess of with pulses at were obtained, which correspond to about 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.
By changing the spectrum of the pump, for instance, using intensity or frequency modulation, one can change  to achieve the maximum delay without distortion for a given bit rate. In  delays of intervals were achieved for a moderate bandwidth of but only in a 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, 25, 26]. Without trying to explain all the intricacies of EIT, one can understand 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 in its response spectrum, just like any harmonic wave whose spectrum contains just one frequency component [Fig. 8(a)]. But if the wave is amplitude modulated with some frequency Ω, there will appear two sidebands at frequencies in its spectrum [Fig. 8(b)]; when the modulation depth reaches 100%, the carrier frequency becomes entirely suppressed, and the spectrum shows just two sidebands separated by [Fig. 8(c)].
Now, according to this analogy, if one strongly modulates the strength of the atomic oscillator with some external frequency Ω, 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 . The material should become transparent as the resonant frequency 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 “Λ” scheme  shown in Fig. 9, in which the ground-to-excited state transition is resonant with the frequency of the optical signal carrier and has a dephasing rate of . In the absence of the pump the absorption spectrum (dashed curve in Fig. 9) is a normal Lorentzian line. There also exists a strong transition coupling the excited 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 at frequency 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 , where the Rabi frequency is3, 4] with a slow down factor [Eq. (29)]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),30], which is not coupled to outside world. Thus typically , 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 incremental time that the light spends inside the slow light medium before its intensity decreases by 1/e, i.e. over one absorption length:
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 excitation 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., , 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  and then even stopped , first in metal vapors and then in a solid-state medium containing rare-earth ions . These achievements are of great importance for physics when it comes to manipulating single photons  and coherent control  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 . 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, 36, 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-down factor S is proportional to —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, 10, 11, 38, 39]. In fact it was shown in  and then in  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 at a given bit rate B. The required length was found in  to be
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 and the transmission window of width that successfully delays the signal of bit rate B by the delay time 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 as shown in Fig. 11(b). Sure enough, the delay time will increase to the required value of , but the differential group delay, Eq. (30),39), by a factor of . That, as shown in Fig. 11(c), will bring the intersymbol interference down to an acceptable value, but, unfortunately the slow down factor11(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 , 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 and the smaller is , the earlier will the saturation take place. Therefore, we can introduce a critical parameter of cutoff bit rate,
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 is typically in the range of few hundreds of gigahertz. For example, in it is , and in it is about . For solid-state slow light media using rare-earth ions, such as , is only . As a result, the maximum bit rate at which meaningful slow down of light with a total delay of, say, can be achieved is only in 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 , 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 Λ,40],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 being41, 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 resonance—strong second-order dispersion . 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 wavelength 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 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 and as shown in Fig. 12(c). Such combination was first suggested for dispersion compensation [43, 44] and then considered for application in electro-optic modulators . 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 and . 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 , of almost equal widths with a narrow passband between them. By choosing the periods and for a given index modulation , 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 is46, 47, 48, 49, 50]. However, cascaded Bragg gratings also have a number of disadvantages, the first of which is a relatively small available 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 , 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 . The ability of a Moiré grating to slow down the light was first predicted in  and demonstrated in . It was also noted that the Moiré grating is only one example of 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). 
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)] , or so-called defect modes in the photonic crystal [Fig. 13(d)] [55, 56]. These and other CRS implementations are discussed at length in the literature [41, 57, 58, 59, 60, 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 κ. The dispersion relation in this chain can be written as13(e) and consist of the series of passbands around resonant frequencies 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 width57) at ,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: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 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)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, 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 , which means that one can ultimately store bits of optically encoded information in roughly length .
As far as the bandwidth limitations go, one can show that the cutoff bit rate for the photonic structures is
The limitations imposed by high-order dispersion can be mitigated by using a combination of CRSs with s of different sign as shown in . Alternatively, one can use the so-called dynamic slow light scheme [65, 66, 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 , 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.
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 shown in Fig. 14(a) and characterized by its peak value and the FWHM linewidth . 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 . 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 of14(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 nonlinear interaction between the pump (at angular frequency ) and a probe (or Stokes) field through the induced acoustic wave of frequency . 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 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 . With the peak values of Brillouin gain of about 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,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 . 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 independently by Song et al.  and Okawachi et al.  In the experiment preformed in  delay could be tuned continuously by as much as by adjusting the intensity of the pump field, and the technique can be applied to pulses as short as . A fractional slow light delay of 1.3 was achieved for the 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, 77, 78, 79, 80, 81] by essentially trying to shape the Brillouin gain with multiple pumps. In numerous experimental demonstrations with multiple closely spaced SBS gain lines generated 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, 83, 84, 85, 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. Herraez et al.  were the first to increase the SBS slow light bandwidth and achieved a bandwidth of about 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 , thereby supporting a data rate of more than . The latest results on expanding bandwidth are summarized in the review article 
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, unlike 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 Brillouin gain, but is also much smaller in absolute value. Integrated gain is similar for SBS and SRS, but in SRS it is spread out over much wider .
Sharping et al. demonstrated an ultrafast all-optical controllable delay in a fiber Raman amplifier . In this experiment a pulse is delayed by 85% of its pulse width by using SRS in a long high-nonlinearity fiber. The ability to accommodate the bandwidth of pulses shorter than 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 . 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 for in a very short 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  in the optical fiber as well as taking advantage of EIT in hollow optical fibers .
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 experiencing rapid growth, and entirely new directions, such as, for instance using metamaterials to achieve slow light , 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 miniature 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 New York, where he received a Ph.D. in Electro-physics in January 1987. In January 1988 he joined the Electrical Engineering Department of John Hopkins University, where he is currently a Professor. 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.