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Advances in Optics and Photonics

| BRINGING REVIEWS AND TUTORIALS TO LIGHT

  • Editor: Bahaa E. A. Saleh
  • Vol. 2, Iss. 3 — Sep. 30, 2010
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Slow light in various media: a tutorial

Jacob B. Khurgin  »View Author Affiliations


Advances in Optics and Photonics, Vol. 2, Issue 3, pp. 287-318 (2010)
http://dx.doi.org/10.1364/AOP.2.000287


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Abstract

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

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

1. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–596 (1999). [CrossRef]

] and stopping the light entirely [2

2. D. F. Philips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86, 783–786 (2001). [CrossRef]

]. 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 [3

3. R. W. Boyd and D. J. Gauthier, “Slow and fast light,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2002), Vol. 43, pp. 497–530. [CrossRef]

] and Milonni [4

4. P. W. Milonni, “Controlling the speed of light pulses,” J. Phys. B 35, R31–R56 (2002). [CrossRef]

], who also published a book in 2005 including chapters on slow, fast, and left-handed light [5

5. P. W. Milonni, Fast Light, Slow Light, and Left-Handed Light (Institute of Physics, 2005).

]. The first comprehensive description of the state-of-the-art slow light research has been given in our recent book [6

6. J. B. Khurgin and R. S. Taylor, Slow Light: Science and Applications (Taylor & Francis, 2009).

], 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 [7

7. H. A. Lorentz, “Uber die Beziehung zwischen der Fortpflanzung des Lichtes und der Körperdichte,” Wiedemann Ann. 9, 641–664 (1880). [CrossRef]

] and others. Slow wave propagation was also observed and widely used in the microwave range as early as the 1940s [8

8. J. R. Pierce, Traveling Wave Tubes (Van Nostrand, 1950).

]. 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

9. S. L. McCall and E. L. Hahn, “Self-induced transparency by pulsed coherent light,” Phys. Rev. Lett. 18, 908–911 (1967). [CrossRef]

]. Soon afterward Grischkowsky and others observed slow light in the linear regime [10

10. D. Grischkowsky, “Adiabatic following and slow optical pulse propagation in rubidium vapor,” Phys. Rev. A 7, 2096 (1973). [CrossRef]

]. 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 ω12 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:
ϵ(ω)=ϵ¯+Ωp2ω122ω2jωγ12,
(1)
where ϵ¯=n¯2 is the nonresonant part, or “background” part, of the dielectric constant, and the damping constant γ12 is the decay rate of the polarization. The expression in the numerator Ωp is the “plasma frequency,” which can be found as
Ωp2=Nae2ϵ0m0f12,
(2)
where e is an electron charge, ϵ0 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,
f12=2m0ω12d122.
(3)
Here ed12 is the dipole matrix element of the transition and is the only quantum mechanical 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 γ12 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 small, such that Ωp2ωγ121, and when we introduce the complex refractive index as ñ(ω)=ϵ12(ω), we can immediately use a series expansion of it around the background value, and for the frequencies relatively close to the resonance |ω12ω|ω we obtain
ñ(ω)=(n¯2+Ωp2ω122ω2jωγ12)12n¯+12n¯Ωp2ω122ω2jωγ12n¯+14n¯ωΩp2(ω12ω)(ω12ω)2+γ1224+j8n¯ωΩp2γ12(ω12ω)2+γ1224.
(4)
From real and imaginary parts of relation (4) one immediately obtains the expressions for the absorption coefficient shown in Fig. 1(b),
α(ω)=2ωcIm(ñ)14n¯cΩp2γ12(ω12ω)2+γ1224,
(5)
and the refractive index [Fig. 1(c)]
n(ω)=Re(ñ)n¯+14n¯ωΩp2(ω12ω)(ω12ω)2+γ1224=n¯+cωα(ω)ω12ωγ12.
(6)
The absorption spectrum presents a Lorentzian peak with a full width at half-maximum (FWHM) of exactly γ12. In the vicinity of the resonance the refractive index first increases, then rapidly decreases to the point where n(ω)<n¯, 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,
Re(ϵ(ω))=1+1πPIm(ϵ(ω))ωω,
Im(ϵ(ω))=1πPRe(ϵ(ω))1ωω,
(7)
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 ω propagating in the medium defined by relation (6),
E(z,t)=E0exp(jkzjωt),
(8)
with the wave vector
k(ω)=n(ω)ωc.
(9)
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(ω)=ωk=cn(ω),
(10)
deviates somewhat from its background value cn¯ 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 Δt:
E(t)=E0e2ln2(t2Δt2)ejω0t.
(11)
The power spectrum of the Gaussian wave packet is also Gaussian:
|E(ω)|2=E02e4ln2[(ωω0)2Δω122]ejω0t,
(12)
where the spectral FWHM Δω12 is
Δω12=4ln2Δt.
(13)
The envelope of the wave packet propagates with the group velocity defined as the slope of the dispersion curve of Fig. 1(d):
vg=ωk.
(14)
It is more convenient to deal with the variable that is the inverse of the group velocity,
vg1(ω)=k(ω)ω=1c(n(ω)ω+ωn(ω)ω)=vp1(ω)+ωn(ω)ω.
(15)
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:
S(ω)=cn¯vg(ω)=n(ω)n¯+ωn¯n(ω)ω.
(16)
Substituting Eq. (6) into Eq. (16), we obtain
S(ω)1+Ωp24n¯2(ω12ω)2γ1224[(ω12ω)2+γ1224]2=1+cα(ω)n¯γ12(ω12ω)2γ1224(ω12ω)2+γ1224,
(17)
shown in Fig. 1(e). One can now see that there are two regions in which S>0, i.e., the slow light regime and one region where S<1, which can be either the fast light regime (0<S<1) or the negative group velocity regime (S<0), 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, |ω12ω|γ12, this relation becomes particularly simple:
S(ω)1+cα(ω)n¯γ12.
(18)
Equation (18) can be interpreted in an interesting way—parameter Lc=cn¯γ12 is a coherence length in the medium, the length over which the material polarization (dipole moments of individual atoms) preserve the phase, while La=1α is the absorption length. Therefore, Eq. (18) can be rewritten as
S(ω)1+LcLa(ω),
(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., γ12=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 ωp=ckn¯. The second mode is the atomic polarization characterized by a resonance frequency ω12. The dispersion curve of atomic polarization is a horizontal line ωa=ω12, 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.

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

11. R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximum time delay achievable on propagation through a slow-light medium,” Phys. Rev. A 71, 023801 (2005). [CrossRef]

, 12

12. A. B. Matsko, D. V. Strekalov, and L. Maleki, “On the dynamic range of optical delay lines based on coherent atomic media,” Opt. Express 13, 2210–2223 (2005). [CrossRef] [PubMed]

, 13

13. J. B. Khurgin, “Optical buffers based on slow light in EIT media and coupled resonator structures—comparative analysis,” J. Opt. Soc. Am. B 22, 1062–1074 (2005). [CrossRef]

]. To ascertain the importance of the group-velocity dispersion (GVD), one only needs to use the Taylor series expansion of the dispersion relation k(ω) near the signal frequency ω0 to obtain the expression for the group velocity:
vg1(ω)=vg1(ω0)+β2(ω0)(ωω0)+12β3(ω0)(ωω0)2+,
(23)
where βn(ω)=nk(ω)ωn. The delay is also a function of frequency:
Td(ω)=vg1(ω)L=vg1(ω0)L+β2(ωω0)L+12β3(ωω0)2L+.
(24)
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 Fig. 3(a), whose FWHM is equal to one half of the bit interval, Δt=12B1, and the bandwidth [Fig. 3(b)], according to Eq. (13), is Δω12=8ln2B 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 ω0±Δω122,
ΔTd(L)β2Δω12L.
(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.,
β2Δω12L=β28ln2BL<12B,
(26)
which leads to the condition relating the maximum allowable length and bit rate:
|β2|B2L<116ln2.
(27)
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 β2, 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

12. A. B. Matsko, D. V. Strekalov, and L. Maleki, “On the dynamic range of optical delay lines based on coherent atomic media,” Opt. Express 13, 2210–2223 (2005). [CrossRef] [PubMed]

, 13

13. J. B. Khurgin, “Optical buffers based on slow light in EIT media and coupled resonator structures—comparative analysis,” J. Opt. Soc. Am. B 22, 1062–1074 (2005). [CrossRef]

, 14

14. S. Chu and S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. 48, 738–741 (1982). [CrossRef]

, 15

15. S. Chu and S. Wong, “Linear pulse propagation in an absorbing medium—response,” Phys. Rev. Lett. 49, 1293 (1982). [CrossRef]

, 16

16. B. Segard and B. Macke, “Observation of negative velocity pulse propagation,” Phys. Lett. A 109, 213–216 (1985). [CrossRef]

], it was only in 1990s when double-resonance schemes with complete cancellation of β2 had been discovered [17

17. S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optics using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990). [CrossRef]

] 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

17. S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optics using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107–1110 (1990). [CrossRef]

, 18

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

], 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 Rb85 [19

19. R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys. Rev. A 73, 063812 (2006). [CrossRef]

], where two D2 resonances near 780nm separated by ν32=3GHz 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 β2 has opposite signs below and above resonance. Then, if one can combine two resonances as in Fig. 4(a), only the third-order GVD β3 will be a factor for signals centered at frequency ω0 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:
n(ω)n¯+18n¯ωΩp2ω31ω+18n¯ωΩp2ω21ω=n¯+Ωp24n¯ωω0ω(ω0ω)2ω3224,
(28)
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 perfectly antisymmetric around ω0; 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 ω322:
S(ω)1+Ωp24n¯2(ω0ω)2+ω3224[(ω0ω)2ω3224]2.
(29)

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 ν32, 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

11. R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximum time delay achievable on propagation through a slow-light medium,” Phys. Rev. A 71, 023801 (2005). [CrossRef]

, 20

20. R. M. Camacho, M. V. Pack, J. C. Howell, A. Schweinsberg, and R. W. Boyd, “Wide-bandwidth, tunable, multiple-pulse-width optical delays using slow light in cesium vapor,” Phys. Rev. Lett. 98, 153601 (2007). [CrossRef] [PubMed]

]. According to Eq. (24), we obtain the limitation for the relative delay between the spectral component at the central frequency ω0 and the spectral components at the edges of the signal bandwidth, i.e., at ω0±Δω122
ΔTd(L)12β3(Δω122)2L=8(ln2)2β3B2L<12B1,
(30)
which leads us to the criterion
|β3|B3L<116(ln2)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 ω0 has the shape
α(ω)18n¯cΩp2γ21(ω0ω+ω322)2+18n¯cΩp2γ21(ω0ωω322)2=Ωp2γ214n¯c(ω0ω)2+ω3224[(ω0ω)2ω3224]2=(S(ω)1)n¯γ21c.
(32)
The absorption reaches its minimum value at ω0,
α0=α(ω0)=Ωp2γ12n¯cω322
(33)
and then increases as
α(ω)α0+12α0(ω0ω)2ω322.
(34)
The transmission of the length L of the double-resonant atomic medium thus can be described as
T(ω)=eα0Le12α0L[(ω0ω)2ω322]=T0e4ln2[(ωω0)2Δωt2],
(35)
where FWHM of the transmission window is
Δωt=ω32(ln23α0L)12.
(36)
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
1Δω12,L2=1Δω122+1Δωt2
(37)
When the transmission window equals the original bandwidth of the signal, Δω12=8ln2B, the spectrum narrows by a factor of 212, 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
α0LB2ω322<ln2192.
(38)
It is important to note that to a certain extent small absorption can actually mitigate the deleterious effect of the dispersion, as reduced FWHM leads to smaller third-order dispersion [Eq. (30)]. This effect was indeed observed in [19

19. R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys. Rev. A 73, 063812 (2006). [CrossRef]

], where group delays in excess of 100ns with 2ns pulses at 780nm 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

21. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90, 113903 (2003). [CrossRef] [PubMed]

, 22

22. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003). [CrossRef] [PubMed]

]. As shown in Fig. 7(a), a strong pump pulse creates a situation in which the absorption in the frequency range Δω 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 Δω [23

23. R. M. Camacho, M. V. Pack, and J. C. Howell, “Slow light with large fractional delays by spectral hole-burning in rubidium vapor,” Phys. Rev. A 74, 0338012006. [CrossRef]

] to achieve the maximum delay without distortion for a given bit rate. In [23

23. R. M. Camacho, M. V. Pack, and J. C. Howell, “Slow light with large fractional delays by spectral hole-burning in rubidium vapor,” Phys. Rev. A 74, 0338012006. [CrossRef]

] delays of 2 bit intervals were achieved for a moderate bandwidth of 100MHz but only in a 40cm 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

24. K-J. Boller, A. Imamoglu, and S. E. Harris, “Observation of EIT,” Phys. Rev. Lett. 66, 2593–2596 (1991). [CrossRef] [PubMed]

, 25

25. S. E. Harris, J. E. Field, and A. Kasapi, “Dispersive properties of EIT,” Phys. Rev. A. 46, R39–R321992. [CrossRef]

, 26

26. A. Kasapi, M. Jain, G. Y. Jin, and S. E. Harris, “EIT: propagation dynamics,” Phys. Rev. Lett. 74, 2447–2450 (1995). [CrossRef] [PubMed]

]. 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 ω12 in its response spectrum, just like any harmonic wave whose spectrum contains just one frequency component ω0 [Fig. 8(a)]. But if the wave is amplitude modulated with some frequency Ω, there will appear two sidebands at frequencies ω12±Ω in its spectrum [Fig. 8(b)]; when the modulation depth reaches 100%, the carrier frequency ω0 becomes entirely suppressed, and the spectrum shows just two sidebands separated by 2Ω [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 2Ω. The material should become transparent as the resonant frequency ω0 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 [1

1. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–596 (1999). [CrossRef]

] shown in Fig. 9, in which the ground-to-excited state transition ω12 is resonant with the frequency of the optical signal carrier ω0 and has a dephasing rate of γ21. 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 Ipump at frequency ω23 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 ω0±Ω, where the Rabi frequency is
Ω=(f234παfIpumpmn¯ω23)12,
(39)
where αf=e24πϵ0c=1137 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

3. R. W. Boyd and D. J. Gauthier, “Slow and fast light,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2002), Vol. 43, pp. 497–530. [CrossRef]

, 4

4. P. W. Milonni, “Controlling the speed of light pulses,” J. Phys. B 35, R31–R56 (2002). [CrossRef]

] with a slow down factor [Eq. (29)]
S(ω)1+Ωp24n¯2(ω0ω)2+Ω2[(ω0ω)2Ω2]2.
(40)
If one introduces the density of pump phonons inside the medium as
Npump=n¯Ipumpcω23,
(41)
one can obtain a different expression for the Rabi frequency,
Ω=(f23Npumpe2mϵ¯ϵ0)12,
(42)
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),
S(ω0)1+Ωp24n¯2Ω2=1+Na4Npump.
(43)
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,
α(ω0)=1n¯cΩp24Ω2γ31=[S(ω0)1]n¯γ31c,
(44)
is proportional to the dephasing rate of the intra-atomic excitation [30

30. O. Firstenberg, M. Shuker, N. Davidson, and A. Ron, “Elimination of the diffraction of arbitrary images imprinted on slow light,” Phys. Rev. Lett. 102, 043601 (2009). [CrossRef] [PubMed]

], which is not coupled to outside world. Thus typically γ31γ21, 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:
ΔTd=LavgLacn¯=n¯[S(ω0)1]α(ω0)c=γ311,
(45)
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 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., γ31γ21, 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

3. R. W. Boyd and D. J. Gauthier, “Slow and fast light,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2002), Vol. 43, pp. 497–530. [CrossRef]

] and then even stopped [4

4. P. W. Milonni, “Controlling the speed of light pulses,” J. Phys. B 35, R31–R56 (2002). [CrossRef]

], first in metal vapors and then in a solid-state medium containing rare-earth ions [27

27. A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, 2002. “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2002). [CrossRef] [PubMed]

]. These achievements are of great importance for physics when it comes to manipulating single photons [28

28. M. D. Lukin and A. Imamoglu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419–1422 (2000). [CrossRef] [PubMed]

] and coherent control [29

29. F. E. Zimmer, A. Andre, M. D. Lukin, and M. Fleischhauer, “Coherent control of stationary light pulses,” Opt. Commun. 264, 441–453 (2006). [CrossRef]

] Also important are imaging applications of slow light, where indeed impressive results were observed by a number of groups [30

30. O. Firstenberg, M. Shuker, N. Davidson, and A. Ron, “Elimination of the diffraction of arbitrary images imprinted on slow light,” Phys. Rev. Lett. 102, 043601 (2009). [CrossRef] [PubMed]

, 31

31. R. M. Camacho, C. J. Broadbent, I. Ali-Khan, and J. C. Howell, “All-optical delay of images using slow light,” Phys. Rev. Lett. 98, 043902 (2007). [CrossRef] [PubMed]

], extrahigh-resolution interferometers [32

32. Z. Shi and R. W. Boyd, “Slow-light interferometry: practical limitations to spectroscopic performance,” J. Opt. Soc. Am. 25, 136–139 (2008). [CrossRef]

, 33

33. Z. Shi, R. W. Boyd, D. J. Gauthier, and C. C. Dudley, “Enhancing the spectral sensitivity of interferometers using slow-light media,” Opt. Lett. 32, 915–917 (2007). [CrossRef] [PubMed]

], and rotation sensors [34

34. M. S. Shahriar, G. S. Pati, R. Tripathi, V. Gopal, M. Messall, and K. Salit, “Ultrahigh enhancement in absolute and relative rotation sensing using fast and slow light,” Phys. Rev. A 75, 053807 (2007). [CrossRef]

]. 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

35. P. K. Kondratko, S. W. Chang, H. Su, and S. L. Chuang, “Slow light with tunable bandwidth in p-doped and intrinsic quantum dot electro-absorbers,” Appl. Phys. Lett. 90, 251108 (2007). [CrossRef]

, 36

36. F. Ohman, K. Yivind, and J. Mork, “Voltage-controlled slow light in an integrated semiconductor structure with net gain,” Opt. Express 14, 9955–9962 (2006). [CrossRef] [PubMed]

, 37

37. F. G. Sedgwick, B. Pesala, J. Y. Lin, W. S. Ko, X. X. Zhao, and C. J. Chang-Hasnain, “THz-bandwidth tunable slow light in semiconductor optical amplifiers,” Opt. Express 15, 747–753 (2007). [CrossRef] [PubMed]

]. But overall the best results in terms of the delay–bit-rate product were achieved in metal vapors [19

19. R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys. Rev. A 73, 063812 (2006). [CrossRef]

, 24

24. K-J. Boller, A. Imamoglu, and S. E. Harris, “Observation of EIT,” Phys. Rev. Lett. 66, 2593–2596 (1991). [CrossRef] [PubMed]

] 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 (ω12ω)2—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

9. S. L. McCall and E. L. Hahn, “Self-induced transparency by pulsed coherent light,” Phys. Rev. Lett. 18, 908–911 (1967). [CrossRef]

, 10

10. D. Grischkowsky, “Adiabatic following and slow optical pulse propagation in rubidium vapor,” Phys. Rev. A 7, 2096 (1973). [CrossRef]

, 11

11. R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximum time delay achievable on propagation through a slow-light medium,” Phys. Rev. A 71, 023801 (2005). [CrossRef]

, 38

38. R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, “Slow-light optical buffers: capabilities and fundamental limitations,” J. Lightwave Technol. 23, 4046–4066 (2005). [CrossRef]

, 39

39. D. A. B. Miller, “Fundamental limit to linear one-dimensional slow light structures,” Phys. Rev. Lett. 99, 203903 (2007). [CrossRef]

]. In fact it was shown in [11

11. R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximum time delay achievable on propagation through a slow-light medium,” Phys. Rev. A 71, 023801 (2005). [CrossRef]

] and then in [39

39. D. A. B. Miller, “Fundamental limit to linear one-dimensional slow light structures,” Phys. Rev. Lett. 99, 203903 (2007). [CrossRef]

] 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

11. R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximum time delay achievable on propagation through a slow-light medium,” Phys. Rev. A 71, 023801 (2005). [CrossRef]

] to be
LB(Nst)cBNst2[ΩP2π]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 LB(Nst) and the transmission window of width 2Ω that successfully delays the signal of bit rate B by the delay time Td(Nst)=NstB1 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 LB(2Nst)=LB(Nst)×2 as shown in Fig. 11(b). Sure enough, the delay time will increase to the required value of Td(2Nst)=2NstB1, but the differential group delay, Eq. (30),
ΔTd(2LB(Nst))=8(ln2)2β3B2LB(2Nst)=2ΔTd(LB(Nst)),
(47)
will also increase by a factor of two and cause unacceptable intersymbol interference. The third-order dispersion,
β3(ω0)=32n¯cΩp2Ω4,
(48)
can be reduced twofold by increasing the Rabi frequency by a factor of 214, which can be conveniently achieved by reducing the pump intensity Ipump, Eq. (39), by a factor of 212. That, as shown in Fig. 11(c), will bring the intersymbol interference down to an acceptable value, but, unfortunately the slow down factor
S(ω0)1+Ωp24n¯2Ω2
(49)
will now decrease by about a factor of 212 with a commensurate decrease in the delay time. Therefore, the length will have to be further increased by a factor of 212, which in turn will cause additional distortion, and the process will continue iteratively until the required length converges at
LB(2Nst)=LB(Nst)×2×212×214.=LB(Nst)×22
(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 and the smaller is Ωp2, the earlier will the saturation take place. Therefore, we can introduce a critical parameter of cutoff bit rate,
Bcut(atom)(Nst)[ΩP4πn¯]Nst12,
(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, when the bit rate approaches Bcut(atom)(Nst), the slow light delay line has to be just as long as the length of a conventional delay line, i.e., cn¯NstB1

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 ΩP2π is typically in the range of few hundreds of gigahertz. For example, in Rb87 it is 100GHz, and in Pb205 it is about 440GHz. For solid-state slow light media using rare-earth ions, such as Pr:Y2SiO5, ΩP2π is only 40GHz. 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 10Gbits 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 1Gbits, 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,
n=n¯+δncos(2πΛz).
(52)
As a result a photonic bandgap opens in the vicinity of Bragg frequency [40

40. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997). [CrossRef]

],
ωB=πcΛn¯,
(53)
and the dispersion law becomes modified as
kkBkB=(ωωBωB)2(δn2n¯)2.
(54)
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
S=|ωωBωB|(ωωBωB)2(δn2n¯)2.
(55)
Furthermore, the width of the forbidden gap is Δωgap=ωBδnn¯, and hence the index contrast δnn¯ 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

41. T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008). [CrossRef]

, 42

42. T. F. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D 40, 2666–2670 (2007). [CrossRef]

] 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 β2. 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 ν32 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 Λ1 and Λ2 as shown in Fig. 12(c). Such combination was first suggested for dispersion compensation [43

43. N. M. Litchinitser, B. J. Eggleton, and G. P. Agrawal, “Dispersion of cascaded fiber gratings in WDM lightwave systems,” J. Lightwave Technol. 16, 1523–1529 (1999). [CrossRef]

, 44

44. S. Wang, H. Erlig, H. R. Fetterman, E. Yablonovitch, V. Grubsky, D. S. Starodubov, and Feinberg, “Group velocity dispersion cancellation and additive group delays by cascaded fiber Bragg gratings in transmission,” IEEE Microw. Guid. Wave Lett. 8, 327–329 (1998). [CrossRef]

] and then considered for application in electro-optic modulators [45

45. J. B. Khurgin, J. U. Kang, and Y. J. Ding, “Ultrabroad-bandwidth electro-optic modulator based on a cascaded Bragg grating,” Opt. Lett. 25, 70–72 (2000). [CrossRef]

]. 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 Λ1 and Λ2. 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 ωB,i=πcΛin¯, of almost equal widths Δωgap,i=ωB,iδnn¯ω0δnn¯ with a narrow passband Δω between them. By choosing the periods Λ1 and Λ2 for a given index modulation δn, one can design Δω 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
S(ω0)[1+Δωgap2Δω]12.
(56)
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

46. D. Mori and T. Baba, “Wideband and low dispersion slow light by chirped photonic crystal coupled waveguide.,” Opt. Express 13, 9398–9408 (2005). [CrossRef] [PubMed]

, 47

47. M. L. Povinelli, S. G. Johnson, and J. D. Joannopoulos, “Slow-light, band-edge waveguides for tunable time delays,” Opt. Express 13, 7145–7159 (2005). [CrossRef] [PubMed]

, 48

48. D. Mori, S. Kubo, H. Sasaki, and T. Baba, “Experimental demonstration of wideband dispersion-compensated slow light by a chirped photonic crystal directional coupler,” Opt. Express 15, 5264–5270 (2007). [CrossRef] [PubMed]

, 49

49. S. C. Huang, M. Kato, E. Kuramochi, C. P. Lee, and M. Notomi, “Time-domain and spectral-domain investigation of inflection-point slow-light modes in photonic crystal coupled waveguides,” Opt. Express 15, 3543–3549 (2007). [CrossRef] [PubMed]

, 50

50. T. Kawasaki, D. Mori, and T. Baba, “Experimental observation of slow light in photonic crystal coupled waveguides,” Opt. Express 15, 10274–10281 (2007). [CrossRef] [PubMed]

]. 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 [51

51. J. B. Khurgin, “Light slowing down in Moiré fiber gratings and its implications for nonlinear optics,” Phys Rev A 62, 3821–3824 (2000). [CrossRef]

], 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 2πd. The ability of a Moiré grating to slow down the light was first predicted in [51

51. J. B. Khurgin, “Light slowing down in Moiré fiber gratings and its implications for nonlinear optics,” Phys Rev A 62, 3821–3824 (2000). [CrossRef]

] and demonstrated in [52

52. S. Longhi, D. Janner, G. Galzerano, G. Della Valle, D. Gatti, and P. Laporta, “Optical buffering in phase-shifted fibre gratings,” Electron. Lett. 41, 1075–1077 (2005). [CrossRef]

]. 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). [53

53. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999). [CrossRef]

]

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

54. C. K. Madsen and G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photon. Technol. Lett. 10, 994–9961998. [CrossRef]

], or so-called defect modes in the photonic crystal [Fig. 13(d)] [55

55. A. Melloni, F. Morichetti, and M. Martnelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35, 365–378 (2003). [CrossRef]

, 56

56. Z. Wang and S. Fan, “Compact all-pass filters in photonic crystals as the building block for high-capacity optical delay lines,” Phys. Rev. E 68, 066616 (2003). [CrossRef]

]. These and other CRS implementations are discussed at length in the literature [41

41. T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008). [CrossRef]

, 57

57. D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69, 63804 (2004). [CrossRef]

, 58

58. J. E. Heebner and R. W. Boyd, “‘Slow’ and ‘fast’ light in resonator-coupled waveguides,” J. Mod. Opt. 49, 2629–2636 (2002). [CrossRef]

, 59

59. S. Mookhejea, “Dispersion characteristics of coupled-resonator optical waveguides,” Opt. Lett. 30, 2406–2408 (2005). [CrossRef]

, 60

60. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001). [CrossRef] [PubMed]

, 61

61. F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2007). [CrossRef]

], and here I shall give only a short description of their properties and compare them with the EIT-like photonic structures.

Using the Taylor expansion of dispersion relation (58), one obtains the group velocity:
vg1=τdκ=n¯cκ1.
(60)
Hence the slow down factor is simply
S=κ1.
(61)
At the same time the third-order dispersion is
β3=vg1(τκ)2(1κ2),
(62)
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:
LB(Nst)cNst32(ωδnn¯)1.
(63)
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 32 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)
S=(πτΔνpass)1
(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,
β3=n¯cτ(πΔνpass)3,
(65)
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

61. F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2007). [CrossRef]

, 62

62. A. B. Matsko, A. A. Savchenlov, and L. Maleki, “Vertically coupled whispering-gallery-mode resonator waveguide,” Opt. Lett. 3066–3068 (2005). [CrossRef] [PubMed]

] or photonic crystals [41

41. T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008). [CrossRef]

], which means that one can ultimately store Nst bits of optically encoded information in roughly LNst32λ length [63

63. J. B. Khurgin, “Dispersion and loss limitations on the performance of optical delay lines based on coupled resonant structures,” Opt. Lett. 32, 163–165 (2007). [CrossRef]

].

As far as the bandwidth limitations go, one can show that the cutoff bit rate for the photonic structures is
Bcut(Nst)(Nst)(νδnn¯)Nst12,
(66)
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 exceed a few dozen times.

The limitations imposed by high-order dispersion can be mitigated by using a combination of CRSs with β3s of different sign as shown in [64

64. J. B. Khurgin, “Expanding the bandwidth of slow-light photonic devices based on coupled resonators,” Opt. Lett. 30, 513–515 (2005). [CrossRef] [PubMed]

]. Alternatively, one can use the so-called dynamic slow light scheme [65

65. M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92, 083901 (2004). [CrossRef] [PubMed]

, 66

66. M. F. Yanik and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93, 233903 (2004). [CrossRef] [PubMed]

, 67

67. M. F. Yanik and S. Fan, “Dynamic photonic structures: stopping, storage, and time-reversal of light,” Stud. Appl. Math. 115, 233–253 (2005). [CrossRef]

] 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

68. Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, “Distortionless light pulse delay in quantum-well Bragg structures,” Opt. Lett. 30, 2790–2792 (2005). [CrossRef] [PubMed]

, 69

69. Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, “Stopping, storing, and releasing light in quantum-well Bragg structures,” J. Opt. Soc. Am B 22, 2144–2156 (2005). [CrossRef]

] 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

70. J. B. Khurgin, “Adiabatically tunable optical delay lines and their performance limitations,” Opt. Lett. 30, 2778–2780 (2005). [CrossRef] [PubMed]

, 71

71. R. S. Tucker, “The role of optics and electronics in high-capacity routers,” J. Lightwave Technol. 24, 4655–4673 (2006). [CrossRef]

].

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 S12, 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 g(ω) shown in Fig. 14(a) and characterized by its peak value g0=g(ω0) and the FWHM linewidth δω12. 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 γg0δω12. 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
S(ω0)1+g0cn¯δω121+γcn¯δω122,
(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 nonlinear interaction between the pump (at angular frequency ωp) and a probe (or Stokes) field (ωs) through the induced acoustic wave of frequency (ΩA) [72

72. R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003).

]. 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 30MHz. With the peak values of Brillouin gain of about g00.02m1 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,
ΔTdg0Lδω12.
(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

73. J. B. Khurgin, “Performance limits of delay lines based on optical amplifiers,” Opt. Lett. 31, 948–950 (2006). [CrossRef] [PubMed]

]. 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

73. J. B. Khurgin, “Performance limits of delay lines based on optical amplifiers,” Opt. Lett. 31, 948–950 (2006). [CrossRef] [PubMed]

]. 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. [74

74. K. Y. Song, M. G. Herráez, and L. Thevenaz, “Observation of pulse delaying and advancementin optical fibers using stimulated Brillouin scattering,” Opt. Express 13, 82–88 (2005). [CrossRef] [PubMed]

] and Okawachi et al. [75

75. Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. oyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005). [CrossRef] [PubMed]

] In the experiment preformed in [74

74. K. Y. Song, M. G. Herráez, and L. Thevenaz, “Observation of pulse delaying and advancementin optical fibers using stimulated Brillouin scattering,” Opt. Express 13, 82–88 (2005). [CrossRef] [PubMed]

] delay could be tuned continuously by as much as 25ns by adjusting the intensity of the pump field, and the technique can be applied to pulses as short as 15ns. A fractional slow light delay of 1.3 was achieved for the 15ns 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

76. M. D. Stenner, M. A. Neifeld, Z. Zhu, A. M. C. Dawes, and D. J. Gauthier, “Distortion management in slow-light pulse delay,” Opt. Express 13, 9995–10002 (2005). [CrossRef] [PubMed]

, 77

77. A. Minardo, R. Bernini, and L. Zeni, “Low distortion Brillouin slow light in optical fibers sing AM modulation,” Opt. Express 14, 5866–5876 (2006). [CrossRef] [PubMed]

, 78

78. Z. Shi, R. Pant, Z. Zhu, M. D. Stenner, M. A. Neifeld, D. J. Gauthier, and R. W. Boyd, “Design of a tunable time-delay element using multiple gain lines for increased fractional delay with high data fidelity,” Opt. Lett. 32, 1986–1988 (2007). [CrossRef] [PubMed]

, 79

79. K. Y. Song, K. S. Abedin, K. Hotate, M. G. Herraez, and L. Thevenaz, “Highly efficient Brillouin slow and fast light using As2Se3 chalcogenide fiber,” Opt. Express 14, 5860–5865 (2006). [CrossRef] [PubMed]

, 80

80. A. Zadok, A. Eyal, and M. Tur, “Extended delay of broadband signals in stimulated rillouin cattering slow light using synthesized pump chirp,” Opt. Express 14, 8498–8505 (2006). [CrossRef] [PubMed]

, 81

81. T. Schneider, R. Henker, K. U. Lauterbach, and M. Junker, “Comparison of delay enhancement mechanisms for SBS-based slow light systems,” Opt. Express 15, 9606–9613 (2007). [CrossRef] [PubMed]

] 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

82. M. G. Herraez, K. Y. Song, and L. Thevenaz, “Arbitrary-bandwidth Brillouin slow light in optical fibers,” Opt. Express 14, 1395–1400 (2006). [CrossRef]

, 83

83. Z. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “Broadband SBS slow light in an optical fiber,” J. Lightwave Technol. 25, 201–206 (2007). [CrossRef]

, 84

84. T. Schneider, M. Junker, and K.-U. Lauterbach, “Potential ultra wide slow-light bandwidth enhancement,” Opt. Express 14, 11082–11087 (2006). [CrossRef] [PubMed]

, 85

85. K. Y. Song and K. Hotate, “25GHz bandwidth Brillouin slow light in optical fibers,” Opt. Lett. 32, 217–219 (2007). [CrossRef] [PubMed]

, 86

86. L. Yi, L. Zhan, W. Hu, and Y. Xia, “Delay of broadband signals using slow light in stimulated Brillouin scattering with phase-modulated pump,” IEEE Photon. Technol. Lett. 19, 619–621 (2007). [CrossRef]

]. 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. [82

82. M. G. Herraez, K. Y. Song, and L. Thevenaz, “Arbitrary-bandwidth Brillouin slow light in optical fibers,” Opt. Express 14, 1395–1400 (2006). [CrossRef]

] were the first to increase the SBS slow light bandwidth and achieved a bandwidth of about 325MHz 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 12.6GHz, thereby supporting a data rate of more than 10Gbitss [83

83. Z. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “Broadband SBS slow light in an optical fiber,” J. Lightwave Technol. 25, 201–206 (2007). [CrossRef]

]. The latest results on expanding bandwidth are summarized in the review article [87

87. L. Thevenaz, “Slow light in optical fibers,” Nat. Photonics 2, 474–481 (2008). [CrossRef]

]

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 γg0δω12 is similar for SBS and SRS, but in SRS it is spread out over much wider δω12.

Sharping et al. demonstrated an ultrafast all-optical controllable delay in a fiber Raman amplifier [88

88. J. E. Sharping, Y. Okawachi, and A. L. Gaeta, “Wide bandwidth slow light using a Raman fiber amplifier,” Opt. Express 13, 6092–6098 (2005). [CrossRef] [PubMed]

]. In this experiment a 430fs pulse is delayed by 85% of its pulse width by using SRS in a 1km long high-nonlinearity fiber. The ability to accommodate the bandwidth of pulses shorter than 1ps 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

89. Y. Okawachi, M. A. Foster, J. E. Sharping, A. L. Gaeta, Q. Xu, and M. Lipson, “All-optical slow-light on a photonic chip,” Opt. Express 14, 2317–2322 (2006). [CrossRef] [PubMed]

]. 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 3ps for 4ps in a very short (8mm) waveguide. This scheme represents an important step in the development of chip-scale photonics devices for telecommunication and optical signal processing.

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 experiencing rapid growth, and entirely new directions, such as, for instance using metamaterials to achieve slow light [94

94. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450, 397–401 (2007). [CrossRef] [PubMed]

], 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.

Figures

Fig. 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).
Fig. 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.
Fig. 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.
Fig. 4 (a) Double atomic resonance and (b) dispersion and group velocity near it. (c) Absorption spectrum. (d) Refractive index spectrum.
Fig. 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.
Fig. 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.
Fig. 7 (a) Slow light scheme based on spectral hole burning. (b) Absorption spectrum. (c) Refractive index spectrum.
Fig. 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.
Fig. 9 Principle of electromagnetic transparency in atomic Λ scheme.
Fig. 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.
Fig. 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.
Fig. 12 (a) Bragg grating and its index profile. (b) Dispersion of Bragg grating. (c) Cascaded Bragg grating. (d) Dispersion curve of cascaded grating.
Fig. 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.
Fig. 14 Slow light in an optical amplifier: (a) gain, (b) index, (c) slow down factor.
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M. F. Yanik and S. Fan, “Dynamic photonic structures: stopping, storage, and time-reversal of light,” Stud. Appl. Math. 115, 233–253 (2005). [CrossRef]

68.

Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, “Distortionless light pulse delay in quantum-well Bragg structures,” Opt. Lett. 30, 2790–2792 (2005). [CrossRef] [PubMed]

69.

Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, “Stopping, storing, and releasing light in quantum-well Bragg structures,” J. Opt. Soc. Am B 22, 2144–2156 (2005). [CrossRef]

70.

J. B. Khurgin, “Adiabatically tunable optical delay lines and their performance limitations,” Opt. Lett. 30, 2778–2780 (2005). [CrossRef] [PubMed]

71.

R. S. Tucker, “The role of optics and electronics in high-capacity routers,” J. Lightwave Technol. 24, 4655–4673 (2006). [CrossRef]

72.

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003).

73.

J. B. Khurgin, “Performance limits of delay lines based on optical amplifiers,” Opt. Lett. 31, 948–950 (2006). [CrossRef] [PubMed]

74.

K. Y. Song, M. G. Herráez, and L. Thevenaz, “Observation of pulse delaying and advancementin optical fibers using stimulated Brillouin scattering,” Opt. Express 13, 82–88 (2005). [CrossRef] [PubMed]

75.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. oyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005). [CrossRef] [PubMed]

76.

M. D. Stenner, M. A. Neifeld, Z. Zhu, A. M. C. Dawes, and D. J. Gauthier, “Distortion management in slow-light pulse delay,” Opt. Express 13, 9995–10002 (2005). [CrossRef] [PubMed]

77.

A. Minardo, R. Bernini, and L. Zeni, “Low distortion Brillouin slow light in optical fibers sing AM modulation,” Opt. Express 14, 5866–5876 (2006). [CrossRef] [PubMed]

78.

Z. Shi, R. Pant, Z. Zhu, M. D. Stenner, M. A. Neifeld, D. J. Gauthier, and R. W. Boyd, “Design of a tunable time-delay element using multiple gain lines for increased fractional delay with high data fidelity,” Opt. Lett. 32, 1986–1988 (2007). [CrossRef] [PubMed]

79.

K. Y. Song, K. S. Abedin, K. Hotate, M. G. Herraez, and L. Thevenaz, “Highly efficient Brillouin slow and fast light using As2Se3 chalcogenide fiber,” Opt. Express 14, 5860–5865 (2006). [CrossRef] [PubMed]

80.

A. Zadok, A. Eyal, and M. Tur, “Extended delay of broadband signals in stimulated rillouin cattering slow light using synthesized pump chirp,” Opt. Express 14, 8498–8505 (2006). [CrossRef] [PubMed]

81.

T. Schneider, R. Henker, K. U. Lauterbach, and M. Junker, “Comparison of delay enhancement mechanisms for SBS-based slow light systems,” Opt. Express 15, 9606–9613 (2007). [CrossRef] [PubMed]

82.

M. G. Herraez, K. Y. Song, and L. Thevenaz, “Arbitrary-bandwidth Brillouin slow light in optical fibers,” Opt. Express 14, 1395–1400 (2006). [CrossRef]

83.

Z. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “Broadband SBS slow light in an optical fiber,” J. Lightwave Technol. 25, 201–206 (2007). [CrossRef]

84.

T. Schneider, M. Junker, and K.-U. Lauterbach, “Potential ultra wide slow-light bandwidth enhancement,” Opt. Express 14, 11082–11087 (2006). [CrossRef] [PubMed]

85.

K. Y. Song and K. Hotate, “25GHz bandwidth Brillouin slow light in optical fibers,” Opt. Lett. 32, 217–219 (2007). [CrossRef] [PubMed]

86.

L. Yi, L. Zhan, W. Hu, and Y. Xia, “Delay of broadband signals using slow light in stimulated Brillouin scattering with phase-modulated pump,” IEEE Photon. Technol. Lett. 19, 619–621 (2007). [CrossRef]

87.

L. Thevenaz, “Slow light in optical fibers,” Nat. Photonics 2, 474–481 (2008). [CrossRef]

88.

J. E. Sharping, Y. Okawachi, and A. L. Gaeta, “Wide bandwidth slow light using a Raman fiber amplifier,” Opt. Express 13, 6092–6098 (2005). [CrossRef] [PubMed]

89.

Y. Okawachi, M. A. Foster, J. E. Sharping, A. L. Gaeta, Q. Xu, and M. Lipson, “All-optical slow-light on a photonic chip,” Opt. Express 14, 2317–2322 (2006). [CrossRef] [PubMed]

90.

A. Schweinsberg, N. N. Lepeshkin, M. S. Bigelow, R. W. Boyd, and S. Jarabo, “Observation of superluminal and slow light propagation in erbium-doped optical fiber,” Europhys. Lett. 73, 218–224 (2006). [CrossRef]

91.

H. Shin, A. Schweinsberg, G. Gehring, K. Schwertz, H. J. Chang, R. W. Boyd, Q.-H. Park, and D. J. Gauthier, “Reducing pulse distortion in fast-light pulse propagation through an erbium-doped fiber amplifier,” Opt. Lett. 32, 906–908 (2007). [CrossRef] [PubMed]

92.

D. Dahan and G. Eisenstein, “Tunable all optical delay via slow and fast light propagation in a Raman assisted fiber optical parametric amplifier: a route to all optical buffering,” Opt. Express 13, 6234–6249 (2005). [CrossRef] [PubMed]

93.

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

94.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450, 397–401 (2007). [CrossRef] [PubMed]

aop-2-3-287-i001 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.

ToC Category:
Nonlinear Optics

History
Original Manuscript: October 26, 2009
Revised Manuscript: February 4, 2010
Manuscript Accepted: February 8, 2010
Published: March 23, 2010

Virtual Issues
(2010) Advances in Optics and Photonics

Citation
Jacob B. Khurgin, "Slow light in various media: a tutorial," Adv. Opt. Photon. 2, 287-318 (2010)
http://www.opticsinfobase.org/aop/abstract.cfm?URI=aop-2-3-287


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  74. K. Y. Song, M. G. Herráez, L. Thevenaz, “Observation of pulse delaying and advancementin optical fibers using stimulated Brillouin scattering,” Opt. Express 13, 82–88 (2005). [CrossRef] [PubMed]
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  83. Z. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, A. E. Willner, “Broadband SBS slow light in an optical fiber,” J. Lightwave Technol. 25, 201–206 (2007). [CrossRef]
  84. T. Schneider, M. Junker, K.-U. Lauterbach, “Potential ultra wide slow-light bandwidth enhancement,” Opt. Express 14, 11082–11087 (2006). [CrossRef] [PubMed]
  85. K. Y. Song, K. Hotate, “25 GHz bandwidth Brillouin slow light in optical fibers,” Opt. Lett. 32, 217–219 (2007). [CrossRef] [PubMed]
  86. L. Yi, L. Zhan, W. Hu, Y. Xia, “Delay of broadband signals using slow light in stimulated Brillouin scattering with phase-modulated pump,” IEEE Photon. Technol. Lett. 19, 619–621 (2007). [CrossRef]
  87. L. Thevenaz, “Slow light in optical fibers,” Nat. Photonics 2, 474–481 (2008). [CrossRef]
  88. J. E. Sharping, Y. Okawachi, A. L. Gaeta, “Wide bandwidth slow light using a Raman fiber amplifier,” Opt. Express 13, 6092–6098 (2005). [CrossRef] [PubMed]
  89. Y. Okawachi, M. A. Foster, J. E. Sharping, A. L. Gaeta, Q. Xu, M. Lipson, “All-optical slow-light on a photonic chip,” Opt. Express 14, 2317–2322 (2006). [CrossRef] [PubMed]
  90. A. Schweinsberg, N. N. Lepeshkin, M. S. Bigelow, R. W. Boyd, S. Jarabo, “Observation of superluminal and slow light propagation in erbium-doped optical fiber,” Europhys. Lett. 73, 218–224 (2006). [CrossRef]
  91. H. Shin, A. Schweinsberg, G. Gehring, K. Schwertz, H. J. Chang, R. W. Boyd, Q.-H. Park, D. J. Gauthier, “Reducing pulse distortion in fast-light pulse propagation through an erbium-doped fiber amplifier,” Opt. Lett. 32, 906–908 (2007). [CrossRef] [PubMed]
  92. D. Dahan, G. Eisenstein, “Tunable all optical delay via slow and fast light propagation in a Raman assisted fiber optical parametric amplifier: a route to all optical buffering,” Opt. Express 13, 6234–6249 (2005). [CrossRef] [PubMed]
  93. S. Ghosh, A. R. Bhagwat, C. K. Renshaw, S. Goh, A. L. Gaeta, B. J. Kirby, “Low-light level optical interactions with rubidium vapor in a photonic band-gap fiber,” Phys. Rev. Lett. 97, 023603 (2006). [CrossRef]
  94. K. L. Tsakmakidis, A. D. Boardman, O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450, 397–401 (2007). [CrossRef] [PubMed]

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