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

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2009

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

2008

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

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

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

2007

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

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]

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]

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

Z. Shi, R. Pant, Z. Zhu, M. D. Stenner, M. A. Neifeld, D. J. Gauthier, 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]

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

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]

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

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

D. Mori, S. Kubo, H. Sasaki, 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]

S. C. Huang, M. Kato, E. Kuramochi, C. P. Lee, 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]

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

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]

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

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

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

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

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

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

R. M. Camacho, M. V. Pack, J. C. Howell, A. Schweinsberg, 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]

2006

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

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

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

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

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

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

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

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]

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

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

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

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Figures (14)

Fig. 1
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
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
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
Fig. 4

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

Fig. 5
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
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
Fig. 7

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

Fig. 8
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
Fig. 9

Principle of electromagnetic transparency in atomic Λ scheme.

Fig. 10
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
Fig. 11

Why the length of EIT delay line growth quadratically with delay capacity. (a) A delay line with delay of N st 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 2 N st bits with acceptable distortion is achieved in the delay line that is four times as long as original one.

Fig. 12
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
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
Fig. 14

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

Equations (69)

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ϵ ( ω ) = ϵ ¯ + Ω p 2 ω 12 2 ω 2 j ω γ 12 ,
Ω p 2 = N a e 2 ϵ 0 m 0 f 12 ,
f 12 = 2 m 0 ω 12 d 12 2 .
n ̃ ( ω ) = ( n ¯ 2 + Ω p 2 ω 12 2 ω 2 j ω γ 12 ) 1 2 n ¯ + 1 2 n ¯ Ω p 2 ω 12 2 ω 2 j ω γ 12 n ¯ + 1 4 n ¯ ω Ω p 2 ( ω 12 ω ) ( ω 12 ω ) 2 + γ 12 2 4 + j 8 n ¯ ω Ω p 2 γ 12 ( ω 12 ω ) 2 + γ 12 2 4 .
α ( ω ) = 2 ω c Im ( n ̃ ) 1 4 n ¯ c Ω p 2 γ 12 ( ω 12 ω ) 2 + γ 12 2 4 ,
n ( ω ) = Re ( n ̃ ) n ¯ + 1 4 n ¯ ω Ω p 2 ( ω 12 ω ) ( ω 12 ω ) 2 + γ 12 2 4 = n ¯ + c ω α ( ω ) ω 12 ω γ 12 .
Re ( ϵ ( ω ) ) = 1 + 1 π P Im ( ϵ ( ω ) ) ω ω ,
Im ( ϵ ( ω ) ) = 1 π P Re ( ϵ ( ω ) ) 1 ω ω ,
E ( z , t ) = E 0 exp ( j k z j ω t ) ,
k ( ω ) = n ( ω ) ω c .
v p ( ω ) = ω k = c n ( ω ) ,
E ( t ) = E 0 e 2 ln 2 ( t 2 Δ t 2 ) e j ω 0 t .
| E ( ω ) | 2 = E 0 2 e 4 ln 2 [ ( ω ω 0 ) 2 Δ ω 1 2 2 ] e j ω 0 t ,
Δ ω 1 2 = 4 ln 2 Δ t .
v g = ω k .
v g 1 ( ω ) = k ( ω ) ω = 1 c ( n ( ω ) ω + ω n ( ω ) ω ) = v p 1 ( ω ) + ω n ( ω ) ω .
S ( ω ) = c n ¯ v g ( ω ) = n ( ω ) n ¯ + ω n ¯ n ( ω ) ω .
S ( ω ) 1 + Ω p 2 4 n ¯ 2 ( ω 12 ω ) 2 γ 12 2 4 [ ( ω 12 ω ) 2 + γ 12 2 4 ] 2 = 1 + c α ( ω ) n ¯ γ 12 ( ω 12 ω ) 2 γ 12 2 4 ( ω 12 ω ) 2 + γ 12 2 4 ,
S ( ω ) 1 + c α ( ω ) n ¯ γ 12 .
S ( ω ) 1 + L c L a ( ω ) ,
U = P v g = ( P n ¯ c ) S ,
U = 1 4 ϵ 0 ( ω ϵ ) ω E 2 + 1 4 μ 0 H 2 ,
U = 1 4 ϵ 0 ϵ ¯ E 2 + 1 4 ω ϵ 0 n 2 ( ω ) ω E 2 + 1 4 ϵ 0 ϵ ¯ E 2 = 1 2 ϵ 0 ϵ ¯ E 2 + 1 2 ϵ 0 n ¯ ω n ( ω ) ω E 2 = 1 2 ϵ 0 ϵ ¯ E 2 ( 1 + ω n ¯ n ( ω ) ω ) = 1 2 ϵ 0 ϵ ¯ E 2 S
v g 1 ( ω ) = v g 1 ( ω 0 ) + β 2 ( ω 0 ) ( ω ω 0 ) + 1 2 β 3 ( ω 0 ) ( ω ω 0 ) 2 + ,
T d ( ω ) = v g 1 ( ω ) L = v g 1 ( ω 0 ) L + β 2 ( ω ω 0 ) L + 1 2 β 3 ( ω ω 0 ) 2 L + .
Δ T d ( L ) β 2 Δ ω 1 2 L .
β 2 Δ ω 1 2 L = β 2 8 ln 2 B L < 1 2 B ,
| β 2 | B 2 L < 1 16 ln 2 .
n ( ω ) n ¯ + 1 8 n ¯ ω Ω p 2 ω 31 ω + 1 8 n ¯ ω Ω p 2 ω 21 ω = n ¯ + Ω p 2 4 n ¯ ω ω 0 ω ( ω 0 ω ) 2 ω 32 2 4 ,
S ( ω ) 1 + Ω p 2 4 n ¯ 2 ( ω 0 ω ) 2 + ω 32 2 4 [ ( ω 0 ω ) 2 ω 32 2 4 ] 2 .
Δ T d ( L ) 1 2 β 3 ( Δ ω 1 2 2 ) 2 L = 8 ( ln 2 ) 2 β 3 B 2 L < 1 2 B 1 ,
| β 3 | B 3 L < 1 16 ( ln 2 ) 2 .
α ( ω ) 1 8 n ¯ c Ω p 2 γ 21 ( ω 0 ω + ω 32 2 ) 2 + 1 8 n ¯ c Ω p 2 γ 21 ( ω 0 ω ω 32 2 ) 2 = Ω p 2 γ 21 4 n ¯ c ( ω 0 ω ) 2 + ω 32 2 4 [ ( ω 0 ω ) 2 ω 32 2 4 ] 2 = ( S ( ω ) 1 ) n ¯ γ 21 c .
α 0 = α ( ω 0 ) = Ω p 2 γ 12 n ¯ c ω 32 2
α ( ω ) α 0 + 12 α 0 ( ω 0 ω ) 2 ω 32 2 .
T ( ω ) = e α 0 L e 12 α 0 L [ ( ω 0 ω ) 2 ω 32 2 ] = T 0 e 4 ln 2 [ ( ω ω 0 ) 2 Δ ω t 2 ] ,
Δ ω t = ω 32 ( ln 2 3 α 0 L ) 1 2 .
1 Δ ω 1 2 , L 2 = 1 Δ ω 1 2 2 + 1 Δ ω t 2
α 0 L B 2 ω 32 2 < ln 2 192 .
Ω = ( f 23 4 π α f I pump m n ¯ ω 23 ) 1 2 ,
S ( ω ) 1 + Ω p 2 4 n ¯ 2 ( ω 0 ω ) 2 + Ω 2 [ ( ω 0 ω ) 2 Ω 2 ] 2 .
N pump = n ¯ I pump c ω 23 ,
Ω = ( f 23 N pump e 2 m ϵ ¯ ϵ 0 ) 1 2 ,
S ( ω 0 ) 1 + Ω p 2 4 n ¯ 2 Ω 2 = 1 + N a 4 N pump .
α ( ω 0 ) = 1 n ¯ c Ω p 2 4 Ω 2 γ 31 = [ S ( ω 0 ) 1 ] n ¯ γ 31 c ,
Δ T d = L a v g L a c n ¯ = n ¯ [ S ( ω 0 ) 1 ] α ( ω 0 ) c = γ 31 1 ,
L B ( N st ) c B N st 2 [ Ω P 2 π ] 2 ,
Δ T d ( 2 L B ( N st ) ) = 8 ( ln 2 ) 2 β 3 B 2 L B ( 2 N st ) = 2 Δ T d ( L B ( N st ) ) ,
β 3 ( ω 0 ) = 3 2 n ¯ c Ω p 2 Ω 4 ,
S ( ω 0 ) 1 + Ω p 2 4 n ¯ 2 Ω 2
L B ( 2 N st ) = L B ( N st ) × 2 × 2 1 2 × 2 1 4 . = L B ( N st ) × 2 2
B cut ( atom ) ( N st ) [ Ω P 4 π n ¯ ] N st 1 2 ,
n = n ¯ + δ n cos ( 2 π Λ z ) .
ω B = π c Λ n ¯ ,
k k B k B = ( ω ω B ω B ) 2 ( δ n 2 n ¯ ) 2 .
S = | ω ω B ω B | ( ω ω B ω B ) 2 ( δ n 2 n ¯ ) 2 .
S ( ω 0 ) [ 1 + Δ ω gap 2 Δ ω ] 1 2 .
sin ω τ = κ sin k d .
Δ ν pass = ( π τ ) 1 sin 1 ( κ ) .
d τ = ω k = c n ¯ ,
v g 1 = τ d κ = n ¯ c κ 1 .
S = κ 1 .
β 3 = v g 1 ( τ κ ) 2 ( 1 κ 2 ) ,
L B ( N st ) c N st 3 2 ( ω δ n n ¯ ) 1 .
S = ( π τ Δ ν pass ) 1
β 3 = n ¯ c τ ( π Δ ν pass ) 3 ,
B cut ( N st ) ( N st ) ( ν δ n n ¯ ) N st 1 2 ,
S ( ω 0 ) 1 + g 0 c n ¯ δ ω 1 2 1 + γ c n ¯ δ ω 1 2 2 ,
Δ T d g 0 L δ ω 1 2 .

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