Broadband time-reversal of optical pulses using a switchable photonic-crystal mirror

doi:10.1364/OE.19.014502

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Broadband time-reversal of optical pulses using a switchable photonic-crystal mirror

Yonatan Sivan and John B. Pendry »View Author Affiliations

Optics Express, Vol. 19, Issue 15, pp. 14502-14507 (2011)
http://dx.doi.org/10.1364/OE.19.014502

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– Abstract
1. Principles of time-reversal using a switchable mirror
2. Analysis
3. Implementation
– References and links

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Abstract

Recently, Chumak et al. have demonstrated experimentally the time-reversal of microwave spin pulses based on non-adiabatically tuning the wave speed in a spatially-periodic manner [Nat. Comm. 1, 141 (2010)]. Here, we solve the associated wave equations analytically, and give an explicit formula for the reversal efficiency. We discuss the implementation for short optical electromagnetic pulses and show that the new scheme may lead to their accurate time-reversal with efficiency higher than before.

Time-reversal is one of the most spectacular and useful wave phenomena. A time-reversed pulse evolves as if time runs backward, thus eliminating any distortions or scattering that occurred at earlier times. This enables light detection, imaging and focusing through complex media [1

M. Fink, “Time-reversed acoustics,” Scientific American 91, (November 1999).

, 2

M. Fink, “Acoustic time-reversal mirrors: Imaging of complex media with acoustic and seismic waves,” Topics in Applied Physics 84, 17–43 (2002). [CrossRef]

, 3

J. Aullbach, B. Gjonaj, P. M. Johnson, A. M. Mosk, and A. Lagendijk, “Control of light transmission through opaque scattering media in space and time,” Phys. Rev. Lett. 106, 103901 (2011). [CrossRef]

, 4

O. Katz, Y. Bromberg, E. Small, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Phot. 5, 372–377 (2011). [CrossRef]

] with applications in diverse fields such as medical ultrasound [1

M. Fink, “Time-reversed acoustics,” Scientific American 91, (November 1999).

, 2

M. Fink, “Acoustic time-reversal mirrors: Imaging of complex media with acoustic and seismic waves,” Topics in Applied Physics 84, 17–43 (2002). [CrossRef]

], communication systems and adaptive optics [5

D. M. Pepper, Laser handbook , Vol. 4, (North-Holland Physics, Amsterdam 1988).

], superlensing [6

J. B. Pendry, “Time-reversal and negative refraction,” Science 322, 71–73 (2008). [CrossRef] [PubMed]

], ultrafast plasmonics [7

X. Li and M. I. Stockman, “Highly efficient spatio-temporal coherent control in nanoplasmonics on a nanometer-femtosecond scale by time-reversal,” Phys. Rev. B 77, 195109 (2008). [CrossRef]

], biological and THz imaging [8

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Phot. 2, 110–115 (2008). [CrossRef]

], spintronics [9

Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, “Transmission of electrical signals by spin-wave interconversion in a magnetic insulator,” Nature 464, 262–266 (2010). [CrossRef] [PubMed]

] and quantum information and computing [10

F. M. Cucchietti, “Time-reversal in an optical lattice,” J. Opt. Soc. Am. B 27, 30–35 (2010). [CrossRef]

For low frequency waves (e.g., in acoustics, microwave spin/electromagnetic waves etc.), time-reversal can be accomplished by electronic sampling, recording, and playing back [1

M. Fink, “Time-reversed acoustics,” Scientific American 91, (November 1999).

, 2

M. Fink, “Acoustic time-reversal mirrors: Imaging of complex media with acoustic and seismic waves,” Topics in Applied Physics 84, 17–43 (2002). [CrossRef]

]. This is possible since in this frequency range, the pulse oscillates on a scale slower than electronic sampling speed. On the other hand, for high frequency waves, specifically, for optical electromagnetic waves, the pulse oscillations (and even the pulse envelope itself for sufficiently short pulses) are too short to be sampled properly by even the fastest electronic detector.

The standard solution in the optical regime is to use nonlinear processes such as Three-Wave or Four-Wave Mixing (3WM or 4WM, respectively), see e.g., [5

D. M. Pepper, Laser handbook , Vol. 4, (North-Holland Physics, Amsterdam 1988).

, 11

A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, “Femtosecond spectral holography,” IEEE J. Qu. Electron. 28, 2251–2261 (1992). [CrossRef]

, 12

D. Marom, D. Panasenko, R. Rokitski, P.-C. Sun, and Y. Fainman, “Time-reversal of ultrafast waveforms by wave mixing of spectrally decomposed waves,” Opt. Lett. 25, 132–134 (2000). [CrossRef]

, 13

O. Kuzucu, Y. Okawachi, R. Salem, M. A. Foster, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Spectral phase conjugation via temporal imaging,” Opt. Exp. 17, 20605–20614 (2009). [CrossRef]

]. However, while such techniques have been demonstrated experimentally, they usually suffer from one or several disadvantages. In particular, they require fairly high intensities, thus, limiting on-chip integration; almost all existing schemes are narrowband whereas the schemes which are applicable to relatively short pulses are in general complex, requiring complicated setups and sometimes even cryogenic temperatures. Finally, some schemes may be difficult to apply to 2-and 3-dimensional systems.

Recent suggestions to overcome these limitations involved optical periodic structures (photonic crystals, PhCs) which are dynamically-tuned [14

M. F. Yanik and S. Fan, “Time-reversal of light with linear optics and modulators,” Phys. Rev. Lett. 93, 173903 (2004). [CrossRef] [PubMed]

, 15

S. Longhi, “Stopping and time-reversal of light in dynamic photonic structures via Bloch oscillations,” Phys. Rev. E 75, 026606 (2007). [CrossRef]

]. However, despite being very efficient, the structures and/or modulations required for these schemes were quite challenging to realize; in addition, these schemes allow for reversal of only relatively narrow pulses.

These obstacles were recently lifted in a study by Chumak et al. [16

A. V. Chumak, V. S. Tiberkevich, A. D. Karenowska, A. A. Serga, J. F. Gregg, A. N. Slavin, and B. Hillebrands, “All-linear time-reversal by a dynamic artificial crystal,” Nat. Comm. 1, 141 (2010). [CrossRef]

], performed in the context of magnonic crystals, a field which has much to contribute, and gain from, the field of PhCs. In particular, it was shown in [16

] that a pulse propagating in a homogeneous material can be time-reversed if the material is subject to a spatially-periodic, time-localized non-adiabatic modulation. This scheme is closely related to other non-adiabatic optical pumping based schemes which were studied theoretically earlier in slab waveguides [17

D. A. B Miller, “Time reversal of optical pulses by four-wave mixing,” Opt. Lett. 5, 300–302 (1980). [CrossRef] [PubMed]

, 18

M. Tsang and D. Psaltis, “Spectral phase conjugation with cross-phase modulation compensation,” Opt. Exp. 12, 2207–2219 (2004). [CrossRef]

], atomic vapours [19

A. B. Matsko, Y. V. Rostovtsev, O. Kocharovskaya, A. S. Zibrov, and M. O. Scully, “Nonadiabatic approach to quantum optical information storage,” Phys. Rev. A 64, 043809 (2001). [CrossRef]

], spin wave systems [20

G. A. Melkov, A. A. Serga, V. S. Tiberkevich, A. N. Oliynyk, and A. N. Slavin, “Wave Front Reversal of a Dipolar Spin Wave Pulse in a Non-Stationary Three-Wave Parametric Interaction,” Phys. Rev. Lett. 84, 3438–3441 (2000). [CrossRef] [PubMed]

], high-dimension PhCs [21

L. Tkeshelashvili and K. Busch, “Nonlinear three-wave interaction in photonic crystals,” Appl. Phys. B 81, 225–229 (2005). [CrossRef]

] and especially to our recent detailed studies of zero-gap photonic-crystals [22

Y. Sivan and J. B. Pendry, “Time-reversal in dynamically-tuned zero-gap periodic systems,” Phys. Rev. Lett. , 106, 193902 (2011). [CrossRef] [PubMed]

, 23

Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” submitted; available on ArXiv at http://arxiv.org/abs/1105.5583.

In this article, we study the time-reversal scheme of [16

] theoretically and compare it to previous non-adiabatic modulation-based schemes; we focus on time-reversal of ultrashort optical pulses, which as noted above, still has not been demonstrated experimentally. We first give a simple heuristic explanation of the dynamically-tuned PhC-based schemes [16

, 22

Y. Sivan and J. B. Pendry, “Time-reversal in dynamically-tuned zero-gap periodic systems,” Phys. Rev. Lett. , 106, 193902 (2011). [CrossRef] [PubMed]

, 23

Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” submitted; available on ArXiv at http://arxiv.org/abs/1105.5583.

] showing that the reversal originates from switching-on a bulk PhC mirror (Section 1). Then, we study the reversal process in [16

] analytically, derive a simple explicit formula for the reversal efficiency in the weak-coupling limit and compare it with existing techniques (Section 2). We then briefly discuss the implementation for optical electromagnetic pulses (Section 3).

1. Principles of time-reversal using a switchable mirror

In order to understand the reversal schemes of [16

, 22

Y. Sivan and J. B. Pendry, “Time-reversal in dynamically-tuned zero-gap periodic systems,” Phys. Rev. Lett. , 106, 193902 (2011). [CrossRef] [PubMed]

, 23

Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” submitted; available on ArXiv at http://arxiv.org/abs/1105.5583.

], it is beneficial to adopt a somewhat heuristic interpretation. Recall that when a pulse is reflected by a (standard) mirror, its spatial components change their propagation direction at different times, i.e., the leading edge first and trailing edge last. Thus, the pulse effectively undergoes a U-turn, i.e., the leading edge remains the leading edge etc.. Now, imagine that one could change of direction of the pulse propagation at all points in space and at the same time. Then, obviously, the leading edge will become the trailing edge, and vice versa, i.e., the pulse is (time-) reversed.

In order to perform such an extreme manipulation, one needs to abruptly reduce the transmissivity of the medium, preferably to zero, everywhere is space and for a spectral band as wide as possible. Possibly the simplest way to do that would be to open a frequency bandgap by periodically modulating the material properties. Heuristically, when the bandgap is turned on, the wave cannot propagate in any direction. Instead, the forward waves are then repeatedly converted to backward waves, then back to forward waves and vice versa. If one re-establishes the transmissivity once most of the energy of the forward wave has been converted to a backward wave, then a time-reversed pulse is released backwards. In a sense, this procedure transforms a perfectly transmitting medium into a “volume” mirror. Accordingly, in what follows we refer to these schems as switchable mirror (SM) -based reversal schemes.

This heuristic explanation clarifies why the zero-gap-based switchable-mirror (ZGSM) is equivalent to a homogeneous-medium-based switchable-mirror (HSM). Indeed, unlike a finite-gap system, in both structures all the incident light is perfectly admitted. In addition, in both structures a gap is opened due to the modulation (see [16

, 23

Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” submitted; available on ArXiv at http://arxiv.org/abs/1105.5583.

]). It is also implied by the heuristic explanation (and later proved analytically in Section 2), that in both ZGSM and HSM, although the wave-front is reversed, it is not conjugated. Thus, the scheme can lead to perfect time-reversal only if it is complemented by a consequent step of phase-conjugation, e.g., via nearly-degenerate 4WM [22

Y. Sivan and J. B. Pendry, “Time-reversal in dynamically-tuned zero-gap periodic systems,” Phys. Rev. Lett. , 106, 193902 (2011). [CrossRef] [PubMed]

, 23

Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” submitted; available on ArXiv at http://arxiv.org/abs/1105.5583.

]. As shown in Section 2, this may also be beneficial in order to make the scheme more efficient.

The advantage of the SM-based schemes is allowing to reverse pulses of almost unlimited wide spectrum, the only limitation being the shortness of the modulation. Also, as noted in [16

, 22

Y. Sivan and J. B. Pendry, “Time-reversal in dynamically-tuned zero-gap periodic systems,” Phys. Rev. Lett. , 106, 193902 (2011). [CrossRef] [PubMed]

], the new reversal schemes, which require only a periodic modulation rather than complex optics-specific concepts, open the way for time-reversal in other wave systems for which time-reversal was not accessible before, such as atomic physics [19

A. B. Matsko, Y. V. Rostovtsev, O. Kocharovskaya, A. S. Zibrov, and M. O. Scully, “Nonadiabatic approach to quantum optical information storage,” Phys. Rev. A 64, 043809 (2001). [CrossRef]

] and quantum computing [10

F. M. Cucchietti, “Time-reversal in an optical lattice,” J. Opt. Soc. Am. B 27, 30–35 (2010). [CrossRef]

Importantly, the HSM has several advantages over the ZGSM. First, the former is obviously simpler as essentially no fabrication is required; its performance is also practically insensitive to the modulation spatial pattern. Second, our analysis (Eq. (16) below) shows that the reversal effeciency in a HSM is about an order of magnitude higher than in a ZGSM. Third, while the frequency conversion performed in a ZGSM is purely vertical [22

Y. Sivan and J. B. Pendry, “Time-reversal in dynamically-tuned zero-gap periodic systems,” Phys. Rev. Lett. , 106, 193902 (2011). [CrossRef] [PubMed]

, 23

Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” submitted; available on ArXiv at http://arxiv.org/abs/1105.5583.

], the frequency conversion in a HSM involves also a change in the sign of the carrier wavevector, i.e., the dynamic grating provides the momentum to allow a horizontal transition between positive to negative momentum [16

]. As a result, the out-coupling of pulses from the HSM is not plagued by reflections at the boundaries, as may happen in a ZGSM. A final advantage of the HSM is that by choosing the proper modulation pattern, the reversal can be performed for any angle of incidence, for plane-waves as well as for beams, for high dimensional gratings or waveguide structures as well as for any incident carrier frequency.

2. Analysis

In this Section we derive the coupled envelope equations and solve them analytically in the weak-coupling limit. Since the former step is very similar to the derivations in [24

C. M. de Sterke and J. E. Sipe, in Prog. in Opt. , Vol. XXXIV, (North-Holland, Amsterdam 1994).

], and latter step is almost identical to the procedure detailed in [22

Y. Sivan and J. B. Pendry, “Time-reversal in dynamically-tuned zero-gap periodic systems,” Phys. Rev. Lett. , 106, 193902 (2011). [CrossRef] [PubMed]

, 23

Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” submitted; available on ArXiv at http://arxiv.org/abs/1105.5583.

], here, the derivation and solution are given quite briefly.

Consider an electromagnetic plane-wave pulse propagating along the x-direction in a homogeneous medium which is time-modulated in the following manner

ɛ (x, t) = n_{0}^{2} + Δ ɛ m (t - t_{0}) ɛ_{m} (x) .

(1)

Here, n ₀ is the average refractive index, the modulation is spatially periodic, i.e.,

ɛ_{m} (x) = ɛ_{m} (x + d), \int_{0}^{d} ɛ_{m} (x) d x = 0, m a x [ɛ_{m} (x)] = 1,

(2)

and time-localized around t ₀ with max[m(t – t ₀)] = m(t ₀) = 1, i.e., the modulation essentially turns on a periodic grating perpendicular to the direction of propagation of the pulse. In this case, the Maxwell equations reduce to the 1D wave equation

\partial_{x x} E (x, t) = \frac{1}{c^{2}} \partial_{t t} [ɛ (x, t) E (x, t)] .

(3)

In order to solve Eq. (3), we follow the derivation given in [24

C. M. de Sterke and J. E. Sipe, in Prog. in Opt. , Vol. XXXIV, (North-Holland, Amsterdam 1994).

], used previously to study the propagation of optical pulses in a photonic crystal with a cubic (Kerr) nonlinear response. As an aside, we note that unlike our previous studies of time-reversal in a zero-gap photonic crystal system [22

Y. Sivan and J. B. Pendry, “Time-reversal in dynamically-tuned zero-gap periodic systems,” Phys. Rev. Lett. , 106, 193902 (2011). [CrossRef] [PubMed]

, 23

Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” submitted; available on ArXiv at http://arxiv.org/abs/1105.5583.

], which can only be studied within the deep-grating formalism [25

C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E 54, 1969–1989 (1996). [CrossRef]

], here, we can use the shallow-grating analysis [24

C. M. de Sterke and J. E. Sipe, in Prog. in Opt. , Vol. XXXIV, (North-Holland, Amsterdam 1994).

]. The latter is much simpler to derive, and does not require the use of multiple scales analysis and calculation of the Floquet-Bloch modes which are required in [22

Y. Sivan and J. B. Pendry, “Time-reversal in dynamically-tuned zero-gap periodic systems,” Phys. Rev. Lett. , 106, 193902 (2011). [CrossRef] [PubMed]

, 23

Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” submitted; available on ArXiv at http://arxiv.org/abs/1105.5583.

We assume the field can be written as

E (x, t) = [E^{+} (x, t) e^{i k_{0} x} + E^{-} (x, t) e^{- {i k}_{0} x}] e^{- i ω_{0} t} + c . c .,

(4)

where E ^± represent the Slowly-Varying Envelopes (SVEs) of the forward and backward field compoenets, respectively; ω ₀ = ck ₀, k ₀ = 2πn ₀/λ_v and n ₀ are the carrier frequency, wavevector and refractive index of the medium, respectively, with λ_v being the vacuum wavelength. Substituting the ansatz (4) into Eq. (3), neglecting the c.c. terms and the second order derivative terms, and removing the factor e ^{−iω
₀
t} gives

\begin{matrix} 2 i k_{0} \partial_{x} E^{+} & + & 2 i ω_{0} \frac{n_{0}^{2}}{c^{2}} \partial_{t} E^{+} = 2 i (k_{0} \partial_{x} E^{-} - ω_{0} \frac{n_{0}^{2}}{c^{2}} \partial_{t} E^{-}) e^{- 2 i k_{0} x} \\ + & Δ ɛ \frac{ɛ_{m} (x)}{c^{2}} [2 \partial_{t} m (\partial_{t} E^{+} - i ω_{0} E^{+}) - m (2 i ω_{0} \partial_{t} E^{+} + ω_{0}^{2} E^{+})] \\ + & Δ ɛ \frac{ɛ_{m} (x)}{c^{2}} [2 \partial_{t} m (\partial_{t} E^{-} - i ω_{0} E^{-}) - m (2 i ω_{0} \partial_{t} E^{-} + ω_{0}^{2} E^{-})] e^{- 2 i k_{0} x} . \end{matrix}

(5)

The spatial modulation ɛ_m (x) couples the forward and backward field components. When k ₀ is close to the first bandgap, i.e., when k ₀ = k ^(g) + δk (with k ^(g) ≡ π/d or equivalently,

λ_{v}^{(g)} = 2 n_{0} d

), the j = ±1 components of the grating are close to the phase mismatch between the forward and backward field components, so that the coupling becomes most efficient. Following [24

C. M. de Sterke and J. E. Sipe, in Prog. in Opt. , Vol. XXXIV, (North-Holland, Amsterdam 1994).

], we now expand the spatial part of the modulation as a Fourier series as follows

ɛ_{m} (x) = \sum_{j = 1}^{\infty} ɛ_{m}^{(j)} e^{i \frac{2 π j x}{d}} + c . c ., ɛ_{m}^{(j)} = \frac{1}{d} \int_{0}^{d} ɛ_{m} (x) e^{i \frac{2 π j x}{d}} d x .

(6)

For a weak grating, Δɛ ≪ 1, it is justified to take only the j = ±1 components of the grating [24

C. M. de Sterke and J. E. Sipe, in Prog. in Opt. , Vol. XXXIV, (North-Holland, Amsterdam 1994).

]; this is equivalent to setting

ɛ_{m} (x) = 2 ɛ_{m}^{(1)} cos (\frac{2 π x}{d}) .

(7)

Substituting Eq. (7) in Eq. (5), neglecting all the fast-oscillating terms and separating into two sets of equations gives

\begin{matrix} 2 i k_{0} \partial_{x} E^{+} (x, t) & + & 2 (i k_{0} \frac{n_{0}}{c} - (\partial_{t} m - i ω_{0} m) \frac{Δ ɛ ɛ_{m}^{(1)}}{c^{2}} e^{- 2 i δ k x}) \partial_{t} E^{+} \\ = & Δ ɛ \frac{ɛ_{m}^{(1)}}{c^{2}} e^{- 2 i δ k x} ω_{0} (ω_{0} m + 2 i \partial_{t} m) E^{-}, \end{matrix}

(8)

\begin{matrix} - 2 i k_{0} \partial_{x} E^{-} (x, t) & + & 2 (i k_{0} \frac{n_{0}}{c} - (\partial_{t} m - i ω_{0} m) \frac{Δ ɛ ɛ_{m}^{(1) *}}{c^{2}} e^{2 i δ k x}) \partial_{t} E^{-} \\ = & Δ ɛ \frac{ɛ_{m}^{(1) *}}{c^{2}} e^{2 i δ k x} ω_{0} (ω_{0} m + 2 i \partial_{t} m) E^{+} . \end{matrix}

(9)

For modulations longer than the pulse period, the dominant term on the Right-Hand-Side is the first. Furthermore, for a weak modulation, one can neglect the correction to the pulse velocity, i.e., the second term inside the parentheses on the Left-Hand-Side. These assumptions lead to the following coupled equations

\partial_{x} E^{+} (x, t) + \frac{n_{0}}{c} \partial_{t} E^{+} = i κ e^{- 2 i δ k x} m (t) E^{-},

(10)

\partial_{x} E^{-} (x, t) - \frac{n_{0}}{c} \partial_{t} E^{-} = - i κ^{*} e^{2 i δ k x} m (t) E^{+},

(11)

where

κ \equiv Δ ɛ \frac{ɛ_{m}^{(1)}}{2 c n_{0}} ω_{0},

(12)

is a complex coupling coefficient. The final form of the coupled equations (10)–(11) is the same as in [24

C. M. de Sterke and J. E. Sipe, in Prog. in Opt. , Vol. XXXIV, (North-Holland, Amsterdam 1994).

] except for the nature of the modulation. Indeed, whereas in [24

C. M. de Sterke and J. E. Sipe, in Prog. in Opt. , Vol. XXXIV, (North-Holland, Amsterdam 1994).

] the modulation is induced by the traveling pulses and thus is spatio-temporally-localized, in the current context, the modulation occurs everywhere at the same time, but for a brief moment.

In order to solve equations (10)–(11), we follow the procedure outlined in [22

Y. Sivan and J. B. Pendry, “Time-reversal in dynamically-tuned zero-gap periodic systems,” Phys. Rev. Lett. , 106, 193902 (2011). [CrossRef] [PubMed]

, 23

Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” submitted; available on ArXiv at http://arxiv.org/abs/1105.5583.

]. Specifically, we first transform each equation into a frame moving with each of the pulses, namely, we define x ^(f,b) ≡ x ∓ vt. We then assume that the coupling is weak and neglect the coupling term on the RHS of Eq. (10). In this case, the solution of Eq. (10) is simply

E^{+} (x^{(f)}, t) = E^{+} (x - v t, 0) \equiv E_{0}^{+} (t),

(13)

where

E_{0}^{+} (t)

is the incident pulse profile. Substituting in Eq. (11) then gives

E^{-} (x^{(b)}, t) = i v κ^{*} e^{2 i δ k x^{(b)}} \int_{- \infty}^{t} e^{- 2 i δ k v t'} m (t' - t_{0}) E^{+} (x^{(b)} - 2 v t') d t' .

(14)

Eq. (14) shows that the backward wave is given by the convolution of the forward wave with the modulation. Thus, as noted in [22

Y. Sivan and J. B. Pendry, “Time-reversal in dynamically-tuned zero-gap periodic systems,” Phys. Rev. Lett. , 106, 193902 (2011). [CrossRef] [PubMed]

, 23

Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” submitted; available on ArXiv at http://arxiv.org/abs/1105.5583.

], this reversal scheme yields a wave-front reversal rather than a complete time-reversal (which requires also the conjugation of the envelope).

The convolution integral can be solved exactly for a unit amplitude Gaussian pulse and a Gaussian modulation

E_{0}^{+} (\frac{x}{v T_{p}}) = e^{- \frac{x^{2}}{v^{2} T_{p}^{2}}}, m (t - t_{0}) = e^{- \frac{{(t - t_{0})}^{2}}{T_{m o d}^{2}}} .

(15)

In this case, following [23

Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” submitted; available on ArXiv at http://arxiv.org/abs/1105.5583.

], it can be shown that for t ₀ = 0 and δk = 0, the wave-front of the reversed component is given by

| E^{-} (x + v t) | = \sqrt{π} \frac{Δ n}{n_{0}} ω_{0} \frac{T_{m o d} T_{p}}{\sqrt{T_{p}^{2} + 4 T_{m o d}^{2}}} E_{0}^{+} (\frac{x + v t}{v \sqrt{T_{p}^{2} + 4 T_{m o d}^{2}}}) .

(16)

where

Δ n ≅ \frac{| ɛ_{m}^{(1)} | Δ ɛ}{2 n_{0}}

is the depth of the refractive index modulations. Note that at the time at which the modulation is maximal (i.e., at t = t ₀ = 0), the width of the gap opened by the modulation is given approximately by

\frac{2 c}{d} \frac{Δ n}{n_{0}}

[26

P. Yeh, Optical Waves in Layered Media , (Wiley-Interscience, 2nd edition 2005).

]. Thus, the reversal efficiency is proportional to the width of the gap opened by the modulation, in agreement with our interpretation of the scheme (see Section 1).

By comparing the reversal efficiency (16) to that obtained in a 1D zero-gap PhC [23

Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” submitted; available on ArXiv at http://arxiv.org/abs/1105.5583.

, Ch. VI], it is seen that the reversal in the HSM is about one to two orders of magnitude more efficient (in terms of intensity) than in a ZGSM. Furthermore, in comparison to the early suggestion of time-reversal via non-adiabatic 4WM [17

D. A. B Miller, “Time reversal of optical pulses by four-wave mixing,” Opt. Lett. 5, 300–302 (1980). [CrossRef] [PubMed]

, 18

M. Tsang and D. Psaltis, “Spectral phase conjugation with cross-phase modulation compensation,” Opt. Exp. 12, 2207–2219 (2004). [CrossRef]

], the reversal efficiency in our case is almost an order of magnitude more efficient. Thus, by splitting the time-reversal into two consecutive steps (envelope reversal via a SM and phase conjugation), one gains flexibility, simplicity (e.g., by allowing to work with a single pump pulse) and efficiency. In fact, with relatively small intensity, the phase conjugation step can be used also for amplifying the reversed signal, thus allowing for 100% or more time-reversal of ultrashort optical pulses. Finally, the SM-based techniques can be implemented with a wider variety of modulation techniques, including linear modulators [14

M. F. Yanik and S. Fan, “Time-reversal of light with linear optics and modulators,” Phys. Rev. Lett. 93, 173903 (2004). [CrossRef] [PubMed]

, 15

S. Longhi, “Stopping and time-reversal of light in dynamic photonic structures via Bloch oscillations,” Phys. Rev. E 75, 026606 (2007). [CrossRef]

, 16

3. Implementation

As discussed in detail in [23

Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” submitted; available on ArXiv at http://arxiv.org/abs/1105.5583.

], the pulse duration dictates the required modulation technique. For pulses longer than a few picoseconds, the non-adiabatic modulation can be done electronically, e.g., with linear modulators. Thus, a time-reversal mirror can be built from a standard material such as LiNbO ₃ which is spatially-modulated in a periodic manner. A 100% reversal efficiency can be easily obtained using index modulations on the scale of 10⁻³ [22

Y. Sivan and J. B. Pendry, “Time-reversal in dynamically-tuned zero-gap periodic systems,” Phys. Rev. Lett. , 106, 193902 (2011). [CrossRef] [PubMed]

, 23

Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” submitted; available on ArXiv at http://arxiv.org/abs/1105.5583.

For shorter pulses, the required modulations can be performed via a nonlinear process such as Cross-Phase Modulation or carrier injection in a pump-probe configuration. One way to ensure that the pump induces an index modulation in the x direction only is to confine the probe (signal) into a thin waveguide and send a much wider and much shorter intensity-modulated pump at right angles to the waveguide (see e.g., [17

D. A. B Miller, “Time reversal of optical pulses by four-wave mixing,” Opt. Lett. 5, 300–302 (1980). [CrossRef] [PubMed]

, 18

M. Tsang and D. Psaltis, “Spectral phase conjugation with cross-phase modulation compensation,” Opt. Exp. 12, 2207–2219 (2004). [CrossRef]

]). The intensity modulation can be performed by sending a single pump pulse through a waveguide array or by interference of two pump beams. An alternative is to use a structure with a periodically-varying Kerr coefficient [27

G. Fibich, Y. Sivan, and M. I. Weinstein, “Bound states of nonlinear Schrödinger equations with a periodic nonlinear microstructure,” Physica D 217, 31–57 (2006). [CrossRef]

, 28

Y. Sivan, G. Fibich, and M. I. Weinstein, “Waves in nonlinear lattices: ultrashort optical pulses and bose-einstein condensates,” Phys. Rev. Lett. 97, 193902 (2006). [CrossRef] [PubMed]

], preferably, with a changing sign (see [29

B. A. Malomed, Y. V. Kartashov, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–306 (2011). [CrossRef]

] for a review).

References and links

1.	M. Fink, “Time-reversed acoustics,” Scientific American 91, (November 1999).
2.	M. Fink, “Acoustic time-reversal mirrors: Imaging of complex media with acoustic and seismic waves,” Topics in Applied Physics 84, 17–43 (2002). [CrossRef]
3.	J. Aullbach, B. Gjonaj, P. M. Johnson, A. M. Mosk, and A. Lagendijk, “Control of light transmission through opaque scattering media in space and time,” Phys. Rev. Lett. 106, 103901 (2011). [CrossRef]
4.	O. Katz, Y. Bromberg, E. Small, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Phot. 5, 372–377 (2011). [CrossRef]
5.	D. M. Pepper, Laser handbook , Vol. 4, (North-Holland Physics, Amsterdam 1988).
6.	J. B. Pendry, “Time-reversal and negative refraction,” Science 322, 71–73 (2008). [CrossRef] [PubMed]
7.	X. Li and M. I. Stockman, “Highly efficient spatio-temporal coherent control in nanoplasmonics on a nanometer-femtosecond scale by time-reversal,” Phys. Rev. B 77, 195109 (2008). [CrossRef]
8.	Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Phot. 2, 110–115 (2008). [CrossRef]
9.	Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, “Transmission of electrical signals by spin-wave interconversion in a magnetic insulator,” Nature 464, 262–266 (2010). [CrossRef] [PubMed]
10.	F. M. Cucchietti, “Time-reversal in an optical lattice,” J. Opt. Soc. Am. B 27, 30–35 (2010). [CrossRef]
11.	A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, “Femtosecond spectral holography,” IEEE J. Qu. Electron. 28, 2251–2261 (1992). [CrossRef]
12.	D. Marom, D. Panasenko, R. Rokitski, P.-C. Sun, and Y. Fainman, “Time-reversal of ultrafast waveforms by wave mixing of spectrally decomposed waves,” Opt. Lett. 25, 132–134 (2000). [CrossRef]
13.	O. Kuzucu, Y. Okawachi, R. Salem, M. A. Foster, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Spectral phase conjugation via temporal imaging,” Opt. Exp. 17, 20605–20614 (2009). [CrossRef]
14.	M. F. Yanik and S. Fan, “Time-reversal of light with linear optics and modulators,” Phys. Rev. Lett. 93, 173903 (2004). [CrossRef] [PubMed]
15.	S. Longhi, “Stopping and time-reversal of light in dynamic photonic structures via Bloch oscillations,” Phys. Rev. E 75, 026606 (2007). [CrossRef]
16.	A. V. Chumak, V. S. Tiberkevich, A. D. Karenowska, A. A. Serga, J. F. Gregg, A. N. Slavin, and B. Hillebrands, “All-linear time-reversal by a dynamic artificial crystal,” Nat. Comm. 1, 141 (2010). [CrossRef]
17.	D. A. B Miller, “Time reversal of optical pulses by four-wave mixing,” Opt. Lett. 5, 300–302 (1980). [CrossRef] [PubMed]
18.	M. Tsang and D. Psaltis, “Spectral phase conjugation with cross-phase modulation compensation,” Opt. Exp. 12, 2207–2219 (2004). [CrossRef]
19.	A. B. Matsko, Y. V. Rostovtsev, O. Kocharovskaya, A. S. Zibrov, and M. O. Scully, “Nonadiabatic approach to quantum optical information storage,” Phys. Rev. A 64, 043809 (2001). [CrossRef]
20.	G. A. Melkov, A. A. Serga, V. S. Tiberkevich, A. N. Oliynyk, and A. N. Slavin, “Wave Front Reversal of a Dipolar Spin Wave Pulse in a Non-Stationary Three-Wave Parametric Interaction,” Phys. Rev. Lett. 84, 3438–3441 (2000). [CrossRef] [PubMed]
21.	L. Tkeshelashvili and K. Busch, “Nonlinear three-wave interaction in photonic crystals,” Appl. Phys. B 81, 225–229 (2005). [CrossRef]
22.	Y. Sivan and J. B. Pendry, “Time-reversal in dynamically-tuned zero-gap periodic systems,” Phys. Rev. Lett. , 106, 193902 (2011). [CrossRef] [PubMed]
23.	Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” submitted; available on ArXiv at http://arxiv.org/abs/1105.5583.
24.	C. M. de Sterke and J. E. Sipe, in Prog. in Opt. , Vol. XXXIV, (North-Holland, Amsterdam 1994).
25.	C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E 54, 1969–1989 (1996). [CrossRef]
26.	P. Yeh, Optical Waves in Layered Media , (Wiley-Interscience, 2nd edition 2005).
27.	G. Fibich, Y. Sivan, and M. I. Weinstein, “Bound states of nonlinear Schrödinger equations with a periodic nonlinear microstructure,” Physica D 217, 31–57 (2006). [CrossRef]
28.	Y. Sivan, G. Fibich, and M. I. Weinstein, “Waves in nonlinear lattices: ultrashort optical pulses and bose-einstein condensates,” Phys. Rev. Lett. 97, 193902 (2006). [CrossRef] [PubMed]
29.	B. A. Malomed, Y. V. Kartashov, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–306 (2011). [CrossRef]

OCIS Codes
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons
(190.2055) Nonlinear optics : Dynamic gratings
(250.4110) Optoelectronics : Modulators

ToC Category:
Nonlinear Optics

History
Original Manuscript: May 31, 2011
Revised Manuscript: June 20, 2011
Manuscript Accepted: June 20, 2011
Published: July 13, 2011

Citation
Yonatan Sivan and John B. Pendry, "Broadband time-reversal of optical pulses using a switchable photonic-crystal mirror," Opt. Express 19, 14502-14507 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-15-14502

Sort: Author | Year | Journal | Reset

References

M. Fink, “Time-reversed acoustics,” Scientific American 91, (November1999).
M. Fink, “Acoustic time-reversal mirrors: Imaging of complex media with acoustic and seismic waves,” Topics in Applied Physics 84, 17–43 (2002). [CrossRef]
J. Aullbach, B. Gjonaj, P. M. Johnson, A. M. Mosk, and A. Lagendijk, “Control of light transmission through opaque scattering media in space and time,” Phys. Rev. Lett. 106, 103901 (2011). [CrossRef]
O. Katz, Y. Bromberg, E. Small, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Phot. 5, 372–377 (2011). [CrossRef]
D. M. Pepper, Laser handbook , Vol. 4, (North-Holland Physics, Amsterdam1988).
J. B. Pendry, “Time-reversal and negative refraction,” Science 322, 71–73 (2008). [CrossRef] [PubMed]
X. Li and M. I. Stockman, “Highly efficient spatio-temporal coherent control in nanoplasmonics on a nanometer-femtosecond scale by time-reversal,” Phys. Rev. B 77, 195109 (2008). [CrossRef]
Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Phot. 2, 110–115 (2008). [CrossRef]
Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, “Transmission of electrical signals by spin-wave interconversion in a magnetic insulator,” Nature 464, 262–266 (2010). [CrossRef] [PubMed]
F. M. Cucchietti, “Time-reversal in an optical lattice,” J. Opt. Soc. Am. B 27, 30–35 (2010). [CrossRef]
A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, “Femtosecond spectral holography,” IEEE J. Qu. Electron. 28, 2251–2261 (1992). [CrossRef]
D. Marom, D. Panasenko, R. Rokitski, P.-C. Sun, and Y. Fainman, “Time-reversal of ultrafast waveforms by wave mixing of spectrally decomposed waves,” Opt. Lett. 25, 132–134 (2000). [CrossRef]
O. Kuzucu, Y. Okawachi, R. Salem, M. A. Foster, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Spectral phase conjugation via temporal imaging,” Opt. Exp. 17, 20605–20614 (2009). [CrossRef]
M. F. Yanik and S. Fan, “Time-reversal of light with linear optics and modulators,” Phys. Rev. Lett. 93, 173903 (2004). [CrossRef] [PubMed]
S. Longhi, “Stopping and time-reversal of light in dynamic photonic structures via Bloch oscillations,” Phys. Rev. E 75, 026606 (2007). [CrossRef]
A. V. Chumak, V. S. Tiberkevich, A. D. Karenowska, A. A. Serga, J. F. Gregg, A. N. Slavin, and B. Hillebrands, “All-linear time-reversal by a dynamic artificial crystal,” Nat. Comm. 1, 141 (2010). [CrossRef]
D. A. B Miller, “Time reversal of optical pulses by four-wave mixing,” Opt. Lett. 5, 300–302 (1980). [CrossRef] [PubMed]
M. Tsang and D. Psaltis, “Spectral phase conjugation with cross-phase modulation compensation,” Opt. Exp. 12, 2207–2219 (2004). [CrossRef]
A. B. Matsko, Y. V. Rostovtsev, O. Kocharovskaya, A. S. Zibrov, and M. O. Scully, “Nonadiabatic approach to quantum optical information storage,” Phys. Rev. A 64, 043809 (2001). [CrossRef]
G. A. Melkov, A. A. Serga, V. S. Tiberkevich, A. N. Oliynyk, and A. N. Slavin, “Wave Front Reversal of a Dipolar Spin Wave Pulse in a Non-Stationary Three-Wave Parametric Interaction,” Phys. Rev. Lett. 84, 3438–3441 (2000). [CrossRef] [PubMed]
L. Tkeshelashvili and K. Busch, “Nonlinear three-wave interaction in photonic crystals,” Appl. Phys. B 81, 225–229 (2005). [CrossRef]
Y. Sivan and J. B. Pendry, “Time-reversal in dynamically-tuned zero-gap periodic systems,” Phys. Rev. Lett. , 106, 193902 (2011). [CrossRef] [PubMed]
Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” submitted; available on ArXiv at http://arxiv.org/abs/1105.5583 .
C. M. de Sterke and J. E. Sipe, in Prog. in Opt. , Vol. XXXIV, (North-Holland, Amsterdam1994).
C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E 54, 1969–1989 (1996). [CrossRef]
P. Yeh, Optical Waves in Layered Media , (Wiley-Interscience, 2nd edition2005).
G. Fibich, Y. Sivan, and M. I. Weinstein, “Bound states of nonlinear Schrödinger equations with a periodic nonlinear microstructure,” Physica D 217, 31–57 (2006). [CrossRef]
Y. Sivan, G. Fibich, and M. I. Weinstein, “Waves in nonlinear lattices: ultrashort optical pulses and bose-einstein condensates,” Phys. Rev. Lett. 97, 193902 (2006). [CrossRef] [PubMed]
B. A. Malomed, Y. V. Kartashov, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–306 (2011). [CrossRef]

Ando, K.
Aullbach, J.
Bromberg, Y.
Busch, K.
Chumak, A. V.
Cucchietti, F. M.
de Sterke, C. M.
Fainman, Y.
Fan, S.
Feld, M. S.
Fibich, G.
Fink, M.
Foster, M. A.
Gaeta, A. L.
Gjonaj, B.
Gregg, J. F.
Harii, K.
Hillebrands, B.
Johnson, P. M.
Kajiwara, Y.
Karenowska, A. D.
Kartashov, Y. V.
Katz, O.
Kawai, H.
Kocharovskaya, O.
Kuzucu, O.
Lagendijk, A.
Leaird, D. E.
Li, X.
Lipson, M.
Longhi, S.
Maekawa, S.
Malomed, B. A.
Marom, D.
Matsko, A. B.
Melkov, G. A.
Miller, D. A. B
Mizuguchi, M.
Mosk, A. M.
Ohe, J.
Okawachi, Y.
Oliynyk, A. N.
Paek, E. G.
Panasenko, D.
Pendry, J. B.
Pepper, D. M.
Psaltis, D.
Reitze, D. H.
Rokitski, R.
Rostovtsev, Y. V.
Saitoh, E.
Salem, R.
Salinas, D. G.
Scully, M. O.
Serga, A. A.
Silberberg, Y.
Sipe, J. E.
Sivan, Y.
Slavin, A. N.
Small, E.
Stockman, M. I.
Sun, P.-C.
Takahashi, S.
Takanashi, K.
Tiberkevich, V. S.
Tkeshelashvili, L.
Torner, L.
Tsang, M.
Turner-Foster, A. C.
Uchida, K.
Umezawa, H.
Weiner, A. M.
Weinstein, M. I.
Yang, C.
Yanik, M. F.
Yaqoob, Z.
Yeh, P.
Zibrov, A. S.

Appl. Phys. B

L. Tkeshelashvili and K. Busch, “Nonlinear three-wave interaction in photonic crystals,” Appl. Phys. B 81, 225–229 (2005). [CrossRef]

IEEE J. Qu. Electron.

A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, “Femtosecond spectral holography,” IEEE J. Qu. Electron. 28, 2251–2261 (1992). [CrossRef]

J. Opt. Soc. Am. B

F. M. Cucchietti, “Time-reversal in an optical lattice,” J. Opt. Soc. Am. B 27, 30–35 (2010). [CrossRef]

Nat. Comm.

A. V. Chumak, V. S. Tiberkevich, A. D. Karenowska, A. A. Serga, J. F. Gregg, A. N. Slavin, and B. Hillebrands, “All-linear time-reversal by a dynamic artificial crystal,” Nat. Comm. 1, 141 (2010). [CrossRef]

Nat. Phot.

O. Katz, Y. Bromberg, E. Small, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Phot. 5, 372–377 (2011). [CrossRef]
Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Phot. 2, 110–115 (2008). [CrossRef]

Nature

Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, “Transmission of electrical signals by spin-wave interconversion in a magnetic insulator,” Nature 464, 262–266 (2010). [CrossRef] [PubMed]

Opt. Exp.

O. Kuzucu, Y. Okawachi, R. Salem, M. A. Foster, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Spectral phase conjugation via temporal imaging,” Opt. Exp. 17, 20605–20614 (2009). [CrossRef]
M. Tsang and D. Psaltis, “Spectral phase conjugation with cross-phase modulation compensation,” Opt. Exp. 12, 2207–2219 (2004). [CrossRef]

Opt. Lett.

Phys. Rev. A

A. B. Matsko, Y. V. Rostovtsev, O. Kocharovskaya, A. S. Zibrov, and M. O. Scully, “Nonadiabatic approach to quantum optical information storage,” Phys. Rev. A 64, 043809 (2001). [CrossRef]

Phys. Rev. B

X. Li and M. I. Stockman, “Highly efficient spatio-temporal coherent control in nanoplasmonics on a nanometer-femtosecond scale by time-reversal,” Phys. Rev. B 77, 195109 (2008). [CrossRef]

Phys. Rev. E

C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E 54, 1969–1989 (1996). [CrossRef]
S. Longhi, “Stopping and time-reversal of light in dynamic photonic structures via Bloch oscillations,” Phys. Rev. E 75, 026606 (2007). [CrossRef]

Phys. Rev. Lett.

Y. Sivan, G. Fibich, and M. I. Weinstein, “Waves in nonlinear lattices: ultrashort optical pulses and bose-einstein condensates,” Phys. Rev. Lett. 97, 193902 (2006). [CrossRef] [PubMed]
G. A. Melkov, A. A. Serga, V. S. Tiberkevich, A. N. Oliynyk, and A. N. Slavin, “Wave Front Reversal of a Dipolar Spin Wave Pulse in a Non-Stationary Three-Wave Parametric Interaction,” Phys. Rev. Lett. 84, 3438–3441 (2000). [CrossRef] [PubMed]
Y. Sivan and J. B. Pendry, “Time-reversal in dynamically-tuned zero-gap periodic systems,” Phys. Rev. Lett. , 106, 193902 (2011). [CrossRef] [PubMed]
M. F. Yanik and S. Fan, “Time-reversal of light with linear optics and modulators,” Phys. Rev. Lett. 93, 173903 (2004). [CrossRef] [PubMed]
J. Aullbach, B. Gjonaj, P. M. Johnson, A. M. Mosk, and A. Lagendijk, “Control of light transmission through opaque scattering media in space and time,” Phys. Rev. Lett. 106, 103901 (2011). [CrossRef]

Physica D

G. Fibich, Y. Sivan, and M. I. Weinstein, “Bound states of nonlinear Schrödinger equations with a periodic nonlinear microstructure,” Physica D 217, 31–57 (2006). [CrossRef]

Rev. Mod. Phys.

B. A. Malomed, Y. V. Kartashov, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–306 (2011). [CrossRef]

Science

J. B. Pendry, “Time-reversal and negative refraction,” Science 322, 71–73 (2008). [CrossRef] [PubMed]

Scientific American

M. Fink, “Time-reversed acoustics,” Scientific American 91, (November1999).

Topics in Applied Physics

M. Fink, “Acoustic time-reversal mirrors: Imaging of complex media with acoustic and seismic waves,” Topics in Applied Physics 84, 17–43 (2002). [CrossRef]

Other

D. M. Pepper, Laser handbook , Vol. 4, (North-Holland Physics, Amsterdam1988).
Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” submitted; available on ArXiv at http://arxiv.org/abs/1105.5583 .
C. M. de Sterke and J. E. Sipe, in Prog. in Opt. , Vol. XXXIV, (North-Holland, Amsterdam1994).
P. Yeh, Optical Waves in Layered Media , (Wiley-Interscience, 2nd edition2005).

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Broadband time-reversal of optical pulses using a switchable photonic-crystal mirror

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Abstract

1. Principles of time-reversal using a switchable mirror

2. Analysis

3. Implementation

References and links

References

Appl. Phys. B

IEEE J. Qu. Electron.

J. Opt. Soc. Am. B

Nat. Comm.

Nat. Phot.

Nature

Opt. Exp.

Opt. Lett.

Phys. Rev. A

Phys. Rev. B

Phys. Rev. E

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Physica D

Rev. Mod. Phys.

Science

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Topics in Applied Physics

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