Adiabatic transfer of light via a continuum in optical waveguides

doi:10.1364/OL.34.002405

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Adiabatic transfer of light via a continuum in optical waveguides

F. Dreisow, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi »View Author Affiliations

Optics Letters, Vol. 34, Issue 16, pp. 2405-2407 (2009)
http://dx.doi.org/10.1364/OL.34.002405

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Abstract

The optical analog of the stimulated Raman adiabatic passage via a continuum is experimentally demonstrated for photonic tunneling in femtosecond laser written waveguides. The results clearly show that the mechanism of light transfer relies on destructive interference and on the existence of a photonic dark state.

Recently, adiabatic passage phenomena discovered in atomic and molecular physics, such as the stimulated Raman adiabatic passage (STIRAP) [1

K. Bergmann, H. Theuer, and B. W. Shore, Rev. Mod. Phys. 70, 1003 (1998). [CrossRef]

], have been proposed for controlling photonic tunneling. Concepts such as quantum interference, electromagnetically induced transparency, trapped states, and adiabatic passage have thus entered the photonic community (see, e.g., [2

A. E. Miroshnichenko and Y. S. Kivshar, Phys. Rev. E 72, 056611 (2005). [CrossRef]

, 3

E. Paspalakis, Opt. Commun. 258, 30 (2006). [CrossRef]

, 4

S. Longhi, Phys. Rev. E 73, 026607 (2006). [CrossRef]

, 5

P. Ginzburg and M. Orenstein, Opt. Express 14, 11312 (2006). [CrossRef] [PubMed]

]). Experimental demonstrations of robust light transfer among evanescently coupled optical waveguides, based on the optical analogs of three-level atomic and multilevel straddle STIRAPs, have been reported [6

S. Longhi, G. D. Valle, M. Ornigotti, and P. Laporta, Phys. Rev. B 76, 201101 (2007). [CrossRef]

, 7

G. Della Valle, M. Ornigotti, T. T. Fernandez, P. Laporta, S. Longhi, A. Coppa, and V. Foglietti, Appl. Phys. Lett. 92, 011106 (2008). [CrossRef]

], whereas the effects of nonlinearity on photonic STIRAP have been investigated [8

Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 101, 193901 (2008). [CrossRef] [PubMed]

]. Among the various extensions of STIRAP techniques, population transfer via a continuum represents perhaps the most counterintuitive process, which raised a lively debate in atomic physics about its practical feasibility (see, e.g., [9

C. E. Carroll and F. T. Hioe, Phys. Rev. Lett. 68, 3523 (1992). [CrossRef] [PubMed]

, 10

E. Paspalakis, M. Protopapas, and P. L. Knight, J. Phys. B 31, 775 (1998). [CrossRef]

, 11

R. G. Unanyan, N. V. Vitanov, B. W. Shore, and K. Bergmann, Phys. Rev. A 61, 043408 (2000). [CrossRef]

] and references therein), until the recent experimental demonstration reported in [12

T. Peters, L. P. Yatsenko, and T. Halfmann, Phys. Rev. Lett. 95, 103601 (2005). [CrossRef] [PubMed]

]. The STIRAP via a continuum is based on the existence of a trapped state embedded in the continuum and in the adiabatic evolution of the system in this state. Classical and quantum photonic analogs of the population trapping and the STIRAP in the continuum have been proposed very recently [13

S. Longhi, Phys. Rev. A 78, 013815 (2008). [CrossRef]

, 14

S. Longhi, Phys. Rev. A 79, 023811 (2009). [CrossRef]

, 15

S. Longhi, J. Mod. Opt. 56, 729 (2009). [CrossRef]

In this Letter we report what we believe to be the first experimental demonstration of the photonic STIRAP via a continuum. Adiabatic light transfer via a tight-binding continuum [15

S. Longhi, J. Mod. Opt. 56, 729 (2009). [CrossRef]

], reported in the present work, should not be confused with the multilevel straddle STIRAP—previously demonstrated in [7

G. Della Valle, M. Ornigotti, T. T. Fernandez, P. Laporta, S. Longhi, A. Coppa, and V. Foglietti, Appl. Phys. Lett. 92, 011106 (2008). [CrossRef]

]—where the problem of light transfer is basically reduced to a standard three-level STIRAP problem, with a resonance dressed state playing the role of the intermediate discrete state. The photonic structure designed for our experiment follows the proposal of [15

S. Longhi, J. Mod. Opt. 56, 729 (2009). [CrossRef]

] and consists of two (or more) weakly curved single-mode waveguides, which approach a waveguide array A from different sides as depicted in the geometry sketch in Fig. 1a for two waveguides

W_{1}

and

W_{2}

. The modes in the side waveguides are analogous to discrete atomic states, which can decay into a common tight-binding continuum represented by array A. Single-mode waveguides have been realized in a

L = 10 -
                  cm

-long fused silica sample by femtosecond laser waveguide writing [16

S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, J. Mod. Opt. 51, 2533 (2004). [CrossRef]

]. The same writing parameters have been used in the fabrication process so that all waveguides feature identical cross sections. The guides in array A are equally spaced by

d = 14 μ m

. The excitation of a side waveguide (e.g.,

W_{1}

W_{2}

) at

λ = 633 nm

is accomplished by direct fiber butt coupling. The propagation along the structure is monitored transversely [17

F. Dreisow, M. Heinrich, A. Szameit, S. Doering, S. Nolte, A. Tuennermann, S. Fahr, and F. Lederer, Opt. Express 16, 3474 (2008). [CrossRef] [PubMed]

] by acquiring the fluorescence emitted by nonbridging oxygen hole color centers generated during the writing process [18

A. Szameit, F. Dreisow, H. Hartung, S. Nolte, A. Tuennermann, and F. Lederer, Appl. Phys. Lett. 90, 241113 (2007). [CrossRef]

]. Although the height of the structure does not permit one to image the light diffused inside array A with this fluorescence microscopy, a precise measure of the flow of light along the side waveguides is possible owing to their short vertical displacement. To realize light transfer via adiabatic passage, waveguides

W_{1}

and

W_{2}

are weakly curved in the

x - z

plane normal to the array with bending profiles

x (z) = \pm α {(z - L / 2 \pm z_{0} / 2)}^{2} + d_{min}

, where

d_{min}

is the minimal

W - A

separation at the apex point and

2 α

is the inverse of the curvature radius. Correspondingly, the coupling rates

κ_{1} (z)

and

κ_{2} (z)

of waveguides

W_{1}

and

W_{2}

with their neighboring waveguides of the array change slowly with the propagation distance z. Light transport in the photonic structure of Fig. 1a is analogous to the quantum-mechanical decay of two discrete states coupled to a common tight-binding continuum as schematically shown in Fig. 1b. Efficient transfer of light from waveguide

W_{1}

W_{2}

with small losses in the continuum requires the existence of a dark (or trapped) photonic state, arising from a Fanolike destructive interference of decay channels, and the adiabatic evolution of this state under a counterintuitive pulse sequence of the atomic STIRAP. The latter is simply mimicked by assuming

κ_{1} / κ_{2} ⪡ 1

at the input plane and

κ_{1} / κ_{2} ⪢ 1

at the output plane of the sample as in [6

S. Longhi, G. D. Valle, M. Ornigotti, and P. Laporta, Phys. Rev. B 76, 201101 (2007). [CrossRef]

]. Indicating the modal amplitudes of light in waveguides

W_{1}

and

W_{2}

c_{1} (z)

and

c_{2} (z)

, the corresponding coupled-mode equations may be derived from the weak coupling limit after elimination of the continuum degrees of freedom and neglecting memory effects [13

S. Longhi, Phys. Rev. A 78, 013815 (2008). [CrossRef]

, 15

S. Longhi, J. Mod. Opt. 56, 729 (2009). [CrossRef]

]. For two identical waveguides, as in Fig. 1a, the coupled-mode equations read as [15

S. Longhi, J. Mod. Opt. 56, 729 (2009). [CrossRef]

]

i {\dot{c}}_{1} = - Δ_{11} c_{1} - Δ_{12} c_{2}, i {\dot{c}}_{2} = - Δ_{21} c_{1} - Δ_{22} c_{2},

(1)

where we have set

Δ_{11} = \frac{κ_{1}^{2} (z)}{2 κ}, Δ_{22} = \frac{κ_{2}^{2} (z)}{2 κ}, Δ_{12} = Δ_{21} = i^{m} \frac{κ_{1} (z) κ_{2} (z)}{2 κ} .

(2)

In Eq. (2), κ is the nearest-neighbor coupling rate of waveguides within array A,

4 κ

is the width of the tight-binding continuum, and the integer m is the number of waveguides that separate waveguides

W_{1}

and

W_{2}

in the array (

m = 0

W_{1}

and

W_{2}

dock to the same guide but from opposite sides of array A). As a single discrete state would decay irreversibly into the continuum [this follows from Eqs. (1) with either

κ_{1} = 0

κ_{2} = 0

], coupling of the continuum with the second discrete state may suppress the decay channels provided that a trapping state in the continuum is created. From Eqs. (1), it readily follows that a trapped state, given by

c_{1} = - i^{m} \frac{κ_{2} (z)}{\sqrt{κ_{1}^{2} (z) + κ_{2}^{2} (z)}}, c_{2} = \frac{κ_{1} (z)}{\sqrt{κ_{1}^{2} (z) + κ_{2}^{2} (z)}},

(3)

does exist whenever the integer m is an even number. Regarding Eqs. (3) with

κ_{1} / κ_{2} ⪡ 1

z = 0

and

κ_{1} / κ_{2} ⪢ 1

z = L

, a light beam injected into waveguide

W_{1}

at the input plane is transferred to waveguide

W_{2}

at the output plane by the adiabatic passage, with a small excitation of the continuum. This adiabatic transfer mechanism, analogous to the atomic STIRAP via a continuum, is clearly demonstrated in the fabricated system, in which we use the geometric parameters

α = 7 \times 10^{- 6} {mm}^{- 1}

z_{0} = 25 mm

, and

d_{min} = 12 μ m

. Figures 2a, 2b, 2c show a high-efficiency light transfer from waveguide

W_{1}

W_{2}

, with a small fraction of light power lost in the continuum, for the case of nondisplaced (i.e.,

m = 0

) waveguides [left panel in Fig. 1c]. The light transfer takes place mainly in the middle of the sample as the system adiabatically evolves once initially excited in its dark state as shown in the fluorescence image of Fig. 2a. The near-field image of the output facet in Fig. 2b yields a transfer efficiency estimate of 87%—where 13% of the total power lacks according to theory—into the continuum. Figure 2c depicts the evolution predicted by coupled-mode equations for

κ = 0.18 {mm}^{- 1}

and for coupling rates

κ_{1, 2}

calculated from the curvature profile

x (z)

using space-coupling dependence adapted from [19

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 101, 143602 (2008). [CrossRef] [PubMed]

], with maximum coupling rates

max [κ_{1}] = max [κ_{2}] = 0.27 {mm}^{- 1}

. The excitation of waveguide

W_{2}

(rather than

W_{1}

) at the input plane completely changes the transfer dynamics, and most of the light is lost in the continuum, as demonstrated in Figs. 2d, 2e, 2f. This case corresponds in fact to an intuitive pulse sequence of the atomic STIRAP, where the system is no more excited in its dark state at the input plane.

For

m = 1

[central panel in Fig. 1c], using the same geometrical parameters as above, no efficient light transfer can be observed neither in counterintuitive nor in intuitive regimes (see Fig. 3 ), with most of the light power rapidly decaying into the continuum. This result is in agreement with theory and demonstrates that, in the absence of a trapped state, population transfer via a continuum is not possible. Adapting the geometrical parameters would help in improving the transfer only very marginally. The STIRAP transfer process becomes again efficient when the displacement m is increased to

m = 2

, according to Eq. (3). Interestingly, this process works not only for a single output channel

W_{2}

but also for two output channels

W_{2}

and

W_{3}

, which are equally displaced by

m = \pm 2

as shown in the right panel of Fig. 1c. An efficient light passage to the two channels is demonstrated in Fig. 4 , which shows equalized light transfer from

W_{1}

to the two waveguides

W_{2}

and

W_{3}

with an experimentally observed power transfer of

\sim 2 \times 28 %

with only 2% light power remaining in the initial state and the residual power being lost in the continuum [Figs. 4a, 4b]. The experimental results are in reasonable agreement with theoretical predictions of power flow in the waveguides shown in the inset of Fig. 4. In the current geometrical setting, the transfer efficiency is mainly limited by the adiabaticity of the process. An increase in the transfer efficiency is predicted by numerical simulations for longer samples and smoother curving of side waveguides, which would improve the adiabaticity of the STIRAP evolution. To understand the adiabatic transfer to two channels via a continuum demonstrated in Fig. 4, we note that in this case there are three states

| 1 ⟩

| 2 ⟩

, and

| 3 ⟩

coupled to a common tight-binding continuum with

m = \pm 2

; and the transfer dynamics is ruled by three reduced coupled-mode equations for the modal amplitudes

c_{1}

c_{2}

, and

c_{3}

in the three waveguides, which are an extension of Eqs. (1) [see Eqs. (12) and (13) of [15

S. Longhi, J. Mod. Opt. 56, 729 (2009). [CrossRef]

]]. In this case, the photonic structure admits two trapped states. For our geometric setting corresponding to

κ_{3} = κ_{2}

(

κ_{3}

is the coupling rate of waveguide

W_{3}

with its neighboring guide in array A) and for the initial condition

c_{2} (0) = c_{3} (0) = 0

, the only relevant nondecaying (trapped) state reads as

c_{1} = \frac{2 κ_{2} (z)}{\sqrt{κ_{1}^{2} (z) + 4 κ_{2}^{2} (z)}}, c_{2} = c_{3} = \frac{κ_{1} (z)}{\sqrt{κ_{1}^{2} (z) + 4 κ_{2}^{2} (z)}} .

(4)

The efficient light transfer observed in Fig. 4 thus results from the adiabatic evolution of the system initially prepared in the dark state.

In conclusion, we demonstrated adiabatic light transfer via a continuum using femtosecond laser written waveguide structures, with a transfer efficiency of up to 87%. Strong reduction in the transfer efficiency observed when the photonic structure does not support a trapped state in the continuum clearly indicates the major role played by the destructive Fanolike interference in the transfer process.

Acknowledgments

We acknowledge support by the Deutsche Forschungsgemeinschaft (DFG) (Leibniz-Programm and R.U. 532 “Nonlinear spatio-temporal dynamics in dissipative and discrete optical systems”) and the German Academy of Science Leopoldina (grant LPDS 2009–13).

References and links

1.	K. Bergmann, H. Theuer, and B. W. Shore, Rev. Mod. Phys. 70, 1003 (1998). [CrossRef]
2.	A. E. Miroshnichenko and Y. S. Kivshar, Phys. Rev. E 72, 056611 (2005). [CrossRef]
3.	E. Paspalakis, Opt. Commun. 258, 30 (2006). [CrossRef]
4.	S. Longhi, Phys. Rev. E 73, 026607 (2006). [CrossRef]
5.	P. Ginzburg and M. Orenstein, Opt. Express 14, 11312 (2006). [CrossRef] [PubMed]
6.	S. Longhi, G. D. Valle, M. Ornigotti, and P. Laporta, Phys. Rev. B 76, 201101 (2007). [CrossRef]
7.	G. Della Valle, M. Ornigotti, T. T. Fernandez, P. Laporta, S. Longhi, A. Coppa, and V. Foglietti, Appl. Phys. Lett. 92, 011106 (2008). [CrossRef]
8.	Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 101, 193901 (2008). [CrossRef] [PubMed]
9.	C. E. Carroll and F. T. Hioe, Phys. Rev. Lett. 68, 3523 (1992). [CrossRef] [PubMed]
10.	E. Paspalakis, M. Protopapas, and P. L. Knight, J. Phys. B 31, 775 (1998). [CrossRef]
11.	R. G. Unanyan, N. V. Vitanov, B. W. Shore, and K. Bergmann, Phys. Rev. A 61, 043408 (2000). [CrossRef]
12.	T. Peters, L. P. Yatsenko, and T. Halfmann, Phys. Rev. Lett. 95, 103601 (2005). [CrossRef] [PubMed]
13.	S. Longhi, Phys. Rev. A 78, 013815 (2008). [CrossRef]
14.	S. Longhi, Phys. Rev. A 79, 023811 (2009). [CrossRef]
15.	S. Longhi, J. Mod. Opt. 56, 729 (2009). [CrossRef]
16.	S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, J. Mod. Opt. 51, 2533 (2004). [CrossRef]
17.	F. Dreisow, M. Heinrich, A. Szameit, S. Doering, S. Nolte, A. Tuennermann, S. Fahr, and F. Lederer, Opt. Express 16, 3474 (2008). [CrossRef] [PubMed]
18.	A. Szameit, F. Dreisow, H. Hartung, S. Nolte, A. Tuennermann, and F. Lederer, Appl. Phys. Lett. 90, 241113 (2007). [CrossRef]
19.	F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 101, 143602 (2008). [CrossRef] [PubMed]

Fig. 1 (a) Sketch of waveguide arrangement, showing two waveguides

W_{1}

and

W_{2}

(solid curved rod) side coupled to a common array A. (b) Quantum-mechanical analog of the photonic system. (c) Microscopy image of manufactured waveguide systems (output facets). From left to right:

m = 0

, 1, and 2 with two output channels.

Fig. 2 Light transfer between two side waveguides

W_{1}

and

W_{2}

coupled to a common guide of array A

(m = 0)

. (a)–(c) Counterintuitive (STIRAP) regime,

W_{1}

is illuminated: (a) fluorescence image, (b) output near field, (c) numerically computed evolution of

{| c_{1} |}^{2}

and

{| c_{2} |}^{2}

. (d)–(f) Intuitive (non-STIRAP) regime,

W_{2}

is illuminated: (d) numerically computed evolution of

{| c_{1} |}^{2}

and

{| c_{2} |}^{2}

, (e) fluorescence image, (f) output near field. The circles indicate the excitation waveguide at input plane.

Fig. 3 Same as Fig. 2 but for

m = 1

. In this case the photonic structure does not have a nondecaying (trapped) state.

Fig. 4 Same as Fig. 2 but for

m = 2

, two output channels, and counterintuitive (STIRAP) regime. The curvature of the bent waveguides is

α = 3 \times 10^{- 6} {mm}^{- 1}

; the apices of

W_{1}

and

W_{2, 3}

are at 60 and 15 mm from the input plane, respectively; the minimal spacing between the waveguides and A is

d_{min} = 11 μ m

OCIS Codes
(000.1600) General : Classical and quantum physics
(130.2790) Integrated optics : Guided waves
(230.3120) Optical devices : Integrated optics devices
(230.7370) Optical devices : Waveguides

ToC Category:
Optical Devices

History
Original Manuscript: April 27, 2009
Revised Manuscript: June 11, 2009
Manuscript Accepted: July 9, 2009
Published: August 4, 2009

Virtual Issues
August 12, 2009 Spotlight on Optics

Citation
F. Dreisow, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, "Adiabatic transfer of light via a continuum in optical waveguides," Opt. Lett. 34, 2405-2407 (2009)
http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-16-2405

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References

K. Bergmann, H. Theuer, and B. W. Shore, Rev. Mod. Phys. 70, 1003 (1998). [CrossRef]
A. E. Miroshnichenko and Y. S. Kivshar, Phys. Rev. E 72, 056611 (2005). [CrossRef]
E. Paspalakis, Opt. Commun. 258, 30 (2006). [CrossRef]
S. Longhi, Phys. Rev. E 73, 026607 (2006). [CrossRef]
P. Ginzburg and M. Orenstein, Opt. Express 14, 11312 (2006). [CrossRef] [PubMed]
S. Longhi, G. D. Valle, M. Ornigotti, and P. Laporta, Phys. Rev. B 76, 201101 (2007). [CrossRef]
G. Della Valle, M. Ornigotti, T. T. Fernandez, P. Laporta, S. Longhi, A. Coppa, and V. Foglietti, Appl. Phys. Lett. 92, 011106 (2008). [CrossRef]
Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 101, 193901 (2008). [CrossRef] [PubMed]
C. E. Carroll and F. T. Hioe, Phys. Rev. Lett. 68, 3523 (1992). [CrossRef] [PubMed]
E. Paspalakis, M. Protopapas, and P. L. Knight, J. Phys. B 31, 775 (1998). [CrossRef]
R. G. Unanyan, N. V. Vitanov, B. W. Shore, and K. Bergmann, Phys. Rev. A 61, 043408 (2000). [CrossRef]
T. Peters, L. P. Yatsenko, and T. Halfmann, Phys. Rev. Lett. 95, 103601 (2005). [CrossRef] [PubMed]
S. Longhi, Phys. Rev. A 78, 013815 (2008). [CrossRef]
S. Longhi, Phys. Rev. A 79, 023811 (2009). [CrossRef]
S. Longhi, J. Mod. Opt. 56, 729 (2009). [CrossRef]
S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, J. Mod. Opt. 51, 2533 (2004). [CrossRef]
F. Dreisow, M. Heinrich, A. Szameit, S. Doering, S. Nolte, A. Tuennermann, S. Fahr, and F. Lederer, Opt. Express 16, 3474 (2008). [CrossRef] [PubMed]
A. Szameit, F. Dreisow, H. Hartung, S. Nolte, A. Tuennermann, and F. Lederer, Appl. Phys. Lett. 90, 241113 (2007). [CrossRef]
F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 101, 143602 (2008). [CrossRef] [PubMed]

Appl. Phys. Lett.

G. Della Valle, M. Ornigotti, T. T. Fernandez, P. Laporta, S. Longhi, A. Coppa, and V. Foglietti, Appl. Phys. Lett. 92, 011106 (2008). [CrossRef]
A. Szameit, F. Dreisow, H. Hartung, S. Nolte, A. Tuennermann, and F. Lederer, Appl. Phys. Lett. 90, 241113 (2007). [CrossRef]

J. Mod. Opt.

S. Longhi, J. Mod. Opt. 56, 729 (2009). [CrossRef]
S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, J. Mod. Opt. 51, 2533 (2004). [CrossRef]

J. Phys. B

E. Paspalakis, M. Protopapas, and P. L. Knight, J. Phys. B 31, 775 (1998). [CrossRef]

Opt. Commun.

E. Paspalakis, Opt. Commun. 258, 30 (2006). [CrossRef]

Opt. Express

Phys. Rev. A

R. G. Unanyan, N. V. Vitanov, B. W. Shore, and K. Bergmann, Phys. Rev. A 61, 043408 (2000). [CrossRef]
S. Longhi, Phys. Rev. A 78, 013815 (2008). [CrossRef]
S. Longhi, Phys. Rev. A 79, 023811 (2009). [CrossRef]

Phys. Rev. B

S. Longhi, G. D. Valle, M. Ornigotti, and P. Laporta, Phys. Rev. B 76, 201101 (2007). [CrossRef]

Phys. Rev. E

S. Longhi, Phys. Rev. E 73, 026607 (2006). [CrossRef]
A. E. Miroshnichenko and Y. S. Kivshar, Phys. Rev. E 72, 056611 (2005). [CrossRef]

Phys. Rev. Lett.

Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 101, 193901 (2008). [CrossRef] [PubMed]
C. E. Carroll and F. T. Hioe, Phys. Rev. Lett. 68, 3523 (1992). [CrossRef] [PubMed]
T. Peters, L. P. Yatsenko, and T. Halfmann, Phys. Rev. Lett. 95, 103601 (2005). [CrossRef] [PubMed]
F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 101, 143602 (2008). [CrossRef] [PubMed]

Rev. Mod. Phys.

K. Bergmann, H. Theuer, and B. W. Shore, Rev. Mod. Phys. 70, 1003 (1998). [CrossRef]

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