Abstract

We experimentally ascertain the role of non locality in the spectral evolution of multifilament patterns generated by modulational instability in nematic liquid crystals.

© 2005 Optical Society of America

1. Introduction

Non locality plays an important role in several areas of physics, from quantum-mechanics [1] to plasmas [2] to quantum optics, [3] to mention just a few. In optics a non local response may originate from mechanisms such as heat [4] or atom diffusion, [5] charge drift [6–7] or elastic intermolecular forces [8–9]. In nonlinear optics, a few major implications of non locality have been addressed with reference to spatial solitons (SS) and modulational instability (MI), both in general terms [4, 10–12] and with specific emphasis on nematic liquid crystals (NLC), even in the presence of significant walk-off [13]. The non local response of NLC stems from the elastic interaction between elongated molecules subject to thermal or reorientational effects, the latter in the presence of an external (optical or low frequency) electric field. As underlined with specific reference to the reorientational response of undoped NLC, a non local response prevents catastrophic collapse, mediates long-range attractive interactions between solitons and supports breather-like propagation of narrow-waist filaments [14]. In addition, spatial non locality supports the generation of spatial solitons by partially incoherent (speckled) input beams, smoothing out the resulting transverse index profile and allowing the guided-wave propagation of a co-polarized (weak) signal in the corresponding eigenmode(s) [15]. Recently, multifilament formation owing to transverse modulational instability in NLC was discussed as an approach to the generation of multiple spatial solitons or nematicons [13]. When a wide beam is employed in a non local system subject to a self-focusing response, however, the latter “globally” affects the propagation of the whole filament pattern, [16] resembling many-body interactions in a system consisting of several individual solitons.

In this Paper we discuss the role of non locality by investigating the evolution of the transverse spectrum of a filament pattern induced in NLC by one-dimensional modulational instability. At variance with previous investigations, the periodic pattern consists of self-localized nematicons which mutually interact through the medium non local response.

2. Sample and model

The optical nonlinearity of undoped NLC stems from light induced reorientation of highly anisotropic molecules and their optic axis or director n̑.[8–9] Due to the intermolecular forces, however, the refractive index change extends well beyond the excitation region, with a characteristic non local range which entails spectral low-pass filtering in transverse space. Thereby, fast transverse perturbations (i.e., refractive features on a scale smaller than the non local range) are filtered out [17–18]. This has a striking effect on modulational instability (MI), which tends to break a plane wavefront into a quasi-periodic pattern irrespective of the origin of nonlinearity and the degree of non locality. In either Kerr, photorefractive, quadratic or reorientational media, in fact, through frequency domain coupling in Fourier space MI promotes a power exchange from the pump spectrum (centered in the origin) to symmetrically displaced side-bands [19–20]. Additional non locality tends to distorts these sidebands in propagation, feeding low-frequency components at the expense of higher harmonics.

 

Fig. 1. Experimental geometry: a highly elliptical Gaussian beam (U>>V) is injected into a planar nematic liquid crystal cell with wavevector k normal to the input interface.

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To experimentally verify this insight, we adopted a high non local medium, i.e., nematic liquid crystal E7, using a sample and the geometry as sketched in Fig. 1: two parallel glass slides spaced by L=75μm hold the NLC, and a third slide seals the input interface to prevent undesired light depolarization. Proper treatments of the interfaces determine the molecular alignment in the plane yc zc at an angle θ0 to the normal ẑc to the input interface. An elliptic Gaussian beam of wavevector kzc, waists U=200μm along yc and V>20μm along xc, (extraordinary) polarization in the cell plane yczc, is injected into the sample using cylindrical lenses. In such configuration, the flux S propagates with walk-off δ relative to k =(ky, kz) in the yczc plane (see Fig. 1). Considering the rotated reference system (x,y,z) with x//xc, the nonlinear contribution Ψ to the reorientation is governed by: [13]

K2ΨA(θ0)Ψ+ε0Δε4sin[2(θ0δ)]Ee2=0

K being the elastic constant (scalar approximation) and Δε the birefringence. The coefficient A(θ0) relates to non locality and, in our geometry, it takes the form:

A(θ0)=Kπ2L2cos(θ0)(sin(2θ0)2θ0cos(2θ0))

Assuming ∂2/∂x 2 << ∂2/∂y 2 , we can solve eq. (1) in the Fourier domain:[16]

Ψ˜=ε0Δε4sin[2(θ0δ)]Kky2+A(θ0){Ee2}

with ky the transverse wavevector component with respect to the Poynting vector S(//z). From the Lorentzian (3) we can calculate the cutoff wavenumber for the transverse intensity: ky=A(θ0)K. It is important to stress that such cutoff does not depend on the material (Frank coefficient K) but only on the geometry (θ0). For the actual parameters of the liquid crystal mixture E7, we obtain the graph in Fig. 2 with a cutoff ky_cut=0.040μm-1 for θ0=45°, the actual molecular orientation in our samples.

 

Fig. 2. Transverse cutoff wavenumber versus angle θ0 between director n̂ and wavevector k̂.

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3. Experimental results and discussion

Based on (3) we expect that, owing to the non local (attractive) interaction, the transverse intensity of a multifilament pattern evolves in propagation while undergoing Lorentzian filtering. Figure 3 below displays some typical experimental results: the wide input beam propagates in the NLC with wavevector k̂ = ẑc and a walk-off of 7°, in perfect agreement with calculated values from birefringence in E7 and molecular alignment at θ0=45°. Modulational instability occurs at powers as low as 30mW, producing a periodic transverse undulation clearly visible for z≥2mm (Fig. 3(a)). At P=60mW (Fig. 3(b)) MI yields a multisoliton pattern, with soliton-soliton attraction and an overall beam (self-) focusing. As the excitation level is further increased (Fig. 3(c)), the filaments tend to coalesce and agglomerate in more intense beams.

 

Fig. 3. Beam propagation in NLC at (a) 30, (b) 60 and (c) 90mW input powers, respectively. The pattern produced via modulational instability (a) eventually results into several solitons as the power increases (b). At higher excitations, adjacent solitons group owing to non locality and “global” self-focusing (c). In the photographs, birefringent walk-off is artificially compensated by rotating the camera axis by about 7°.

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Such “agglomerates” are seeded by the non-uniform intensity and are wider and more intense than individual filaments, hence the Fourier spectrum of the whole transverse pattern changes as light propagates forward in the medium. Figure 4 maps the measured gain spectra versus ky and propagation distance z, i.e., the time-averaged Fourier-transformed transverse intensity normalized to its input spectrum. At P=30mW modulational instability is clearly visible in Fig. 3(a). The nonlinear process results into a sideband between 0.05μm-1 and 0.2μm-1, in good agreement with the estimates of Ref. [18] even though the geometry is slightly different here. At P=60mW (Fig. 4(b)) MI results in a filament pattern with a specific mean-periodicity highlighted by equally-spaced peaks in the transverse spectrum for z>2mm.

 

Fig. 4. Transverse spectral gain, i. e. the transverse intensity spectrum versus z normalized to the input spectrum in z=0, for three different excitations: (a) P=30, (b) 60 and (c) 90mW, respectively.

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Finally, for P=90mW (Fig. 4(c)) a strong non local interaction takes place between filaments: power gets progressively coupled from side-bands towards a specific narrow band around 0.042μm-1, consistently with the estimated filtering action owing to medium elasticity. Such spectral behavior is underlined in Fig. 5 by comparing transverse intensity spectra at the three powers (Fig. 4) after propagation over 3.5mm.

 

Fig. 5. Transverse intensity spectra in z=3.5mm for three input powers as in Fig. 4.

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We observe that the non local interaction allows the transverse pattern to evolve towards a mean intensity spectrum which complies with the medium cutoff. Otherwise stated, the non local contribution to filament-to-filament interaction is cumulative in propagation. Noticeably, light self-localization tends to preserve the filament identities despite coalescence, impeding the power (spectral) conversion towards a plane wave with ky=0.

While a nonlinear response acts on a plane wave by frequency-domain coupling power towards higher harmonics, we could argue that –conversely- non locality acts on a pattern of parallel filaments by frequency-domain coupling power towards lower harmonics.

4. Conclusions

In conclusion, for the first time, we experimentally studied the spatial spectrum of mutually interacting filaments and its evolution versus propagation in a non local medium. The filaments were generated via modulational instability in spatially non local nematic liquid crystals and evolved through progressive power coupling from high to lower frequencies, consistently with an active filtering action and the existence of a transverse wavevector cutoff.

We are grateful to C. Umeton (LICRYL, Univ. of Calabria) for the sample.

References and links

1. J. J. Sakurai, “Modern Quantum Mechanics” (Addison-Wesley, Reading, MA, 1994).

2. R. A. Stern and J. F. Decker, “Nonlocal Instability of Finite-Amplitude Ion Waves,” Phys. Rev. Lett. 27, 1266–1271 (1971) [CrossRef]  

3. B. Hessmo, P. Usachev, H. Heydari, and G. Björk, “Experimental demonstration of single photon nonlocality,” Phys. Rev. Lett. 92, 180401 (2004). [CrossRef]   [PubMed]  

4. J. P. Gordon, R. C. Leite, R. S. Moore, and J. R. Whinnery, “Long-Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965). [CrossRef]  

5. D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of nonlinear medium,” Phys. Rev. A 48, 4583–4587 (1993). [CrossRef]   [PubMed]  

6. A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric-field,” Phys. Rev. A 51, 1520–1531 (1995). [CrossRef]   [PubMed]  

7. S. Gatz and J. Herrmann, “Anisotropy, nonlocality and space-charge field displacement in (2+1)-dimensional self-trapping in biased photorefractive crystals,” Opt. Lett. 23, 1176–1178 (1998). [CrossRef]  

8. N. V. Tabiryan, A. V. Sukhov, and B. Y Zel’dovich, “Orientational nonlinearity of liquid-crystals,” Mol. Cryst. Liq. Cryst. 136, 1–139 (1986). [CrossRef]  

9. I. C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena, (Wiley, New York, 1995).

10. A. W. Snyder and D. J. Mitchell, “Accessible Solitons,” Science 276, 1538–1541 (1997). [CrossRef]  

11. W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004). [CrossRef]  

12. J. Wyller, W. Krolikowski, O. Bang, and J. J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66, 066615 (2002). [CrossRef]  

13. M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of Highly Anisotropic Spatial Solitons and Modulational Instability in liquid crystals,” Nature 432, 733–737 (2004). [CrossRef]   [PubMed]  

14. C. Conti, M. Peccianti, and G. Assanto, “Observation of optical solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004). [CrossRef]   [PubMed]  

15. M. Peccianti and G. Assanto, “Nematic liquid crystals: a suitable medium for self-confinement of coherent and incoherent light,” Phys. Rev. E 65, 035603 (2002). [CrossRef]  

16. M. Peccianti, C. Conti, and G. Assanto, “Interplay between nonlocality and nonlinearity in nematic liquid crystals,” Opt. Lett. 30, 415–417 (2005). [CrossRef]   [PubMed]  

17. M. Peccianti, C. Conti, E. Alberici, and G. Assanto, “Spatially incoherent modulational instability in a non local medium,” Laser Phys. Lett. 2, 25–29 (2005). [CrossRef]  

18. M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602 (2003). [CrossRef]  

19. V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310 (1966).

20. G. I. Stegeman, “Spatial Beam Instabilities Due to Instantaneous Nonlinear Mechanisms,” Proc. NATO ASI/SUSSP56 on “Ultrafast Photonics,” Ed. A. Miller (Inst. Physics Publishing, London, 2003).

References

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  • |

  1. J. J. Sakurai, �??Modern Quantum Mechanics�?? (Addison-Wesley, Reading, MA, 1994).
  2. R. A. Stern and J. F. Decker, �??Nonlocal Instability of Finite-Amplitude Ion Waves,�?? Phys. Rev. Lett. 27, 1266-1271 (1971)
    [CrossRef]
  3. B. Hessmo, P. Usachev, H. Heydari and G. Björk, "Experimental demonstration of single photon nonlocality," Phys. Rev. Lett. 92, 180401 (2004).
    [CrossRef] [PubMed]
  4. J. P. Gordon, R. C. Leite, R. S. Moore and J. R. Whinnery, �??Long-Transient Effects in Lasers with Inserted Liquid Samples,�?? J. Appl. Phys. 36, 3-8 (1965).
    [CrossRef]
  5. D. Suter and T. Blasberg, �??Stabilization of transverse solitary waves by a nonlocal response of nonlinear medium,�?? Phys. Rev. A 48, 4583-4587 (1993).
    [CrossRef] [PubMed]
  6. A. A. Zozulya and D. Z. Anderson, �??Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric-field,�?? Phys. Rev. A 51, 1520- 1531 (1995).
    [CrossRef] [PubMed]
  7. S. Gatz and J. Herrmann, �??Anisotropy, nonlocality and space-charge field displacement in (2+1)- dimensional self-trapping in biased photorefractive crystals,�?? Opt. Lett. 23, 1176-1178 (1998).
    [CrossRef]
  8. N. V. Tabiryan, A. V. Sukhov and B. Y Zel�??dovich, �??Orientational nonlinearity of liquid-crystals,�?? Mol. Cryst. Liq. Cryst. 136, 1-139 (1986).
    [CrossRef]
  9. I. C. Khoo, "Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena, "(Wiley, New York, 1995).
  10. A. W. Snyder and D. J. Mitchell, �??Accessible Solitons,�?? Science 276, 1538-1541 (1997).
    [CrossRef]
  11. W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen and D. Edmundson, �??Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,�?? J. Opt. B 6, S288-S294 (2004).
    [CrossRef]
  12. J. Wyller, W. Krolikowski, O. Bang and J. J. Rasmussen, �??Generic features of modulational instability in nonlocal Kerr media,�?? Phys. Rev. E 66, 066615 (2002).
    [CrossRef]
  13. M. Peccianti, C. Conti, G. Assanto, A. De Luca and C. Umeton, �??Routing of Highly Anisotropic Spatial Solitons and Modulational Instability in liquid crystals,�?? Nature 432, 733-737 (2004).
    [CrossRef] [PubMed]
  14. C. Conti, M. Peccianti and G. Assanto, �??Observation of optical solitons in a highly nonlocal medium,�?? Phys. Rev. Lett. 92, 113902 (2004).
    [CrossRef] [PubMed]
  15. M. Peccianti and G. Assanto, �??Nematic liquid crystals: a suitable medium for self-confinement of coherent and incoherent light,�?? Phys. Rev. E 65, 035603 (2002).
    [CrossRef]
  16. M. Peccianti, C. Conti and G. Assanto, �??Interplay between nonlocality and nonlinearity in nematic liquid crystals,�?? Opt. Lett. 30, 415-417 (2005).
    [CrossRef] [PubMed]
  17. M. Peccianti, C. Conti, E. Alberici and G. Assanto, "Spatially incoherent modulational instability in a non local medium," Laser Phys. Lett. 2, 25-29 (2005).
    [CrossRef]
  18. M. Peccianti, C. Conti and G. Assanto, �??Optical modulational instability in a nonlocal medium,�?? Phys. Rev. E 68, 025602 (2003).
    [CrossRef]
  19. V. I. Bespalov and V. I. Talanov, �??Filamentary structure of light beams in nonlinear liquids,�?? JETP Lett. 3, 307-310 (1966).
  20. G. I. Stegeman, �??Spatial Beam Instabilities Due to Instantaneous Nonlinear Mechanisms,�?? Proc. NATO ASI/SUSSP56 on �??Ultrafast Photonics,�?? Ed. A. Miller (Inst. Physics Publishing, London, 2003).

J. Appl. Phys. (1)

J. P. Gordon, R. C. Leite, R. S. Moore and J. R. Whinnery, �??Long-Transient Effects in Lasers with Inserted Liquid Samples,�?? J. Appl. Phys. 36, 3-8 (1965).
[CrossRef]

J. Opt. B (1)

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen and D. Edmundson, �??Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,�?? J. Opt. B 6, S288-S294 (2004).
[CrossRef]

JETP Lett. (1)

V. I. Bespalov and V. I. Talanov, �??Filamentary structure of light beams in nonlinear liquids,�?? JETP Lett. 3, 307-310 (1966).

Laser Phys. Lett. (1)

M. Peccianti, C. Conti, E. Alberici and G. Assanto, "Spatially incoherent modulational instability in a non local medium," Laser Phys. Lett. 2, 25-29 (2005).
[CrossRef]

Mol. Cryst. Liq. Cryst. (1)

N. V. Tabiryan, A. V. Sukhov and B. Y Zel�??dovich, �??Orientational nonlinearity of liquid-crystals,�?? Mol. Cryst. Liq. Cryst. 136, 1-139 (1986).
[CrossRef]

Nature (1)

M. Peccianti, C. Conti, G. Assanto, A. De Luca and C. Umeton, �??Routing of Highly Anisotropic Spatial Solitons and Modulational Instability in liquid crystals,�?? Nature 432, 733-737 (2004).
[CrossRef] [PubMed]

Opt. Lett. (2)

Phys. Rev. A (2)

D. Suter and T. Blasberg, �??Stabilization of transverse solitary waves by a nonlocal response of nonlinear medium,�?? Phys. Rev. A 48, 4583-4587 (1993).
[CrossRef] [PubMed]

A. A. Zozulya and D. Z. Anderson, �??Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric-field,�?? Phys. Rev. A 51, 1520- 1531 (1995).
[CrossRef] [PubMed]

Phys. Rev. E (3)

J. Wyller, W. Krolikowski, O. Bang and J. J. Rasmussen, �??Generic features of modulational instability in nonlocal Kerr media,�?? Phys. Rev. E 66, 066615 (2002).
[CrossRef]

M. Peccianti, C. Conti and G. Assanto, �??Optical modulational instability in a nonlocal medium,�?? Phys. Rev. E 68, 025602 (2003).
[CrossRef]

M. Peccianti and G. Assanto, �??Nematic liquid crystals: a suitable medium for self-confinement of coherent and incoherent light,�?? Phys. Rev. E 65, 035603 (2002).
[CrossRef]

Phys. Rev. Lett. (3)

C. Conti, M. Peccianti and G. Assanto, �??Observation of optical solitons in a highly nonlocal medium,�?? Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef] [PubMed]

R. A. Stern and J. F. Decker, �??Nonlocal Instability of Finite-Amplitude Ion Waves,�?? Phys. Rev. Lett. 27, 1266-1271 (1971)
[CrossRef]

B. Hessmo, P. Usachev, H. Heydari and G. Björk, "Experimental demonstration of single photon nonlocality," Phys. Rev. Lett. 92, 180401 (2004).
[CrossRef] [PubMed]

Proc. NATO ASI/SUSSP56 (1)

G. I. Stegeman, �??Spatial Beam Instabilities Due to Instantaneous Nonlinear Mechanisms,�?? Proc. NATO ASI/SUSSP56 on �??Ultrafast Photonics,�?? Ed. A. Miller (Inst. Physics Publishing, London, 2003).

Science (1)

A. W. Snyder and D. J. Mitchell, �??Accessible Solitons,�?? Science 276, 1538-1541 (1997).
[CrossRef]

Other (2)

J. J. Sakurai, �??Modern Quantum Mechanics�?? (Addison-Wesley, Reading, MA, 1994).

I. C. Khoo, "Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena, "(Wiley, New York, 1995).

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Figures (5)

Fig. 1.
Fig. 1.

Experimental geometry: a highly elliptical Gaussian beam (U>>V) is injected into a planar nematic liquid crystal cell with wavevector k normal to the input interface.

Fig. 2.
Fig. 2.

Transverse cutoff wavenumber versus angle θ0 between director n̂ and wavevector k̂.

Fig. 3.
Fig. 3.

Beam propagation in NLC at (a) 30, (b) 60 and (c) 90mW input powers, respectively. The pattern produced via modulational instability (a) eventually results into several solitons as the power increases (b). At higher excitations, adjacent solitons group owing to non locality and “global” self-focusing (c). In the photographs, birefringent walk-off is artificially compensated by rotating the camera axis by about 7°.

Fig. 4.
Fig. 4.

Transverse spectral gain, i. e. the transverse intensity spectrum versus z normalized to the input spectrum in z=0, for three different excitations: (a) P=30, (b) 60 and (c) 90mW, respectively.

Fig. 5.
Fig. 5.

Transverse intensity spectra in z=3.5mm for three input powers as in Fig. 4.

Equations (3)

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K 2 Ψ A ( θ 0 ) Ψ + ε 0 Δ ε 4 sin [ 2 ( θ 0 δ ) ] E e 2 = 0
A ( θ 0 ) = K π 2 L 2 cos ( θ 0 ) ( sin ( 2 θ 0 ) 2 θ 0 cos ( 2 θ 0 ) )
Ψ ˜ = ε 0 Δ ε 4 sin [ 2 ( θ 0 δ ) ] K k y 2 + A ( θ 0 ) { E e 2 }

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