Optical pulses—wave-packets—propagating in a linear medium have a natural tendency to broaden in time (dispersion) and space (diffraction). Such broadening can be eliminated in a nonlinear medium that modifies its refractive index in the presence of light in such a way that dispersion or diffraction effects are counteracted by light-induced lensing1,2. This can allow short pulses to propagate without changing their shape2,3, and the ‘self-trapping’ of narrow optical beams1 whereby a beam of light induces a waveguide in the host medium and guides itself in this waveguide, thus propagating without diffraction4. Self-trapped pulses in space and time have been investigated extensively in many physical systems and, as a consequence of their particle-like behaviour, are known as ‘solitons’ (ref. 5). Previous studies of this phenomenon in various nonlinear media6,7,8,9,10,11,12 have involved coherent light, the one exception being our demonstration13 of self-trapping of an optical beam that exhibited partial spatial incoherence. Here we report the observation of self-trapping of a white-light beam from an incandescent source. Self-trapping occurs in both dimensions transverse to the beam when diffraction effects are balanced exactly by self-focusing in the host photorefractive medium. To the best of our knowledge, this is the first observation of self-trapping for any wave-packet that is both temporally and spatially incoherent.
The goal is to achieve self-trapping of an incoherent beam (wave-packet); that is, we wish to trap the time-averaged envelope made by a rapidly changing multi-mode broad-band optical field. At any given time, the beam contains many ‘speckles’: a random distribution of bright and dark patches caused by the randomly varying phase in space. To achieve self-trapping of the envelope, it is necessary to counteract the diffraction of the quickly changing speckled optical beam. The key issue is to find a nonlinear medium that responds on a timescale much longer than the rate of change of the speckle pattern of the beam. The reason for this is as follows: if an instantaneous optical nonlinearity was used, then the medium would respond to the instantaneous ‘speckled’ beam. Each speckle would form a small ‘positive lens’ and would capture a small fraction of the beam. These bright–dark features on the beam change very fast throughout propagation and these tiny induced-waveguides intersect and cross each other in a random manner. The net effect would be beam breakup into small fragments and self-trapping of the beam's envelope would not occur. For an incoherent beam to self-trap, it is necessary that the beam induces a smooth waveguide that will guide its rapidly changing intensity at every instant. A non-instantaneous nonlinearity allows this to occur, as it reacts to a time-averaged intensity which is temporally and spatially smooth if averaged over a long enough time period. When this self-induced waveguide guides the rapidly changing beam, self-trapping is achieved.
Photorefractive crystals are a convenient choice of such a nonlinear medium, because their response time is fully controlled by the intensity of the beam, and can be made (with low intensities) much longer than the rate of the rapid intensity fluctuations of the beam. The photorefractive self-focusing effects used here have been described in detail in conjunction with photorefractive screening solitons14,15,16,17,18. The formation of a bright screening soliton may be viewed in the following manner. A narrow light beam propagates in the centre of a biased dielectric (photorefractive) medium. In the illuminated region, electrons are optically excited, and therefore the conductivity increases and the resistivity decreases. Thus, the voltage drop occurs primarily in the dark regions leading to a large space-charge field Esc(r) there, r being the coordinates in the plane transverse to the propagation direction of the beam. The change in the refractive index Δn(r) is linearly proportional to thespace-charge field via the electro-optic (Pockels') effect. When the polarity of the field is properly chosen, Δn(r) is proportional to −Esc(r) and this creates a ‘graded index waveguide’ that guides the beam that generated it16.
We have recently reported the first (to our knowledge) experimental observation of self-trapping of a partially spatially incoherent optical beam13; we used a quasi-monochromatic laser beam scattered off a rotating diffuser to generate a speckled beam19 in which the phases of any two well-separated points varied randomly with time. This speckled beam was self-trapped with diffraction being eliminated; the beam envelope maintained a constant diameter throughout propagation13. However, that previous experiment served only as a ‘proof of principle’ as the rotating diffuser generated partially spatially incoherent light only. The light originated from a laser and was temporally coherent (quasi-monochromatic) at all times.
Here we report self-trapping of a broad-band spatially and temporally incoherent light beam, emerging from an incandescent source; a light bulb. We have used a quartz–tungsten–halogen incandescent bulb to generate the white-light beam. The light is initially sent through a spectral filter to limit the frequency band to 380–720 nm (the temporal coherence time of the beam is of the order of a few femtoseconds). The light was collimated into a beam, sent through a polarizer to keep one polarization only, and focused onto the input face of an SBN : 75 photorefractive crystal (Sr0.75 Ba0.25 Nb2O6). The spectrum of the light incident on the crystal is shown in Fig. 1. Our SBN crystal exhibited photorefractivity to wavelengths between roughly 380 and 520 nm, which gives a normalized bandwidth Δν/ν0 of roughly 0.3. In the crystal, the incoherent beam propagates along the crystalline a -axis with its polarization parallel to the c -axis (extraordinary polarization). We used a lens to image the beam at the input and output faces of the crystal onto a CCD (charge-coupled device) camera. As with photorefractive screening solitons18, the magnitude of the nonlinearity is fine-tuned with a uniform background beam by generating a bias level of electrons in the conduction band. For this purpose, we use an ordinarily polarized 488-nm laser beam which was expanded to illuminate the crystal uniformly. Self-focusing occurs with the application to the crystal of an appropriate voltage (magnitude and polarity) which gives rise to a space-charge field that has a large component along the c -axis, thus using the r33 = 1,022 pm V−1 electro-optic coefficient to create the index change required for self-trapping. An input beam of 14 μm (all beam sizes are given here as full-width at half-maximum, FWHM) diffracts to 82 μm after 6 mm of propagation. The large diffraction angle demonstrates the spatial incoherence of the beam. A coherent beam of size 14 μm at 380 nm wavelength would have diffracted to 35.34 μm, whereas the same-size input beam at 720 nm would have diffracted to 63.1 μm. Applying 600 V between the electrodes separated by 6 mm results in self-trapping of the beam, which traps to 12 μm. The total optical power in the white-light input beam is 70.8 nW. As this power is spread over 340 nm, the amount of light at the photorefractively sensitive wavelengths is ∼25 nW. The 488-nm background beam had a power of 400 nW. These very low power levels, when translated to the corresponding beam intensities, result in a very long formation time (which is related to the dielectric relaxation time) for the self-trapped beam.
Our experimental results are shown in Figs 2 and 3. Figure 2a shows the profile of a 14-μm beam at the input face of the crystal. Figure 2b shows the profile of the 82-μm-wide normally diffracting beam in the absence of nonlinearity (zero voltage). Figure 2c–l shows the temporal evolution of the beam at the output face of the crystal that occurs once the nonlinearity (voltage) is turned on. The beam starts to self-focus by going through a quasi-steady-state regime that is reminiscent of quasi-steady-state photorefractive solitons10. Then, the beam breaks up and moves towards the positive c -axis and forms the steady-state self-trapped beam as shown by Fig. 2i–l. The centre of the self-trapped beam of Fig. 2j has moved a distance of 57 μm away from the centre of the initial diffracted beam, towards the c -axis. This ‘displacement’ of the self-trapped beam is closely related to the self-bending effects that all photorefractive solitons experience; it is driven by a diffusion field that is the lowest-order correction to the space-charge field that supports the solitons20. The self-trapped beam roughly maintains its structure for at least eight hours, during which the beam fluctuated slightly in shape and drifted slowly towards the c -axis. It remained a self-trapped entity and did not break up or diminish for the entire duration of our experiment. Possible reasons for the slow fluctuations in the shape of the self-trapped beam are ‘environmental’ changes, such as temperature variations or slow drifting in the optical power emitted from the sources, as we have not used any special means to isolate our system. We emphasize, however, that the beam remained self-trapped (localized) for as long as we have monitored the experiment. We believe that the small fluctuations are not related to the self-trapping mechanism of the incoherent beam.
Quasi-steady-state trapping was observed even without use of background illumination. Figure 3a shows the profile of a 26-μm beam at the input of the crystal. Figure 3b shows the profile of the 100-μm normally diffracting beam in the absence of nonlinearity (zero voltage). Figure 3c–f shows the temporal evolution of the beam at the output face of the crystal. The beam focuses to a size of 26-μm then breaks up and eventually diminishes.
To our knowledge, this is the first observation of a self-trapped beam from a source that is both temporally and spatially incoherent. The phase across the self-trapped beam is varying in a random manner both in time and in space. Unlike the case of a self-trapped coherent beam, knowing (measuring) the phase at a particular point on the incoherent self-trapped beam cannot provide any phase information, even at very short distances away from that point. Furthermore, at each point on the beam, photons of widely varying frequencies coexist, so the absolute phase of the total optical field varies randomly between each two points separated by a distance of the order of the optical wavelength. Yet, this incoherent white-light beam indeed self-traps.
Self-trapping of ‘white’ incoherent light introduces the possibility of using incoherent sources (for example, light-emitting diodes) for optical interconnects, beam steering, and other applications that have been thus far proposed only for (coherent) solitons. On the fundamental level, self-trapping of incoherent light raises many intriguing questions. We believe that the statistics of the self-trapped incoherent beam are affected by the self-trapping: going from the delocalized statistics of the input thermal source (that is, the coherence depends on coordinate difference only) into a state in which the statistics depend also on the absolute coordinate across the self-trapped beam. How are the brightness and entropy, which are related to the coherence properties of the beam, affected by self-trapping? If brightness is improved, it must come at the expense of energy loss, in the form of radiation or absorption. Are self-trapped incoherent beams able to maintain their identities as they undergo collisions with each other? These and other issues are currently under investigation.
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This work was supported by the US Army Research Office and the US National Science Foundation.
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Mitchell, M., Segev, M. Self-trapping of incoherent white light. Nature 387, 880–883 (1997). https://doi.org/10.1038/43136
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