1. Introduction

At the beginning of the 1800s, Thomas Young performed what is believed as one of the most beautiful experiments in physics: the interference of light passing through a double slit [1]. This simple conception has been used hence as a paradigm to unveil the wave nature of a number of physical entities, in particular single electrons [2, 3], neutrons [4], atoms [5, 6] and molecules [7]. On the other hand, experiments like the photoelectric effect, predicted by Einstein in 1905 [8], or those by Millikan (1916) [9] and Compton [10], demonstrate the complementary particle character of light and matter. In the case of macroscopic electromagnetic fields, however, the superposition of a large amount of photon states deems its particle character in favor of the wave nature associated with coherent states.

Although Young’s idea is often reduced conceptually to a simple experiment to evidence the field amplitude as a complex number, it has also deeper roots in the wave theory. In particular it also evidences the property of diffraction and the superposition principle: diffraction makes possible the spatial overlap of the electromagnetic fields coming from the separated slits, and the superposition principle ensures that the resulting field is the sum of the partial fields evolving independently. The relevance of Young’s experiment also relies in its simple geometrical interpretation in terms of optical paths, which gives key ideas to the understanding of more elaborated situations, as diffraction gratings, Fresnel lenses, etcetera. Precisely, the simplicity of this experiment has stimulated us to repeat it in the non-linear regime trying to find more insight on the basic properties of the non-linear propagation of light.

In the general case, the propagation of light through non-linear Kerr media has a very complex dynamics. Fortunately, the non-linear phenomena related with the dynamics of the energy flux, as self-focusing, harmonic generation, etcetera, can be easily identified regardless of the small details of the propagation. This allows for an acceptable description of these phenomena without needing a complete study of each particular case. On the other hand, the interpretation of phenomena related with the dynamics of the field phase is less simple, as the the phase evolution is very sensitive to small-scale field-induced changes of the index of refraction. This complexity is reflected in the interference patterns associated to the non-linear propagation, which are rather complex and difficult to reduce to geometrical descriptions in terms of predefined optical paths. However, the phase description may be of practical application, as recent results in the non-linear propagation of intense fields demonstrate that the diffraction patterns may be used to organize regular arrays of filaments allowing for the control of the transport of high intensities over long distances [11]. In this sense, it is reasonable to reproduce the Young’s experiment with the hope that its simplicity may allow to a deeper description of the interference process in non-linear propagation in air, in the same philosophy as happens in the linear case.

A priori we may expect the non-linear propagation to affect most of the fundamental phenomena involved in Young’s experiment. For instance, in Kerr media diffraction is counterbalanced by self-focusing, the lengths of the optical paths depend on self-phase modulation and their trajectories on the field induced refraction index. In addition, the superposition principle does not longer applies, as the field does not result from the sum of its components evolving independently. Fortunately, the double slit scheme turns out to be also useful in this regime. As happens in the linear case, the simplicity of the interference process permits to develop a conceptual analysis which may constitute a basic understanding of the more complex phenomena. In the following section we will describe the experimental setup and results of the double slit experiment in the linear and non-linear regime. Next, we will develop a simple geometrical picture to explain the divergences with the non-linear case.

 

Fig. 1. Experimental setup.

Download Full Size | PDF

2. Experimental setup and results

We employ a 790 nm Ti:sapphire CPA system with 120 fs pulses of 50 mJ at 10 Hz repetition rate [12 ]. The peak power reached by these pulses, of the order of the Terawatt, permits to observe new non-linear processes even in very weak non-linear media as, for instance, air. Nonlinear effects in the propagation of an ultra-intense laser pulse in air has been investigated with special interest after the report of the experimental observation of the formation of filaments [13]. Later, it has been reported the possibility of beam trapping in air for lower intensities, practically in the absence of ionization, [14].

We have performed our experiment with energies below filamentation (in the distances studied in the experiment), as the formation of plasma may obscure the Kerr-induced phase modulation. The energy of the pulse is controlled with a rotable half-wave plate placed before a beam splitter polarizer at the entrance of the the pulse compression stage. After the compressor we have placed the double-slit and we observe the propagationat different distances by imaging the intensity pattern on a BK7 plate with a bidimensional CCD. As the output beam has a FWHM of about 6.5 mm, we have used a rectangular double-slit of 1.5 mm width, separated one from the other 2.5 mm in the horizontal direction, and with vertical dimensions much larger than the beam waist to avoid diffraction patterns coming form the upper or lower borders. An scheme of this setup is shown in Fig. 1.

The resulting interference pattern at the lower energy (250 μJ after the double-slit) is shown in Fig. 2(a). We have plotted the detected intensity integrated over the vertical direction, at different distances from the slit. As expected from the linear theory, after some distance the propagation approaches the Fraunhofer far field limit and three interference maxima appear. The location of these maxima have been contrasted with the corresponding interference pattern derived theoretically from the Fresnel propagator. This allows us to estimate the initial beam divergence of 5×10^-5 rad in our experiment.

Figure 2(b) shows the results of the same experiment, but with higher field energy (14.0 mJ after the double-slit). As can be noticed, the most remarkable difference between the linear and the non-linear experiments is the absence of the central maximum in the second (in the range of distances explored). This can be understood quite obviously as the reduction of the beam diffraction, as is counterbalanced by the Kerr self-focusing. Depending on this balance, the beams will finally either collapse or expand slowly to overlap and generate the central interference maxima at some distance larger than the ones explored here. A less obvious difference corresponds to the angle of divergence of the two lateral maxima. Figure 3 shows the position of the lateral maxima as a function of the distance from the double slit for the linear and non-linear case. It can be noticed that for distances closer to the slits (i.e. about 3.5 m) both maxima coincide, while for larger distances the divergence appears constant in the linear propagation but gradually reduced in the non-linear case. A qualitative insight about the coincidence of linear and non-linear maxima at shorter distances can be drawn by comparison of Figs. 2(a) and (b): at shorter distances the field distribution is rather similar in both cases. This suggest that the particular features of the non-linear propagation appear at some distance from the slits and not right after them. This is an interesting aspect, as consequently the far field interference pattern of the non-linear propagation reveals topological properties present somewhere in the Fresnel region, while in the linear case the Fraunhofer diffraction can be propagated directly from the field distribution at the slits.

 

Fig. 2. Detected interference patterns vs. distance from the slits for (a) lineal (250 μJ), (b) non-lineal (14 mJ) cases. The patterns shown correspond to the intensities integrated over the vertical direction.

Download Full Size | PDF

 

Fig. 3. Positions of the intensity maximum of the right fringe for the linear (black color) and non-linear case (red color). The points correspond to the experimental maxima position for both cases while the lines have been obtained from the computed Fresnel propagator presented in eq. (1) (linear case) and with its extension according with the model described in the text (non-linear case).

Download Full Size | PDF

 

Fig. 4. Linear interference pattern (black line) at distances (a) 3.5 m and (b) 5 m from the slit, as resulting from the superposition of the field components propagated from the left slit (red line) and the right slit (blue line).

Download Full Size | PDF

3. Theoretical model and interpretation

The qualitative ideas drawn in the later section can be grounded with the help of a simple model, extension of the theory of the linear Young’s experiment. With this goal, let us first analyze the formation of the interference patterns in the linear case. In this situation, the electromagnetic field amplitude E(x,y,z,t) = U(x,y, z)e ^-iωt at any spatial point is given by the sum of the contributions of each slit, U(x,y, z) = U ₊(x,y, z) + U _-(x,y, z), which propagate independently according to (Fresnel approximation)

where in our experiment the dimensions of the slit are a=1.5 mm and b effectively ∞, and their separation c=1.25 mm. f=25.0 m represents the initial beam divergence that we have estimated from the comparison between the experimental results for the lowest energy and the theoretical Fresnel prediction. For convenience, we have differentiated the wavenumber of the two fields, k ₊ and k _-, although in the linear case k ₊ = k _- = n ₀ ω/c with n ₀ the linear index of refraction of air.

The resulting intensity pattern at distances z=3.5 m and z=5 m are plotted in Figs. 4(a) and (b), for the linear case. Also, superimposed to these graphics, we show the intensity profiles corresponding to each of the fields coming from the slits, whose superposition leads to the observed interference pattern. In the following we will refer to diffraction orders as the intensity maxima corresponding to the diffraction pattern of each independent slit, while we will refer to interference orders as the intensity maxima resulting from the overlap of the two fields diffracted at each slit (i.e. the detected intensity patterns).

 

Fig. 5. Model scheme of the interference build up in the non-linear case. The (effectively self-trapped in the experimental distance range) zeroth diffraction orders are represented by shadowed areas and the first order coming from the left fringe by its optical path (the symmetric process has been omitted for simplicity).

Download Full Size | PDF

These two pictures are quite useful in understanding the mechanism leading to the Young’s fringes: the central peak in the intensity pattern is built by the superposition of the zeroth order diffraction maxima of the field at each slit, while the lateral fringes correspond to the superposition of the zeroth diffraction order of one of the slits with the first diffraction order of the other. By definition, the lowest diffraction orders correspond to the smaller divergences or, in mechanical terms, field contributions with smaller lateral momenta. In consequence, the field overlap leading to the central fringe appears at larger distances in comparison with the lateral fringes, which are produced by interference with higher lateral momentum field contributions.

Up to this point, the situation is similar also in the non-linear case. Self-focusing, however, tends to oppose to diffraction affecting specially those field components with small lateral momenta. These components can effectively be trapped in the range of distances of our experiments in the sense that their diffraction can be practically inhibited. Of course, they will probably overlap at some distance outside our experimental range and generate a central interference maximum. In contrast, those field components corresponding to the higher diffraction orders diverge with higher momenta avoiding being trapped by the non-linearity. As a result, the central interference fringe is dimmed, or simply does not exist, while the lateral maxima still appear in Fig. 2(b). To our opinion, this simple picture explains satisfactorily the absence of the central fringe in the non-linear experiments. However, as plotted in Fig. 3, in addition to this feature it is found that the lateral fringes are also affected by the non-linear propagation. This turns out to be the signature of a more subtle mechanism. To elucidate it we must consider that, in addition to self-focusing, the non-linear change of index of refraction induces equally a self-phase modulation. Although this effect may seem less important in the general description of propagation, it can be quite relevant in our experiment, as interferences are grounded in the matching of the field phases. To understand the role of self-phase modulation in the build up of the interference patterns, let us consider the simplified scheme depicted in Fig. 5. In this plot we have represented the zeroth diffraction orders of the field in each slit as shadowed areas, while the first order diffracted field is sketched only through its optical path. As discussed above, the lateral fringes in the diffraction pattern have their origin in the overlap between the zeroth order diffraction of one fringe with the first order diffraction field of the other, since these later components have high enough divergence to escape from the non-linear self-induced trap. In simple terms, we may assume that this field component leaves the trap at some distance z ₁ from the slit. As the intensity of this component is small, after leaving the central part of the field it travels effectively through a linear medium, until reaching the central field belonging to the neighbor slit, at a distance z > z ₁. On the other hand, the central field evolving either to distance z ₁ or z suffers from the self-phase modulation associated with the changes of its intensity profile. In the non-linear case these two different trajectories have associated a change in the optical paths which can be approximated to

where n ₂ = 3.2 × 10^-19 cm ²/W defines the non-linear part of the refraction index for air, and I(z) is the effective local intensity at z, averaged in the transversal coordinates. Note that the small value of n ₂ implies that the effect of self-phase modulation will only be apparent over large distances. Therefore only the z component is relevant. Following these ideas, we may propose a simple extension of the linear propagator to describe the field interference leading to each of the lateral fringes. For instance, the fringe in the positive half-plane (x > 0) results from the overlap between the zeroth diffraction order in U ₊ and the first diffraction order in U _-. According to Eq. (2), we should replace k ₊ z in U ₊ by (ω/c)ΔΓ(z) to account for the field dephase due to the non-linearity.

The key ingredients of this naive model, which intends to identify the basic physical processes behind the observed features, are the assumption of two channel regions of effective non-linear refraction index n ₀ + n ₂ I(z), corresponding to the zeroth diffraction orders of each slit, and the coordinate z ₁ in which the first diffraction orders are effectively detached from them. The appropriate values for these factors have to be estimated from the experiment according to a best fit criteria between the model and the experimental findings. To do this, we treat z ₁ as a fitting parameter, and approximate ΔΓ(z) to first order as ΔΓ(z) = α(z - z ₁). We have, therefore, found a best fit of the parameters z ₁ and α to describe the deviation of the divergence of the lateral fringes in the interference pattern of the non-linear propagation compared with the linear case. The results forz1 = 3.5 m and α = -10^-8 are shown in Fig. 3. Note that the value z ₁ is perfectly consistent with our interpretation as the coordinate where the first order diffraction maxima are detached from the central field (situation close to the represented in Fig. 4(a)). In fact, the actual choice of parameters does not result very critical, except for the sign of α. A positive α leads to an increase of the divergence of the lateral fringes with the intensity, which has not been observed in our experiments. Moreover, a negative α is consistent with the intuition of a decrease of the intensity in the radiation channels with the distance to the slits, which should lead also to a gradual decrease in the non-linear refraction index.

4. Discussion

The simple theoretical model presented in section 3 not only has helped us to identify the effects that are important to explain the non-linear Young’s experiment, but also has clarified the interpretation of the experiment regarding the wave or particle nature of the observed phenomena.

It is clear from the previous discussion, that our experiments reflect the interference process associated to the wave nature. In contrast, the particle behavior has been reported by Mitchell and coworkers [15] in a similar situation. They concentrate on higher power situations where the self-focusing results on the generation of self-guided beams, and demonstrate that the non-linearity generated by one of the beams changes the propagation of the other. Therefore, the non-linear intensity gradients affect the trajectory of the beams as if they were interacting particles. We have looked carefully our experimental results in the light of Mitchell’s point of view, and concluded that in our case the intensity maxima cannot be considered as self-guided beams but the result of an interference process. The change of the divergence of the maxima should be attributed to the self-induced phase mechanism and not to gradients of the non-linear index. In this sense our experiment still reflects the wave nature of the process, despite it exists a strong limitation in the diffraction of the zeroth diffraction orders. In conclusion, we place our experiment in between the linear case, where the wave nature is apparent, and the high non-linear case (Mitchell’s situation), where the pure particle interpretation can be done.

5. Conclusion

We performed the Young’s double slit experiments for fields intense enough to induce a nonlinear refraction index in air. Our results clarify the role of the non-linear propagation in the build-up of an interference pattern. Besides the disappearance of the central interference maximum from the explored range of distances, which can be attributed to the reduction of the beam diffraction at each slit due to the self-focusing action, the divergence of the lateral maxima is found to be affected. A simple model can be drawn to explain quantitatively this effect as a result of the self-phase modulation of the beams. The resulting interference pattern is explained as the superposition of the different diffraction orders of each slit. In the nonlinear case, these diffraction orders are affected differently by the non-linearity producing the phenomena observed in the experiment. We believe that this concept of differential non-linear behavior with the diffraction order can be fundamental in understanding the interference process of more complex systems, or at higher fluencies that would lead to regular filamentation.

Moreover, we have discussed in which side of the wave or particle interpretation should we place this experiment. Since our results can be interpreted in terms of interferences, and phase effects induced by the non-linearity, we conclude that the particle perspective of [15] cannot be applied.

This work has been partially supported by the Spanish Ministerio de Ciencia y Tecnologia (FEDER funds, grant FIS2005-01351) and by the Junta de Castilla y Leon (SA026A05)