Arago spot formation in the time domain

Before discussing our experiments, let us first recall the basis of the analogy between the spatial evolution of light affected by diffraction and the temporal changes experienced by light when dispersion is involved. We consider the simple case where a monochromatic plane wave with wavelength λ and an amplitude a₀ illuminates an opaque strip with a width 2l and an infinite length. In this 1D transverse problem that is illustrated in figure 1(a), the longitudinal evolution of light a(x, z) in the scalar approximation is ruled by the following differential equation:

$$\begin{eqnarray}&&i\displaystyle \frac{\partial a}{\partial z}=-\displaystyle \frac{1}{2{k}_{0}}\displaystyle \frac{{\partial }^{2}a}{\partial {x}^{2}},\end{eqnarray} \tag{ 1 }$$

with x and z being the transverse and longitudinal coordinates, respectively (z being large with respect to l), and k₀ = 2π/λ the wavenumber. The two edges of the opaque strip diffract the light leading to the progressive emergence of a bright spot at the center of the wire shadow, as experimentally reported with simple experiments [15–17]. Note that this spot is less intense and spatially broader than that in the historic Arago experiment conducted in the space domain and using a 2D opaque perfectly circular screen illuminated by a spatially fully-coherent source. Indeed in 2D, the Arago bright spot results as a constructive interference of the light passing close to the edge of the 2D opaque screen.

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The goal of the present paper is to study the temporal equivalent of a 1D Arago experiment where the stop is an opaque strip. We therefore consider a fully-coherent continuous wave where light has been switched off for a duration 2 T₀ as illustrated in figure 1(b1) and modelled by the function a(t, z = 0) = a₀ [1 − rect(t/2T₀)], where rect is the rectangular function of width 1 and t being the temporal coordinate. This shaped temporal waveform then propagates in a dispersive single mode waveguide, typically an optical fiber, that ensures that its spatial transverse profile is unaffected upon propagation. As a first order approximation which is usually valid for waveforms with a duration of a few tens of picoseconds [18], the temporal profile of the light in the approximation of the slowly varying envelope evolves according to:

$$\begin{eqnarray}&&i\displaystyle \frac{\partial a}{\partial z}=\displaystyle \frac{1}{2}{\beta }_{2}\displaystyle \frac{{\partial }^{2}a}{\partial {t}^{2}},\end{eqnarray} \tag{ 2 }$$

with β₂ being the group velocity dispersion. After propagation in the waveguide, dispersion leads to the alteration of the temporal profile (figure 1(b2), blue curve): the initially very sharp edges of the waveform have been smoothened, strong oscillations have appeared and a light intensity increase has emerged at the dark pulse center where initially no light was present. This last feature being the temporal counterpart of the Arago spot in this 1D experiment.

The space-time duality readily appears in the mathematical structure of equations (1) and (2) where space and time are exchanged and the link between diffraction and dispersion is clear [19]. Indeed, both effects fulfill the same normalized differential equation:

$$\begin{eqnarray}&&i\displaystyle \frac{\partial \psi }{\partial \xi }=-\displaystyle \frac{1}{2}\displaystyle \frac{{\partial }^{2}\psi }{\partial {\eta }^{2}},\end{eqnarray} \tag{ 3 }$$

where ξ is a normalized propagation distance ξ = z/L_D where L_D is defined by L_D = −T₀²/β₂ (for dispersion, note that with this convention L_D is negative for normally dispersive fibers and positive for anomalous dispersion) or L_D = k₀ l² (for diffraction). η is the normalized time η =t/T₀ (dispersion) or the normalized transverse coordinate η = x/l (diffraction). ψ = a/a₀ is the field a normalized with respect to the amplitude of the initial plane wave. Consequently, both diffraction and dispersion implies the development of a quadratic spectral phase and lead formally to similar consequences. This space-time duality is useful to better understand the temporal evolution of temporally sculpted short pulses subject to dispersion. As a first rough approximation, we can consider that the pattern created by the opaque temporal zone is the result of the superposition of the patterns created by a semi-infinite edges located at η = ±1. The diffraction pattern of a sharp edge is indeed among the first examples that are taught theoretically and experimentally to students when introducing Fresnel's diffraction [16, 17, 20, 21]. Indeed, the Fraunhofer regime that requires w²/(λ z) (w being the width of the aperture) can never be reached for a semi-infinite screen. Consequently, far-field approximations do not hold here but the edge problem can be solved analytically by involving Fresnel's integrals and the graphical Cornu's spiral plot. Indeed, the field ψ_E diffracted by a single initial sharp edge centered at η = 0 is given by:

$$\begin{eqnarray}&&{\psi }_{E}(u)=\,0.5+C(u)+i\,\left(0.5+S(u)\right),\end{eqnarray} \tag{ 4 }$$

where u = η/(π ξ)^1/2 and C and S are the Fresnel's integrals. This semi-infinite screen (grey dashed line in figure 1(b1)) brings some qualitative insight to understand the softening of the edges as well as the nature of the strong fluctuations that develop on each side of the edge in the blue curve of figure 1(b1). However, the addition of the temporal intensity profiles created by each intensity jump (black line in figure 1(b1)) cannot account for the development of the central spot that is the result of a constructive interference process: given the symmetry of our problem, the patterns created by each edge are in phase at the center of the waveform, leading to an increase by a factor 2 of the intensity profile. This intensity increase is the 1D temporal Arago bright spot.

The field diffracted by the opaque 1D screen can be fully analytically expressed by means of Fresnel integrals:

$$\begin{eqnarray}&&\begin{array}{l}{\psi }_{S}(\eta ,\xi )=\displaystyle \frac{1}{\sqrt{2}}\left[\left(1+C\left(\displaystyle \frac{\eta -1}{\sqrt{\pi \,\xi }}\right)+C\left(\displaystyle \frac{-\eta -1}{\sqrt{\pi \,\xi }}\right)\right)\right.\\ \,\,\,\,\,+\,\left.i\left(1+S\left(\displaystyle \frac{\eta -1}{\sqrt{\pi \,\xi }}\right)+S\left(\displaystyle \frac{-\eta -1}{\sqrt{\pi \,\xi }}\right)\right)\right].\end{array}\end{eqnarray} \tag{ 5 }$$

We consider here the time domain evolution and we have plotted on figure 2(a) the temporal amplitude and phase profiles of the wave at three different propagation distances for an anomalous regime of propagation (positive L_D). We can make out that the amplitude of the ripple observed on the top level of the wave does not evolve much with propagation distance. The temporal position of the maximum of the oscillations tend to move away from the center of the waveform. Those trends are consistent with the diffraction pattern of a straight edge (dash line in figure 2(b1)) and we can note that the maximum value of the oscillation is obtained for η = ±(1 + 2.12 ξ^1/2) with a value of 1.37 expected in this case [22]. We concentrate now on the Arago spot formation located at the center of the diffracted pattern where one can notice the continuous increase of a central spot with increasing propagation distance (figure 2(a1)). The growth of the central spot is better seen on the longitudinal intensity evolution map plotted on figure 2(b1) and it can be quite easily expressed analytically. Using Z = (πξ)^−1/2, the field at the center of the waveform is

$$\begin{eqnarray}&&{\psi }_{S,0}(Z)=\sqrt{2}\left[\left(0.5-C\left(Z\right)\right)+i\,\left(0.5-S\left(Z\right)\right)\right]\end{eqnarray} \tag{ 6 }$$

or, equivalently, using the complex error function erf:

$$\begin{eqnarray}&&{\psi }_{S,0}(Z)=\displaystyle \frac{1+i}{\sqrt{2}}\,\left[1-erf\left(\displaystyle \frac{1-i}{2}\,\sqrt{\pi }\,Z\right)\right].\end{eqnarray} \tag{ 7 }$$

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The intensity profiles at the center of the waveform then evolves as:

$$\begin{eqnarray}&&{\left|{\psi }_{S,0}(Z)\right|}^{2}=\,2\,\left[{\left(C(Z)-0.5\right)}^{2}+{\left(S(Z)-0.5\right)}^{2}\right]\end{eqnarray} \tag{ 8 }$$

and can be graphically predicted using the Cornu's spiral as directly linked to the distance between the point of curvilinear coordinate Z and the fixed point (1/2,1/2) [20, 21]. Equation (8) also confirms analytically that ${\left|{\psi }_{S,0}(Z)\right|}^{2}$ is twice the intensity that may result from the incoherent addition of the intensity linked to each abrupt edge, i.e. ${\left(C(Z)-0.5\right)}^{2}\,+{\left(S(Z)-0.5\right)}^{2}.$ Writing equation (8) in real units leads to the explicit evolution of the temporal bright spot in the shadow according to the propagation distance or to the initial duration of the temporal shadow:

$$\begin{eqnarray}&&\begin{array}{l}{\left|{a}_{S,0}(z)\right|}^{2}=2\,{a}_{0}^{2}\left[{\left(C\left(\displaystyle \frac{{T}_{0}}{\sqrt{\pi \left|{\beta }_{2}\right|z}}\right)-0.5\right)}^{2}\right.\\ \,\,\,\,\,+\,\left.{\left(S\left(\displaystyle \frac{{T}_{0}}{\sqrt{\pi \left|{\beta }_{2}\right|z}}\right)-0.5\right)}^{2}\right].\end{array}\end{eqnarray} \tag{ 9 }$$

The intensity at the center of the waveform therefore continuously increases with the propagation distance z and tends asymptotically to a value equals to the intensity outside of shadow. On the contrary, given the scaling laws that apply, longer durations T₀ (which are formally equivalent to a larger obstacle) will lead, for a fixed propagation distance, to a reduced intensity of the central spot.

We complement this study by plotting the temporal phase profile of the diffracted field (figures 2(a2) and (b2)). Similarly to the diffraction of a straight edge [22], we can notice a significant phase difference between the central part and the plateau. Whereas the intensity profile is insensitive to the sign of the dispersion (normal or anomalous), the sign of the phase offset Δφ that exists between the central part and the plateau is influenced by the sign of the dispersion. Similarly to the intensity profile, the evolution of the phase offset can be expressed analytically as:

$$\begin{eqnarray}&&{\rm{\Delta }}\varphi =\arctan \left(\displaystyle \frac{1-2\,S(Z)}{1-2\,C(Z)}\right)-\displaystyle \frac{\pi }{4}.\end{eqnarray} \tag{ 10 }$$

We can note that the phase difference ∣Δφ∣ tends to decrease asymptotically towards 0.

Contrary to the usual diffraction in free space, propagation in a waveguide can also involve nonlinear effects. Indeed, the temporal evolution of a waveform in an optical fiber is affected by Kerr nonlinearity that can be taken into account through an additional term accounting for self-phase modulation in equation (2), leading to the well-known nonlinear Schrödinger equation (NLSE) [18]:

$$\begin{eqnarray}&&i\displaystyle \frac{\partial a}{\partial z}=\displaystyle \frac{1}{2}{\beta }_{2}\displaystyle \frac{{\partial }^{2}a}{\partial {t}^{2}}-\gamma {\left|a\right|}^{2}a,\end{eqnarray} \tag{ 11 }$$

with γ being the nonlinear coefficient of the fiber. For anomalous dispersive fiber, this equation can be normalized by using the 'soliton' number N defined as N² = ∣L_D∣/L_NL where L_NL is the nonlinear length L_NL = 1/γ∣a₀∣².

$$\begin{eqnarray}&&i\displaystyle \frac{\partial \psi }{\partial \xi }=-\displaystyle \frac{1}{2}\displaystyle \frac{{\partial }^{2}\psi }{\partial {\eta }^{2}}-{N}^{2}{\left|\psi \right|}^{2}\psi .\end{eqnarray} \tag{ 12 }$$

Any change in the duration T₀ therefore affects both the normalized propagation distance ξ as well as the soliton number N: in order to observe the same pattern for an increased value of T₀, it is required to increase the propagation distance and to decrease the input power. In the following, we will solve the scalar NLSE using numerical simulations based on the well-established split-step Fourier method [18] with a symmetrized and adaptive stepsize algorithm [23, 24].

We first consider the temporal evolution in an anomalous fiber for which the combination of dispersion and nonlinearity leads to a focusing behavior. The temporal intensity profile plotted for a fixed propagation distance ξ = 0.5 is shown in figure 3(a1) for three different light power level (expressed here through the soliton number). We can note several important points that stress how nonlinearity affects the temporal pattern. Whereas the positions of the oscillations that emerge on each side of the shadow are marginally affected, the amplitudes of these oscillations dramatically increase with increasing power and the oscillations become more and more abrupt. Such changes in the upper part of the signal have been the subject of recent discussions and can be interpreted in terms of solitons over finite background such as Akhmediev or Peregrine breathers [25, 26]. Similar structures linked to nonlinear Fresnel diffraction have also been observed in the spatial pattern resulting from the propagation in nonlinear Kerr material [27]. However, to the best of our knowledge, no study has reported the impact of nonlinearity on the central spot resulting from the constructive interference of the two edges. The level of this central Arago spot depends of the level of the signal, a higher amplitude of the input signal leading, in the anomalous regime of dispersion, to a decrease of the central Arago spot intensity (figure 3(a1)).

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The case of a defocusing nonlinearity (i.e. a normally dispersive fiber) leads to a very different evolution as shown in figure 3(a2). In this case, the lateral oscillations that emerge from each edge are significantly reduced with increasing power. The central Arago spot intensity tends to become larger with increasing power. Ultimately, for higher propagation distances or power, the gap tends to be filled and dark solitonic structures may emerge as has been shown in the temporal domain in the study of undular bores [28] and the interaction of a pair of delayed pulses [29], as well as in the spatial domain in defocusing materials [30, 31].

Both anomalous and normal regimes of propagation are summarized in figure 3(b) showing that the evolution is monotonic with power, with an increase of the Arago central spot when N increases in a defocusing medium (normal dispersion) and the contrary for the focusing medium (anomalous dispersion).

We first present the experimental results obtained in the linear regime of propagation. Details of three intensity and phase profiles are provided in figure 5 for three propagation distances (4, 8 and 12 km). They are qualitatively in line with the analytical predictions made in section 2.2 and reported in figure 2(a). The dispersive propagation is marked by the emergence of oscillations on each edge. A central Arago spot progressively grows while the phase difference ∣Δφ∣ between the central part and the surrounding plateau decreases.

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A more systematic study made every 500 m and presented in figure 6(a1) enables us to reconstruct the experimental longitudinal evolution of the temporal intensity profile that is found in qualitative agreement with the theoretical predictions presented in figure 2. As propagation distance increases, the oscillations move away from the central part but their amplitude does not evolve significantly. On the contrary, the amplitude of the Arago central spot continuously increases. Some deviations can however be noticed for the initial stage of propagation where ideal conditions lead to the emergence of numerous oscillations is the central part. Figure 6(b) summarizes the quantitative evolution of the peak power in the central spot that is compared with the analytical predictions of equation (8) based on the assumption of an abrupt edge. Whereas the analytical trends are well reproduced by the experiment, some discrepancies are visible for short propagation distances. Indeed, whereas the analytics predicts a level of the Arago central spot that should be already detectable after 2 km, the experiment does not reveal the clear existence of the central spot. When taking into account the exact initial temporal intensity profiles and when using numerical simulations to solve equation (2), we can note that these discrepancies vanish and the agreement regarding the temporal patterns obtained experimentally and numerically becomes quantitative (figures 6(a1) and (a2)). Using a second-order super-Gaussian pulse or perfect steps for the initial conditions leads to the same predicted result for large propagation distances (figure 6(b), red and blue solid lines). We can note that experimentally, after 15 km of propagation, the central Arago spot is already a third of the power present on the plateau in excellent agreement with the theory.

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Figure 7 focuses on the longitudinal evolution of the temporal phase profile. Similar conclusions as the ones drawn for figure 6 can be derived. Whereas the abrupt edge assumption is correct to describe qualitatively the experimental evolution, the value of ∣Δφ∣ predicted analytically by equation (10) is higher than the experimental records, especially for the early stages of propagation. Once again, taking into account the details of the edges of the initial condition enables to reproduce the measured results with a remarkable accuracy.

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Our second set of experiments focused on the impact of the nonlinearity. The output intensity profiles obtained after propagation in 10 km long fibers with normal or anomalous dispersion are plotted for three levels of input power in figure 8(a). These results confirm the different trends that have been previously identified and discussed in section 2.3. For the focusing nonlinearity case (Kerr nonlinearity with anomalous dispersion), an increasing input power leads to the progressive development of the lateral oscillations. Note, however, that compared to numerical simulations of figure 3(a1), the level of the lateral peaks is decreased and their temporal duration increased as compared to theory, due to the finite bandwidth of our electrical detection. Following the predicted trends, we observe a decrease of the central Arago spot when increasing the input power. The opposite behavior is observed for normally dispersive fibers: the central spot is increased whereas the lateral oscillations are lowered (see figure 8(a2)).

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A more systematic study of the output pattern according to the input power is reported in figure 9(a) and confirm those trends. In figure 9(b), we quantitatively summarized the evolution of the peak power of the Arago central spot according to the power and the regime of dispersion. The experimental results are in line with the results provided by the numerical integration of NLSE (red line) and are in excellent agreement with the numerical simulations taking into account the fiber loss and the initial second-order super-Gaussian temporal shape.

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