1. Introduction

Nonlinear interactions between two or more laser beams, pulses, and filaments [1] are related to applications ranging from modulation methods in optical communication [2], to coherent combination of beams [3–7], interactions between filaments in atmospheric propagation [8], focusing of multiple speckle patterns [9] and ignition of nuclear fusion using up to 192 beams [10]. In the integrable one-dimensional cubic case, such interactions can only lead to phase and lateral shifts, which can be computed analytically using the Inverse Scattering Transform [11–13]. In the non-integrable case, however, richer dynamics are possible, including beam repulsion, breakup, fusion and spiraling [1, 14–16]. Since the outcome of the interaction strongly depends on the relative phases of the beams [17], one can use the initial phase to control the interaction dynamics [18]. Nonlinear interactions between solitary waves were also studied in other physical systems [19, 20], such as fiber optics [21, 22], waveguide arrays [23], water waves [24,25], plasma waves [26] and Bose-Einstein condensates [27].

In previous studies it was shown, both theoretically and experimentally, that when a laser beam undergoes an optical collapse, its initial phase is “lost”, in the sense that the small shot-to-shot variations in the input beam lead to large changes in the nonlinear phase shift of the collapsing beam [28,29]. Therefore, if two such beams intersect after they collapsed, one cannot predict whether they will intersect in- or out-of-phase, and so post-collapse interactions between beams become “chaotic” and cannot be controlled [30]. Loss of phase due to input noise can also interfere with imaging and holography in nonlinear medium [31, 32]. Note that loss of phase does not imply a loss of coherence, but rather that at any given propagation distance, the coherent beam can only be determined up to an unknown constant phase. Thus, if the unperturbed beam profile is given by ψ = Ae^iS, then in the presence of input noise ψ̃ = e^iS̃, where Ã(z, x) ≈ A(z, x), S̃(z, x) ≈ S(z, x) + θ(z), and θ is an unknown O(1) perturbation of the nonlinear phase shift which varies with the propagation distance z but is independent of the transverse coordinates x.

In this study we show that loss of phase is ubiquitous in nonlinear optics. Thus, while collapse accelerates the loss of phase process, non-collapsing or mildly-collapsing beams can also undergo a loss of phase. The loss of phase builds up gradually with the propagation distance, i.e., the shot-to-shot variations of the beam’s nonlinear phase shift increase with the propagation distance, and approach a uniform distribution in [0, 2π] at sufficiently large distances. As noted, because of the loss of phase, deterministic predictions and control of interactions between laser beams become impossible in the presence of noise. We show, however, that loss of phase allows for accurate predictions of the statistical properties of these stochastic interactions, even without any knowledge of the noise source and characteristics. Indeed, because the relative phase between the beams becomes uniformly distributed in [0, 2π], the statistics of the interaction are universal, and can be computed using a “universal model” in which the only noise source is a uniformly distributed phase difference between the input beams. These computations can be efficiently performed using a polynomial-chaos based approach.

2. Loss of phase

The propagation of laser beams in a homogeneous medium is governed by the dimensionless nonlinear Schrödinger equation (NLS) in d + 1 dimensions

2.1. Numerical observations

In a physical system the input beam always changes from shot to shot due to noise and instabilities of the laser system. To model this shot to shot variation, we write

For example, consider the one-dimensional cubic-quintic NLS (2) with the Gaussian initial condition with a random power

We obtained similar results for the same equation and initial condition in two dimensions. We found that loss of phase occurs much faster in two dimensions, so that the φ̃ becomes uniformly distributed already after two diffraction lengths (z ≈ 2). See Appendix A for further details.

 

Fig. 1 The cubic-quintic NLS (2) with d = 1, ∊ = 10⁻³, and the initial condition (5) at (a₁)–(a₄) z = 0.15, (b₁)–(b₄) z = 3, and (c₁–c₄) z = 11. (a₁)–(c₁) Cumulative on-axis phase as a function of α. (a₂)–(c₂) Non-cumulative on-axis phase. (a₃)–(c₃) The PDF of φ̃. (a₄)–(c₄) Transverse profile for α = 1 (solid) and α = −1 (dot-dash).

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2.2. Theory

To understand the emergence of loss of phase in Fig. 1, we note that after an initial transient, the beam core evolves into a stable solitary wave, see e.g., Fig. 2(a), and so,

 

Fig. 2 Same as Fig. 1. (a) The intensity |ψ(z, x)|² for α = 0. (b) The on-axis phase for α = 1 (solid) and α = −1 (dots).

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Since α is randomly distributed, then so is κ(α). More generally, for any initial noise, the beam core evolves into a solitary wave with a random propagation constant κ(α), and so φ is given by (7) [note that this is also the case with multi-parameter noises, e.g., when the amplitude, the phase and the tilt angle are all random]. Consequently, the initial on-axis phase is completely lost as z → ∞:

Lemma 1 Let α be a random variable which is distributed in [α_min, α_max] with an absolutely-continuous measure dμ, let κ(α) be a continuously differentiable, piece-wise monotone function on [α_min, α_max], let η₀(α) be continuously differentiable on [α_min, α_max], and let φ be given by (7). Then

Lemma 1 provides a new road to the emergence of loss of phase. Indeed, in previous studies [28–30], the loss of phase was caused by the large self-phase modulations (SPM) that accumulate during the initial beam collapse (i.e., by the variation of η₀ in α). Briefly, when a beam undergoes collapse, then in the absence of a collapse-arresting mechanism, φ₀(α) → ∞ as z → Z_c (α), where Z_c is the collapse point [28]. Therefore, if a beam undergoes a considerable self-focusing before its collapse is arrested, then it accumulates significant SPM, i.e., η₀(α) ≫ 2π. In that case, although small changes in α lead to small relative changes in η₀(α), those are O(1) absolute changes in η₀(α). In this study, however, we consider non-collapsing beams of the 1D NLS, or self-trapped beams of the 2D NLS. Therefore Δη₀ := η₀(α_max) − η₀(α_min) ≪ 2π. In such cases, the loss of phase builds up gradually with the propagation distance z, and not abruptly during the initial collapse, as in the previous studies.

Loss of phase is related to the fact that NLS solitary waves can “only” be orbitally stable; i.e., stable up to multiplication by e^iθ for some θ ∈ [0, 2π], see [39,40]. Loss of phase is a genuinely nonlinear phenomenon. Indeed, in the linear propagation regime, $\psi (z,\mathbf{x})={\left(2iz\right)}^{-\frac{1}{2}}{e}^{u\frac{{\left|\mathbf{x}\right|}^2}{4z}}*{\psi }_0(\mathbf{x})$. Therefore if ψ₀(α₁) − ψ₀(α₂) ≪ 1 then ψ(α₁) − ψ(α₂) ≪ 1 as well.

Lemma 1 is reminiscent of classical results in ergodic theory of irrational rotations of the circle [41]. Unlike these results, however, Lemma 1 does not describe the trajectory of a single point on the circle under consecutive discrete phase additions, but rather the convergence of a continuum of points under with continuous linear change with a varying rate κ.

By (7), the maximal difference in the cumulative phase between solutions grows linearly in z, i.e.,

3. Stochastic interactions and the universal model

A priori, the loss of initial phase has no physical implications, since the NLS (1) is invariant under the transformation ψ → e^iβψ. When the NLS (1) is non-integrable, however, the relative phase between two beams [16,17,30,42] or condensates [27] can have a dramatic effect on their interaction, thus making the loss of initial phase physically important. To illustrate that, consider again the cubic-quintic NLS (2) for d = 1 with the two crossing beams initial condition

 

Fig. 3 The 1d cubic-quintic NLS (2) with ∊ = 2· 10⁻² and the initial condition (9) with κ₀ = 8. (a) κ₁ = 8, η₀ = 0. (b) κ₁ = 8.1, η₀ = 0. (c) κ₁ = 8.1, η₀ ≈ −0.48π.

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In what follows, we consider interactions between the two crossing beams

 

Fig. 4 Solutions of the 1d cubic-quintic NLS (2) with ∊ = 2 · 10⁻². (a₁)–(d₁): the exit intensity |ψ(z_f = 17, x; α)|², (a₂)–(d₂): the probability of the number of output beams, and (e) the mean (▵, ★, ○, □) and standard deviation of the lateral location of the output beams, for the noisy initial conditions (11a)–(11d), respectively. Here a = 12, $\theta =\frac{\pi }{8}$, and κ₀ = 8.

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In a physical setting the noise distribution is typically unknown. Nevertheless, the on-axis phase of each beam core is given by (7), where κ(α) is a random variable. Therefore, by Lemma 1, for z_cross ≫ Z_lop, the phase of each beam at (z_cross, x_cross), and hence also the relative phase between them, is uniformly distributed in [0, 2π]. Hence, to leading order, the statistics of the interactions are given by a universal model in which the only noise source is a random relative phase, which is uniformly distributed in [0, 2π], i.e.,

4. Novel numerical method

The above results show that the statistics of long-range interactions between laser beams are independent of the noise source and its characteristics, and can be computed using the “universal model” (12) in which the only randomness comes from the addition of a random constant phase to one of the input beams, which is uniformly distributed in [0, 2π], c.f. (11d). To efficiently use the universal model, we developed a polynomial-chaos based numerical method, which is both spectrally accurate and makes use of any deterministic NLS solver. Although in the universal model the noise is uniformly distributed, we allow for a more general noise distribution, so that we can e.g., produce results such as figure 1 for non-uniform noise distributions.

Let ψ(z, x; α) be the solution of the NLS (1) with the random initial condition (5). In what follows, we introduce an efficient numerical method for computing the statistics of g(ψ), e.g., the average intensity over many shots 𝔼_α[|ψ|²].

The standard numerical method for this problem is Monte-Carlo, in which one draws N random values of α and approximates $E_{\alpha }\left[g(\alpha )\right]\approx \frac{1}{N}\sum _{n=1}^{N}g({\alpha }_n)$. The main drawback of this method is its slow $O(1/\sqrt{N})$ convergence rate, where N is the number of NLS simulations. If g(α) : = g(ψ(·; α)) is smooth in α, however, we can use orthogonal polynomials as a spectrally accurate basis for interpolation [43] and numerical integration. Let α is distributed in [α_min, α_max] according to a PDF c(α), and let ${\left\{p_n(\alpha )\right\}}_{n=0}^{\infty }$ be the corresponding sequence of orthogonal polynomials, in the sense that $\underset{{\alpha }_{\text{min}}}{\overset{{\alpha }_{\text{max}}}{\int }}{p}_n(\alpha )p_m(\alpha )c(\alpha )\hspace{0.17em}d\alpha ={\delta }_{n,m}$. For example, if α is uniformly distributed in [−1, 1], then {p_n} are the Legendre polynomials, and if α is normally distributed in (−∞, ∞), then {p_n} are the Hermite polynomials. Recall that for smooth solutions one has the spectrally accurate quadrature formula $E_{\alpha }\left[g(\alpha )\right]\approx \sum _{j=1}^{N}g\left({\alpha }_j^N\right)w_j^N$, where ${\left\{{\alpha }_j^N\right\}}_{j=1}^N$ and ${\left\{w_j^N\right\}}_{j=1}^N$ are the roots of the orthogonal polynomial p_N (α) and their respective weights $w_j^N=\underset{{\alpha }_{\text{min}}}{\overset{{\alpha }_{\text{max}}}{\int }}\prod _{i=1,i\ne j}^N\frac{\alpha -{\alpha }_i^N}{{\alpha }_j^N-{\alpha }_i^N}c(\alpha )\hspace{0.17em}d\alpha$. See [44] for a numerically efficient and stable algorithm for computing the quadrature points and weights ${\left\{{\alpha }_j^N,w_j^N\right\}}_{j=1}^N$. We apply the collocation Polynomial Chaos Expansion (PCE) method as follows [45,46]:

This method is “non-intrusive”, i.e., it does not require any changes to the deterministic NLS solver. Moreover, the orthogonality of {p_n} leads to direct formulae for the mean and standard deviation of g:

As noted, the PCE method, presented in its basic form in steps 1–2, has a spectral convergence rate for smooth functions. For example, the results in Figs. 1 and 5 were computed using N = 10 and N = 31 NLS simulations, respectively. To reach a similar accuracy with the Monte Carlo method would require more than 1000 NLS simulations. Some quantities of interest, however, such as the number of filaments (Fig. 4(a₂)–4(d₂)), or the non-cumulative on-axis phase φ̃ = arg (ψ(z, x = 0; α)) mod(2π), (Fig. 1(a₃)–1(c₃) and Fig. 5(a₃)–5(c₃)) are non-smooth. Therefore, a straightforward application of the PCE method for such quantities requires O (10³) simulations to converge. In such cases, we begin with stages (1)–(2) and calculate the PCE approximation (13) of the smooth function ψ(z, x; α) using ${\left\{\psi (z,\mathbf{x};{\alpha }_j^N)\right\}}_{j=1}^N$ with a relatively small N. Then we proceed as follows:

 

Fig. 5 Same as figure 1 for d = 2.

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For example, when we computed the number of beams at z = z_f in Fig. 4, we first computed the PCE interpolant ψ_N (z_f, x; α) with N = 71. Then we computed ψ(z_f, x; α̃_m) ≈ ψ_N (z_f, x; α̃_m) for m = 1, . . . , M = 801. For each α̃_m, we count the number of filaments and used this to produce the histogram in figure 4(a₂)–4(d₂). The additional computational cost of sampling ψ_N (13) at M ≫ N grid points in step (3) is negligible compared to directly solving the NLS for N times in stage (1).

5. Conclusions

In this study, we showed that when two (or more) laser beams interact after a sufficiently long propagation distance, their relative phase at the crossing point varies so much from shot to shot, that the outcome of their interaction cannot be deterministically predicted or controlled. In such cases, the notion of a “typical experiment” or a “typical solution” may be misleading, and one should adopt a stochastic approach. The loss of phase can explain some of the difficulties in phase-dependent methods in optical communications such as Quadrature Amplitude Modulation (QAM) [2], and in coherent combining of hundreds of laser beams in a small space for ignition of nuclear fusion [15], and for creating a more powerful laser beam [6]. In these applications, controlling the phases of the input beams or pulses might not provide a good control over their interaction or combination, due to the loss of phase. Our study suggests that controlling the relative phases at the intersection point may be achieved by either shortening the propagation distance, or by coupling the beams throughout the propagation. Loss of phase can also explain the high sensitivity to initial perturbations of interactions between topologically-charged solitons [47]. Loss of phase is also relevant to the loss of polarization for elliptically-polarized beams [48].

Appendix

A. Loss of phase in two dimensions

We repeated the simulations of Fig. 1 in two dimensions, see Fig. 5. As in the one-dimensional case, the phase variation increases with the propagation distance, and becomes uniformly distributed around z = 2, see Fig. 5(c₃).

To compare the rates at which the PDFs of φ̃ converge to the uniform distribution $f_{\text{uni}}(y):=\frac{1}{2\pi }$ on [0, 2π], we plot in Fig. 6(a) the distance $\Vert f-f_{\text{uni}}\Vert :=\underset{0}{\overset{2\pi }{\int }}\left|f(y)-\frac{1}{2\pi }\right|\hspace{0.17em}dy$. Note that the convergence is not monotone, because the distance has a local minimum whenever Δφ = 2πk for an integer k.

 

Fig. 6 The cubic-quintic NLS (2) with ∊ = 10⁻³, and the initial condition (5) in one (dot-dash) and two (solid) dimensions. (a) Distance between the PDF of φ̃ and the uniform distribution on [0, 2π]. (b) The propagation constant of the beam core, see (6), as a function of α.

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The convergence is much faster for d = 2 than for d = 1. This is because typically, Δκ is much larger in 2d than in 1d. For example, in Fig. 6(b) Δκ ≈ 25 in 2d, and Δκ ≈ 1.8 in 1d. Intuitively, this is because the input beam evolves into a solitary wave, and over a given power range, the propagation constant of the solitary wave changes considerably less in 1d than in 2d. Hence, by (8), the loss of phase occurs much faster in the two-dimensional case than in the one-dimensional case, thus explaining Fig. 6(a).

The reason why Δκ is much larger in 2d than in 1d is as follows: denote the solitary-wave power by P(κ) := ∫ |R_κ|² dx. When ∊ = 0 in (2), then $\frac{dP}{d\kappa }=0$ for d = 2 but $\frac{dP}{d\kappa }>0$ for d = 1 [33]. Hence, if 0 < ∊ ≪ 1, then $\frac{dP}{d\kappa }=O(∊)$ for d = 2, but $\frac{dP}{d\kappa }=O(1)$ for d = 1. Therefore $\frac{d\kappa }{dP}=O(1)$ for d = 1 but $\frac{d\kappa }{dP}=O\left(\frac{1}{∊}\right)$ for d = 2.

B. Proof of Lemma 1

For a given z ≥ 0, denote ${\phi }_z(\alpha )=\kappa (\alpha )+\frac{{\eta }_0(\alpha )}{z}$, then φ̃(α) = zφ_z (α) mod (2π). We prove that $\underset{z\to \infty }{\text{lim}}\tilde{\phi }(\alpha )~U\left([0,2\pi ]\right)$ by showing that for every 0 ≤ a < b ≤ 2π,

(14)$$\underset{z\to \infty }{\text{lim}}\mu \left({\tilde{\phi }}^{-1}\left([a,b]\right)\right)=\frac{b-a}{2\pi },$$

where φ̃ [a, b] : = {α ∈ [α_min, α_max] |ũ_z (α) ∈ [a, b]}.

We first prove the lemma for a strictly monotone function κ on (α_min, α_max). For sufficiently large z, φ_z is also monotone. Let $x_k^z$ and $y_k^z$ be the solutions of

(15)$${\phi }_z\left(x_k^z\right)=\frac{2k\pi +a}{z},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\phi }_z\left(y_k^z\right)=\frac{2k\pi +b}{z},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}k\in \mathbb{Z}.$$

There exists k_min ≤ k_max such that $x_{k_{\text{min}}-1}^z$ and $y_{k_{\text{max}}+1}^z$ do not exist, and for clarity we suppressed the dependence of k_min and k_max on z. By definition,

(16)$$\begin{array}{c}\mu \left({\tilde{\phi }}^{-1}([a,b])\right)=\sum _{k=k_{\text{min}}}^{k_{\text{max}}}\mu \left(x_k^z,y_k^z\right)+E(z)=\\ =\sum _{k=k_{\text{min}}}^{k_{\text{max}}}\mu \left({\phi }_z^{-1}\left(\frac{2\pi k+a}{z}\right),{\phi }_z^{-1}\left(\frac{2\pi k+b}{z}\right)\right)+E(z),\end{array}$$

where the error term E(z) exists if either $y_{k_{\text{min}}-1}^z>{\alpha }_{\text{min}}$ or $x_{k_{\text{max}}+1}^z<{\alpha }_{\text{max}}$ exist. In such cases, since dμ is continuous,

$$E(z)=\mu \left(\left[{\alpha }_{\text{min}},y_{k_{\text{min}}-1}^z\right]\right)+\mu \left(\left[x_{k_{\text{max}}+1}^z,{\alpha }_{\text{max}}\right]\right).$$

We now show that if $y_{k_{\text{min}}-1}^z$ exists, then $\underset{z\to \infty }{\text{lim}}\mu \left({\alpha }_{\text{min}},y_{k_{\text{min}}-1}^z\right)=0$ (a similar proof holds also for $x_{k_{\text{max}}+1}^z$). It is enough to show that $\underset{z\to \infty }{\text{lim}}{x}_{k_{\text{min}}}^z={\alpha }_{\text{min}}$, because $y_{k_{\text{min}}-1}^z<x_{k_{\text{min}}}$ and μ is a continuous measure. Let δ > 0, then

$$\begin{array}{c}z{\phi }_z({\alpha }_{\text{min}}+\delta )-z{\phi }_z({\alpha }_{\text{min}})=\\ \left({\eta }_0({\alpha }_{\text{min}}+\delta )-{\eta }_0({\alpha }_{\text{min}})\right)+z\left(\kappa ({\alpha }_{\text{min}}+\delta )-{\kappa }_{\text{min}}\right)\end{array}$$

goes to infinity as z → ∞. Therefore, for large enough z, $x_{k_{\text{min}}}^z\in ({\alpha }_{\text{min}},{\alpha }_{\text{min}}+\delta )$. Thus, for all δ > 0,

$${\alpha }_{\text{min}}\le \underset{z\to \infty }{\text{lim}}{x}_{k_{\text{min}}}^z<{\alpha }_{\text{min}}+\delta .$$

Since φ_z is strictly monotone, then by the inverse function theorem ${\phi }_z^{-1}\in C^1$, and so by substituting $\alpha ={\phi }_z^{-1}(y)$,

$$\begin{array}{l}\mu \left({\phi }_z^{-1}\left(\frac{2\pi k+a}{z}\right),{\phi }_z^{-1}\left(\frac{2\pi k+b}{z}\right)\right)=\\ \underset{{\phi }_z^{-1}\left(\frac{2\pi k+a}{z}\right)}{\overset{{\phi }_z^{-1}\left(\frac{2\pi k+b}{z}\right)}{\int }}c(\alpha )d\alpha =\underset{\frac{2\pi k+a}{z}}{\overset{\frac{2\pi k+b}{z}}{\int }}{g}_z(y)dy,\end{array}$$

where $g_z(y):=c\left({\phi }_z^{-1}(y)\right){\left({\phi }_z^{-1}\right)}^{\prime }(y)$. By Lagrange mean-value theorem, for each index k, there exists ${\xi }_k^z\in (a,b)$ such that

$$\begin{array}{c}\mu \left({\phi }_z^{-1}\left(\frac{2\pi k+a}{z}\right),{\phi }_z^{-1}\left(\frac{2\pi k+b}{z}\right)\right)=\\ g_z\left(\frac{2\pi k+{\xi }_k^z}{k}\right)\frac{b-a}{z}.\end{array}$$

Substituting the above into (16) yields

(17)$$\mu \left({\phi }_z^{-1}([a,b])\right)=\frac{b-a}{z}\sum _{k=k_{\text{min}}}^{k_{\text{max}}}{g}_z\left(\frac{2\pi k+{\xi }_k^z}{z}\right)+E(z).$$

Next, consider the integrals

(18a)$$\begin{array}{l}I:=\underset{{\alpha }_{\text{min}}}{\overset{{\alpha }_{\text{max}}}{\int }}c(\alpha )\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}dy=\mu ({\alpha }_{\text{min}},{\alpha }_{\text{max}})=1,\\ I_2:=\underset{x_{k_{\text{min}}}^z}{\overset{y_{k_{\text{max}}}^z}{\int }}c(\alpha )d\alpha =\underset{\frac{2\pi k_{\text{min}}+a}{z}}{\overset{\frac{2\pi k_{\text{max}}+b}{z}}{\int }}{g}_z(y)dy.\end{array}$$

Using Riemann sums

(18b)$$I_2=\frac{2\pi }{z}\sum _{k=k_{\text{min}}}^{k_{\text{max}}}{g}_z\left(\frac{2\pi k+{\xi }_k^z}{z}\right)+O\left(z^{-1}\right).$$

Denoting φ_z (α_min) : = φ_z,min and φ_z (α_max) : = φ_z,max Since

$$\underset{{\phi }_{z,\text{min}}}{\overset{{\phi }_{z,\text{max}}}{\int }}=\underset{{\phi }_{z,\text{min}}}{\overset{{\phi }_z\left(x_{k_{\text{min}}}^z\right)}{\int }}+\underset{{\phi }_z\left(x_{k_{\text{min}}}^z\right)}{\overset{{\phi }_z\left(y_{k_{\text{max}}}^z\right)}{\int }}+\underset{{\phi }_z\left(y_{k_{\text{max}}}^z\right)}{\overset{{\phi }_{z,\text{max}}}{\int }},$$

then I = I₂ + O(z⁻¹). Equating (18b) and (18a), and substituting into (17), yields

$$\mu \left({\phi }_z^{-1}([a,b])\right)=\frac{b-a}{2\pi }+o(1),$$

by which we prove (14)

Finally, if κ, hence φ_z is piece-wise monotone, we apply the above proof for each sub-interval over which φ_z is monotone, and by additivity of measure have the result.

Funding

The research of AS and GF was partially supported by grant #177/13 from the Israel Science Foundation (ISF).

Acknowledgment

We thank B.A. Malomed and A.L. Gaeta for useful discussions.

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