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Abstract
The nonlinear propagation of incoherent optical pulses is studied using a normalized nonlinear Schrödinger equation and statistical analysis, demonstrating various regimes that depend on the field’s coherence time and intensity. The quantification of the resulting intensity statistics using probability density functions shows that, in the absence of spatial effects, nonlinear propagation leads to an increase in the likelihood of high intensities in a medium with negative dispersion, and a decrease in a medium with positive dispersion. In the latter regime, nonlinear spatial self-focusing originating from a spatial perturbation can be mitigated, depending on the coherence time and amplitude of the perturbation. These results are benchmarked against the Bespalov–Talanov analysis applied to strictly monochromatic pulses.
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1. Introduction
Most applications of lasers rely on the high coherence of the generated radiation. High-energy lasers used for laser–matter interaction are typically narrowband, i.e., their coherence time is a significant fraction of their duration. Various laser–plasma instabilities (LPI’s) are detrimental to the interaction, e.g., by reducing the fraction of energy coupled into the target. This is an important limitation in high-energy-density plasma-physics experiments, and in particular for inertial confinement fusion. The use of incoherent pulses with coherence time much shorter than their duration has been shown via simulations to be a promising path to mitigate LPI [1–4], but the typical fractional bandwidth (ratio of the bandwidth to the central frequency) is out of reach for currently operating laser systems. Developing sources of low-coherence high-energy pulses has been of sustained interest to the high-energy laser community [5–8], but many developments to support a new generation of laser systems producing incoherent pulses with much larger bandwidth have recently occurred. These developments span a large range of technologies, including front-end development [9,10], laser amplification in broadband media [11,12], optical parametric amplification [13,14], broadband nonlinear mixing [15,16], and spectral broadening [17,18].
Small-scale self-focusing, as opposed to whole-beam self-focusing, has been shown early on to be the leading cause of intensity increase during the nonlinear propagation of optical pulses in liquids [19,20]. This phenomenon, caused by the combination of the Kerr effect and diffraction, can significantly impact the operation of laser systems [21,22]. The combination of diffraction and nonlinear phase in a material with positive n2 can lead to the growth of the spatial modulation followed by beam breakup and optical damage [19,22,23]. The canonical approach to study small-scale nonlinear self-focusing is to perform a variational analysis of the field at a given intensity, as originally presented by Bespalov and Talanov [19]. This analysis shows that the amplitude of a small sinusoidal perturbation on a beam with constant intensity can increase with a gain that depends on its period. It allows one to identify safe operating intensities for a high-energy laser system and implementation of far-field filtering to remove the spatial frequencies that are the most likely to self-focus [24].
Understanding the nonlinear propagation of incoherent pulses that have a strongly modulated temporal profile resulting from an incoherent mix of spectral components is of general interest. Because self-focusing is inherently an intensity-dependent process, it is likely to impact a constant-intensity coherent field and a strongly modulated incoherent field differently. The peak intensity within an incoherent pulse can be significantly higher than the average intensity, and such a situation was documented to be the source of damage in a multimoding Q-switched Nd:YAG oscillator [25]. Unlike for the monochromatic pulses considered in the Bespalov–Talanov analysis, the chromatic dispersion of the medium in which the pulses propagate must be considered for incoherent pulses. Previously presented simulations and experiments are generally indicative of self-focusing mitigation via bandwidth. The expected self-focusing in a laser system was simulated to be lower at higher bandwidths, and nonlinear propagation was shown to lead to a reduction of the high-intensity values within the pulse [26]. A linearized theory and experiments pertaining to the nonlinear interaction of two beams at different optical frequencies crossing at an angle (which is relevant to lasers having angular dispersion for beam smoothing) showed that the growth of the modulation via four-wave mixing depends on the difference in frequency and chromatic dispersion, allowing for the identification of conditions avoiding significant nonlinear growth [27]. Mitigation of the maximal intensity reached within a speckle of an incoherent field has been linked to the combination of chromatic dispersion and bandwidth, and similar intensities at the input and output of the nonlinear medium have been simulated at a fractional bandwidth equal to 1% [28]. The self-focusing properties of phase-modulated angularly dispersed pulses have been numerically studied [29]. Coherent and incoherent pulses obtained either by random phase modulation or use of a multimode seed source have been experimentally found to be impacted differently by self-focusing [30]. A statistical model pertaining to amplification in the presence of a Kerr nonlinearity has been developed [31]. The spatial contrast of a broadband incoherent pulse from a Ti:sapphire system propagating in CS2 was experimentally found to be consistent with lower values of the B-integral than expected for a coherent pulse [32]. These simulations and experiments indicate, either directly or indirectly, that higher bandwidth leads to better mitigation of nonlinear self-focusing, but drawing general conclusions is difficult considering the fact that they were performed over a limited range of parameters. The inherent random nature of the temporal intensity fluctuations also complicates the analysis and simulations.
In this work, a normalized version of the nonlinear Schrödinger equation supports simulations over a large parameter space of interest to laser development. These simulations are used to build statistically relevant data sets, e.g., probability density function of the intensity profile, for quantitative analysis. This analysis quantifies the importance of chromatic dispersion on the evolution of the intensity profile and identifies different regimes for self-focusing. In particular, it is shown that nonlinear propagation can lead to mild mitigation of self-focusing, for which the occurrence probability of high-output intensities is lower than the prediction from a direct application of the Bespalov–Talanov gain, and strong mitigation, for which the occurrence probability of high-output intensities is lower than in the input incoherent pulse. The critical parameter to assess the impact of nonlinear propagation is shown to be the normalized coherence time (which depends on both the bandwidth of the pulse and the chromatic dispersion of the medium), and not the bandwidth itself. Section 2 presents the normalized nonlinear Schrödinger equation supporting the simulations and discusses the Bespalov–Talanov analysis. Section 3 describes simulation results pertaining to the propagation of a plane wave with an incoherent time-domain field. Section 4 studies the occurrence and mitigation of self-focusing for incoherent pulses.
2. Normalized nonlinear Schrödinger equation
2.1 Derivation
A normalized nonlinear Schrödinger equation is derived to draw general conclusions about the nonlinear propagation of spectrally incoherent pulses. For propagation of a field E defined as a function of the longitudinal variable z, transverse variables x and y, and delayed time τ, the physical Schrödinger equation is
In Eq. (1), k and β2 are the wave number and second-order dispersion at the central frequency, respectively, and γ is the nonlinear Kerr coefficient. For propagation of the field over a distance L in the medium, the normalization intensity I0 is defined so that the accumulated nonlinear phase for a monochromatic plane wave of intensity I0 is equal to 1 radian, i.e., γLI0 = 1. The variables defined in Table 1 are used to rewrite Eq. (1) for the propagation of the normalized field e using normalized coordinates T, Z, X, and Y, leading to the normalized Schrödinger equation
where the – or + sign corresponds to a positive or negative dispersion β2, respectively. A similar equation has previously been derived, albeit with different normalized variables [28].
Table 1. Definition of variables for the normalized Schrödinger equation.
2.2 Interpretation of the variables
A change of normalized distance Z between 0 and 1 describes the propagation over the physical thickness of the medium L. Over this distance, in the absence of diffraction and dispersion, a field e verifying |e|2 = 1 accumulates a nonlinear phase equal to 1 radian. Because of the random fluctuations of the time-domain intensity, a time-averaged intensity 〈|e|2〉 is used to calculate an equivalent nonlinear phase Ψ = 〈|e|2〉 as a benchmark for incoherent pulses relative to a monochromatic plane wave.
The change of transverse spatial coordinate is interpreted in the context of Gaussian beam propagation of a monochromatic wave in the linear regime. For a Gaussian beam with waist w0, the field is $\exp ({{{ - {x^2}} / {w_0^2}}} )$ and the confocal parameter (twice the Rayleigh range) is $kw_0^2.$ Hence, a Gaussian beam with a normalized waist size equal to 1 has a confocal parameter equal to the length of the medium L. Therefore, the normalized scale in the transverse domain establishes the order of magnitude over which diffraction effects become significant over the medium’s length: fields with transverse variations over scales smaller than 1 tend to be significantly impacted by diffraction, whereas fields with slow variations over this scale do not experience significant diffraction.
The change of temporal coordinate is interpreted in the context of linear dispersive propagation of a plane wave with a Gaussian temporal power profile $\exp ({{{ - {t^2}} / {\Delta t_0^2}}} ),$ where $2\Delta t_0^2$ is the full width at 1/e2. The duration resulting from the second-order dispersion β2 L accumulated in the medium is $\Delta {t^2} = \Delta t_0^2 + {{{{({{\beta_2}L} )}^2}} / {\Delta t_0^2}},$ which leads to ${{\Delta {t^2}} / {{\beta _2}L}} = {{\Delta t_0^2} / {{\beta _2}L}} + {{{\beta _2}L} / {\Delta t_0^2}}.$ Hence, with the normalized temporal coordinate, chromatic dispersion during propagation in the medium increases the half-width at 1/e2 of a Gaussian pulse from 1 to $\sqrt 2 .$ In normalized units, fields with temporal variations over time scales smaller than 1 therefore experience significant pulse-shape changes due to chromatic dispersion, whereas fields with slow variations over these time scales tend not to be impacted by chromatic dispersion.
2.3 Bespalov–Talanov analysis for a coherent pulse
In this subsection, a coherent flat-in-time pulse is considered to establish a benchmark for self-focusing. Following the Bespalov–Talanov analysis, a monochromatic field with constant intensity Ψ is considered, i.e., the second-order time derivative in Eq. (2) is discarded. A sinusoidal perturbation of the field is introduced at the input of the medium (Z = 0) in the spatial domain:
These analytical results are compared to simulations of the self-focusing of a monochromatic pulse performed by solving Eq. (2) without the time-domain derivative. The input field has a super-Gaussian profile SGX to avoid aliasing issues at the edge of the sampled spatial range during numerical propagation, and its width is chosen to ensure uniform intensity Ψ over a range of transverse variable X large enough to accommodate the spatial frequencies of interest, following
The spatial modulation amplitude m is chosen equal to 0.01%, 0.1%, 1%, and 10%. The period for the maximal increase in output intensity pmax is iteratively determined by maximizing the output intensity calculated from the output field e(X,1), which is calculated using a split-step approach. The gain Gmax is then calculated as the intensity gain for the determined period. These simulated quantities are generally in good agreement with those determined from the Bespalov–Talanov variational analysis (Fig. 1). The largest discrepancies are observed when nonlinear propagation significantly modifies the spatial profile via diffraction, resulting in a non-sinusoidal spatial modulation. In this regime, pmax significantly differs from the value determined from the Bespalov–Talanov analysis, for example, for (m = 10%; Ψ > 3), (m = 1%; Ψ > 5), and (m = 0.1%; Ψ > 7). This typically corresponds to Gmax larger than 5.
Fig. 1. (a) Modulation period pmax and (b) intensity modulation gain Gmax for a coherent pulse of normalized intensity Ψ and a spatial modulation of amplitude equal to 10%, 1%, 0.1%, and 0.01% (solid lines). The black dashed line corresponds to the Bespalov–Talanov analysis.
3. Time-domain simulations of the nonlinear propagation of an incoherent pulse
3.1 Description
In this section, time-domain simulations are presented to elucidate the effect of chromatic dispersion, and in particular its sign, on the nonlinear propagation of spectrally incoherent pulses. This is relevant to the free-space propagation of a plane wave or guided propagation. In this regime, the Laplacian term in Eq. (2) is discarded, and one must only consider the second-order temporal derivative, which accounts for the second-order dispersion of the medium, and the intensity-dependent self-phase-modulation term. A split-step approach based on sampling the propagation length (from Z = 0 to Z = 1) into a large number of steps N is used. At each step (propagation distance dZ = Z/N), dispersion is taken into account by an additive phase term Ω2dZ/2 on the field in the frequency domain. Self-phase modulation is considered by adding the phase term |e|2dZ on the field in the time domain.
A spectrally incoherent pulse is generated by allocating a random spectral phase uniformly distributed between 0 and 2π to each frequency component. The coherence time is defined by
where γ is the Fourier transform of the spectral density [33]. A close-up of one realization of the power of a field with ΔT = 1 is shown in Fig. 2(a). The probability density function (PDF) of the field intensity is a negative exponential, i.e.,Fig. 2. Example of a spectrally incoherent pulse with coherence time ΔT = 1 generated with a super-Gaussian spectrum: (a) close-up on one realization of the intensity profile (solid blue line) and average intensity (dashed black line), and (b) probability density function of the intensity (solid blue line) compared to a negative exponential function of average equal to 1 (black dashed line), with the black round marker indicating the intensity 99th percentile.
Statistical analysis, i.e., simulations over a large number of realizations of the incoherent process, is required. Some useful metrics in the context of this work include the scintillation noise σ, the stretching exponent ε, and the intensity 99th percentile I99. The scintillation noise σ is the square root of the scintillation index defined by
An initial field defined as described above has σ = 1. Fields with σ < 1 have reduced intensity modulations, whereas fields with σ > 1 have increased intensity modulations in the rms sense. The intensity 99th percentile I99 is the intensity for which 99% of the intensity values are between 0 and I99 [e.g., as indicated by the black marker on Fig. 2(b)]. It is calculated as the intensity at which the cumulative density function is equal to 0.99. Because the intensity is higher than I99 during only 1% of the duration of the pulse, I99 is a measure of the concentration of intensities toward lower values. The stretching exponent ε is obtained by fitting the PDF of the intensity over a given range (in this work from I = 0 to I = I99) using the functional form where A and B are two other fit parameters [34,35]. The stretching exponent describes the shape of the PDF, while A and B are needed to ensure correct scaling of the intensity axis and normalization of the integrated PDF to 1. This functional form, which implies that the logarithm of the PDF is a single-term polynomial function of the intensity, has been found suitable to describe the PDF’s simulated in this work. A field composed of a large number of frequency components with uncorrelated phases uniformly distributed between 0 and 2π has a stretching exponent equal to 1. Exponents larger than 1 correspond to short-tail distributions, i.e., distributions that are more concentrated toward I = 0, while exponents smaller than 1 correspond to long-tail distributions, i.e., distributions with higher likelihood of relatively high-intensity values.Fields having a spectral density given by a super-Gaussian function have been chosen for the simulations presented in this article, but running simulations on a limited number of test cases with other spectral density functions did not lead to significant differences for the same coherence time. A number of independent spectral modes approximately between 10 and 1000 covered the range of coherence times of interest between ΔT = 10 and ΔT = 0.1. The temporal field calculated from the spectral density and one realization of the random spectral phases eT spans the entire temporal range. It is therefore gated by a super-Gaussian profile SGT to avoid aliasing effects at the two ends of that range during field propagation, resulting in the input field
at Z = 0. The super-Gaussian envelope has the additional advantage (as compared, for example, to a Gaussian envelope) of providing a relatively large range of times over which the incoherent field has on average a constant intensity, which facilitates data interpretation. A relatively low order (10) was chosen to mitigate the generation of spikes at the leading and trailing edge of the pulse during dispersive propagation. The field eT is normalized so that its average intensity over a temporal range spanning a large number of coherence times is equal to 1. The simulation is performed for 1000 realizations of the random spectral phase at each coherence time and equivalent nonlinear phase Ψ. Each simulation run corresponds to numerically solving Eq. (2) for an input field having the prescribed spectral density and phase given by a random draw between 0 and 2π at each spectral mode. The total run time using MATLAB [36] on a mid-range PC is of the order of 1 minute for the 1000 realizations. The intensity statistical properties of spectrally incoherent pulses propagating in a medium that is either nondispersive (β2 = 0) or purely linear (γ = 0) do not change. For nondispersive propagation, the field acquires an intensity-dependent nonlinear temporal phase but its intensity profile does not change, hence the output and input profiles have identical statistical properties. For purely linear propagation, the field acquires a quadratic spectral phase. This phase is added to the random spectral phase, hence modifying the intensity profile at all times but not inducing changes in the correlation between frequencies. Only the combination of chromatic dispersion and nonlinearity can modify the intensity statistical properties of an incoherent pulse.3.2 Medium with positive dispersion
Nonlinear propagation in a medium with β2 > 0 generally leads to a reduction of the random temporal modulations and lower likelihood of high-intensity values (Fig. 3). For such a medium, the scintillation noise decreases from its input unity value, indicating that the random intensity modulations are being clamped around their average value by the combination of chromatic dispersion and nonlinearity [Fig. 3(a)]. The stretching exponent increases from its input unity value, indicating a decrease in the likelihood of high intensities within the pulse [Fig. 3(b)]. This exponent increase is consistent with the decrease of I99 [Fig. 3(c)], which confirms that the intensities within the modulated pulse are more concentrated toward lower values. For a given nonlinearity, the relative change in these three metrics is the largest for ΔT ≈ 1, with the narrowband pulses $(\Delta T \gg 1)$ and broadband pulses $(\Delta T \ll 1)$ experiencing no significant change in their statistical properties. Narrowband pulses are not significantly impacted by dispersion, hence even the accumulation of significant nonlinear temporal phase does not modify their pulse shape and the resulting statistical properties. The intensity profile of broadband pulses changes significantly during propagation, resulting in no accumulation of the temporal phase at a given time. The incoherent pulses experience spectral broadening, which increases with the equivalent nonlinear phase Ψ and the coherence time ΔT, as can be quantified by the bandwidth enhancement defined as the ratio of their output and input rms bandwidth [Fig. 3(d)]. Although short coherence times correspond to fast intensity variations that can locally induce a large change in temporal phase, chromatic dispersion tends to quickly modify the intensity profile as the pulse propagates in the medium. This randomization of the temporal intensity restricts the bandwidth induced over the full propagation length. By contrast, the intensity profile of fields with long coherence times does not change significantly over the full medium length, allowing for the nonlinear temporal phase shift and induced instantaneous bandwidth to accumulate. Examples of intensity histograms demonstrate the statistical properties of the resulting fields [Fig. 3(e)]. At Ψ = 1, the PDF is close to a negative exponential for ΔT = 0.1 and ΔT = 10 (ε ≈ 1), but it has significantly less spread toward high intensities for ΔT = 1 (ε ≈ 2.2). For this coherence time, slightly stronger clamping of the high intensities is observed at larger nonlinear phases (Ψ = 2 and Ψ = 4). It is observed that the fit given by Eq. (9) performed over the range of intensities between 0 and I99 matches the statistical trends well, but it loses its ability to quantitatively describe the tail of the distributions at large nonlinear phases. In particular, the probability of high-intensity values is somewhat larger than predicted by the fits. It should also be noted that the PDF for large values of the nonlinear phase becomes non-monotonic at low intensity, a behavior that cannot be exactly reproduced by Eq. (9).
Fig. 3. Statistical properties of the incoherent field after nonlinear propagation in a medium with positive dispersion as a function of the coherence time of the initial field ΔT and the equivalent nonlinear phase Ψ: (a) scintillation noise σ, (b) stretching exponent ε, (c) intensity 99th percentile I99, (d) bandwidth enhancement factor ΔΩ/ΔΩ0, and (e) examples of five histograms (solid lines) with the corresponding values of I99 (circles) and the negative exponential PDF for reference (black dashed line). The red line on the color bars in (a)–(d) indicates the value of the plotted quantity for the input field.
3.3 Medium with negative dispersion
Nonlinear propagation in a medium with β2 < 0 generally leads to an increase of the temporal modulations, PDF’s with smaller exponents, and higher likelihood of large intensity values (Fig. 4). These changes in statistical properties are, however, not the largest for ΔT ≈ 1, as is the case for positive dispersion. Instead, the highest scintillation indices, lowest power-law exponent, and highest intensity 99th percentile I99 are observed for ΔT > 1 in the phase space where Ψ is approximately proportional to ΔT2. This behavior is interpreted in the context of the modulation instability that exists during nonlinear propagation in a medium with negative dispersion. The typical framework to analyze modulation instability is the propagation of a cw field with a small sinusoidal temporal modulation [37]. For Eq. (2), in which the spatial Laplacian term is set to 0, the sinusoidal modulation experiences exponential growth for frequencies Ω such as 0 < Ω< Ωc, where ${\Omega _\textrm{c}} = 2\sqrt \Psi ,$ and the gain is the highest for ${\Omega _\textrm{m}} = \sqrt {2\Psi } .$ This scaling law is in qualitative agreement with the simulation results for incoherent pulses. Indeed, it indicates that the bandwidth, which is proportional to the inverse of the coherence time, is proportional to the square root of the nonlinear phase. It is also consistent with the fact that the growth is only observed for coherence time larger than a cutoff value because the modulation frequency must be lower than the cutoff value. Quantitative agreement is not expected since the derivation of the conditions for modulation instability assumes a small temporal modulation of a cw field, whereas the incoherent pulses have large temporal modulations corresponding to an incoherent mix of spectral modes. The changes in statistical properties are particularly large in these regions of the phase space: the stretching exponent reaches 0.2 while the intensity 99th percentile reaches 10. A line corresponding to a nonlinear phase Ψ proportional to the square of the coherence time ΔT has been added on Figs. 4(a)–4(d) as a visual guide. Histogram examples demonstrate the different statistical properties observed for a medium with negative dispersion, in particular the higher likelihood of high intensity values. Nonlinear propagation in an optical fiber with negative dispersion has been experimentally shown to lead to long-tailed distributions of the intensity [38], which is consistent with these simulations. As in a medium with positive dispersion, narrowband pulses $(\Delta T \gg 1)$ and broadband pulses $(\Delta T \ll 1)$ are not significantly impacted by the combination of chromatic dispersion and nonlinearity, and their PDF remains close to a negative exponential.
Fig. 4. Statistical properties of the incoherent field after nonlinear propagation in a medium with negative dispersion as a function of the coherence time of the initial field ΔT and the equivalent nonlinear phase Ψ: (a) scintillation noise σ, (b) stretching exponent ε, (c) intensity 99th percentile I99, (d) bandwidth enhancement factor ΔΩ/ΔΩ0, and (e) examples of five histograms (solid lines) with the corresponding values of I99 (circles) and the negative exponential PDF for reference (black dashed line). The red line on the color bars in (a)–(d) indicates the value of the plotted quantity for the input field. On (a)–(d), a black line corresponding to Ψ proportional to ΔT2 has been added as a visual aid [the same proportionality was used for (a)–(c), a different proportionality was used for (d)].
3.4 Discussion
The dispersion in spatially homogeneous materials is typically positive in the wavelength range of interest for high-energy laser systems in which the optical pulse propagates in laser media, nonlinear crystals, and optical components. In the regime of positive dispersion, clamping of high-intensity values has been consistently observed over a large range of coherence times and nonlinear phases. Guided propagation, for example in optical fibers, leads to additional dispersion terms that can significantly steer the dispersion away from the material’s dispersion. In particular, a standard single-mode fiber has a negative group velocity dispersion around 1550 nm, which leads to modulation instability [37]. In the context of high-energy laser systems, for which the fiber front end typically operates in the near infrared where the group velocity dispersion is positive, nonlinear propagation leads to the effects described in Sec. 3.2. Of particular interest in this regime are the capabilities to increase the optical bandwidth and control the temporal modulation of the intensity.
A significant bandwidth increase is obtained when the initial normalized coherence time is large, i.e., for a relatively narrowband field, and average nonlinear phase [Fig. 3(d)]. An initially broadband field with short normalized coherence time changes significantly during linear propagation in the medium and does not incur significant broadening in the presence of self-phase modulation. Even larger bandwidth increases are observed in a medium with negative dispersion in the regime of modulation instability [Fig. 4(d)], although the fact that such regime corresponds to higher scintillation and long-tail behavior of the PDF most likely makes it less suitable for high-energy laser applications.
Nonlinear propagation results in a clamping of the time-domain modulation in a medium with positive dispersion, as consistently shown by the scintillation noise [Fig. 3(a)], stretching exponent [Fig. 3(b)], and 99th percentile intensity [Fig. 3(c)]. This effect is the strongest for fields that have a normalized coherence time around 1, for which higher intensities are less likely than for the input field. Controlling the PDF of the incoherent fields has several potential advantages in laser systems. In the single-pulse regime, the damage threshold of optical components depends on the intensity and the duration of the pulse. For incoherent pulses, the damage threshold is likely to depend on the coherence time and the intensity PDF, implying that control of the latter might result in the capability to operate at higher energies using fields with reduced temporal modulations. In cases where the incoherent pulses undergo an intensity-dependent nonlinear process, the intensity PDF of the input incoherent field has an impact on the outcome [39–41]. Typical processes of interest for high-energy pulses include frequency doubling, frequency tripling, and optical parametric amplification. Studying these effects in detail is beyond the scope of this article, but shaping the intensity PDF of the input field can probably be used to control the properties of the output field, in particular its energy, spectral density, and intensity PDF.
4. Self-focusing simulations of an incoherent pulse
4.1 Description
Simulations have been performed to assess the self-focusing of incoherent pulses. The input field is described as a separable function of time T and one cartesian coordinate X. The time-domain field component is parametrized by its coherence time and generated according to the procedure described in Sec. 3.1. The spatial domain field component has a sinusoidal modulation parametrized by its amplitude m and period p, similarly to what was used in Sec. 2.3. Super-Gaussian envelopes are used in the time and spatial domain to avoid aliasing effects during propagation. The input field is
For a given modulation amplitude and coherence time, 1000 realizations of the time-domain process are used. For each realization, the period that maximizes the output intensity at X = 0 is determined (this location corresponds to one maximum of the sinusoidal spatial modulation, hence quantifies self-focusing). This maximization is done by an iterative process based on multiple resolutions of Eq. (2) for the same time-domain description and different spatial periods. Such determination typically takes 45 s because of the necessity to solve Eq. (2) multiple times (typically 10) for the input field e(X,T,0). A collection of intensities after nonlinear propagation is assembled from the corresponding time-domain intensities at X = 0 simulated over all the realizations. These intensities are only collected over the time range where the temporal super-Gaussian envelope is relatively flat. The PDF of such intensity collection is then calculated.
In a medium with β2 < 0, self-phase modulation increases the intensity via self-focusing in the spatial domain and temporal compression in the time domain, hence self-focusing cannot be mitigated. The chromatic dispersion of materials is typically positive in the wavelength range of high-energy lasers, but angular dispersion leads to a negative group velocity dispersion [42]. In the latter case, the self-focusing behavior is likely to be impacted by both the chromatic dispersion and the propagation of different spectral components at different angles. Such case requires a more-advanced treatment than what is presented in this article. The simulations are therefore only presented for β2 > 0, which is the most frequent in practice. Because of the large computational time imposed by solving Eq. (2) for one realization of the time-domain component and the number of realizations required for good statistical representation, the simulations were performed for five coherence times (0.1, 0.32, 1, 3.16, and 10) and four modulation amplitudes (0.01%, 0.1%, 1%, and 10%), evenly spaced on a logarithmic scale. The equivalent nonlinear phase Ψ was set to 1, which is a typical value for high-energy systems.
Two PDF’s, ρinput and ρBT are used as benchmarks to classify the simulated PDF’s for an incoherent pulse incurring self-focusing in a dispersive medium. The PDF of the input intensity ρinput is a negative exponential function with average equal to Ψ, i.e., 1 in the present case. The PDF of an incoherent pulse undergoing nonlinear propagation in the absence of dispersion ρBT is calculated in Appendix. It is shown in Fig. 5(a) for four values of the modulation amplitude m. The PDF’s ρinput and ρBT identify different mitigation behaviors, as exemplified on Fig. 5(b) for m = 10%:
- • An output field with PDF following ρBT is a field for which no self-focusing mitigation has occurred, i.e., temporal slices of the intensity profile independently self-focus according to the intensity-dependent Bespalov–Talanov gain.
- • An output field with PDF lower than ρBT but higher than ρinput [phase space identified by (I) in Fig. 5(b)] is a field for which some level of self-focusing mitigation has occurred, i.e., the resulting intensities are lower than for a purely intensity-dependent process. However, such a field still has a higher probability of high intensities than the input field. This behavior is labeled as “mild mitigation.”
- • An output field with PDF following ρinput is a field for which propagation had no impact on the intensity statistical distribution, and in particular a given intensity is as likely in the input and output field.
- • An output field having a PDF lower than ρinput [phase space identified by (II) in Fig. 5(b)] is a field for which propagation has decreased the likelihood of high intensities relative to the input field. This behavior is labeled as “strong mitigation.”
Fig. 5. (a) Intensity PDF ρBT of an incoherent field at different values of m, after propagation in a nondispersive medium for Ψ = 1, calculated using the intensity-dependent gain from the Bespalov–Talanov analysis (solid lines), and intensity PDF ρinput of the input field (black dashed line). (b) Intensity PDF ρinput (dashed line) and ρBT calculated for m = 10% (solid line) defining two self-focusing mitigation regimes: mild mitigation for which the likelihood of high intensities is lower than in the coherent regime but higher than in the input field [light gray shaded area, labeled as (I)], and strong mitigation for which the likelihood of high intensities is lower than in the input field [dark gray shaded area, labeled as (II)].
Simulations show that, depending on the parameters of the incoherent pulses and the medium in which they propagate, no significant mitigation, mild mitigation, or strong mitigation can occur.
4.2 Results
Figure 6 displays the simulated PDF’s for the intensity after nonlinear propagation, where each subfigure corresponds to five coherence times ΔT and one modulation amplitude m. For the largest modulation amplitude [m = 10%, Fig. 6(a)], the PDF for long coherence times is similar to ρBT, indicating that there is no self-focusing mitigation. Mild mitigation occurs for ΔT ≤ 1, and the mitigation clearly increases as the coherence time decreases. Stronger mitigation occurs for smaller modulation amplitudes. At m = 1% [Fig. 6(b)], there is no mitigation at long coherence times, but shorter coherence times ΔT ≤ 1 correspond to PDF’s that are similar to the input PDF. At m = 0.1% [Fig. 6(c)], ΔT = 10 does not lead to any mitigation, whereas ΔT = 3.16 leads to an output PDF that is similar to the input PDF, showing significant mitigation. Shorter coherence times lead to strong mitigation, and the overall shape of the resulting PDF’s is consistent with stretching exponents larger than 1, as observed in Sec. 3.2. The strongest mitigation is observed at ΔT = 1, which is consistent with the strongest intensity mitigation identified in Sec. 3.2. This indicates that spatial self-focusing is not a dominant factor in this regime, and the output intensity statistics are dominated by time-domain effects occurring as a result of the interplay between nonlinearity and dispersion. A similar conclusion is drawn for m = 0.01% [Fig. 6(d)], for which strong mitigation is observed at all coherence times, with the strongest mitigation observed at ΔT = 1.
Fig. 6. Intensity PDF of an incoherent pulse after propagation for (a) m = 10%, (b) m = 1%, (c) m = 0.1%, and (d) m = 0.01%. On each figure, the PDF is shown for the five coherence times indicated in the legend. The corresponding regimes of mild mitigation and strong mitigation are identified by the light gray and dark gray shaded areas, respectively.
The scintillation noise, stretching exponent, and intensity 99th percentile corresponding to the previously discussed test cases are plotted in Fig. 7. The scintillation noise σ generally shows that the rms variation of the intensity is increased by nonlinear propagation at large values of m, with a larger increase for narrowband fields [Fig. 7(a)]. At lower values of m, the rms variation decreases relative to that of the input field, and the largest decrease is observed at ΔT = 1. The latter behavior is consistent with what has been observed in the time-domain simulations presented in Sec. 3.2, confirming that the interplay between nonlinearity and positive group velocity dispersion is the main contributor to the intensity statistics in the absence of significant spatial self-focusing seeded by the low-amplitude spatial perturbation. The stretching exponent [Fig. 7(b)] and intensity 99th percentile [Fig. 7(c)] confirm the trends observed with the scintillation noise. Exponents smaller than 1 indicating an increase in the likelihood of high intensities relative to the input field are observed at high modulation indices, but exponents larger than 1 indicating intensity PDF’s more concentrated toward low intensities are observed at low modulation indices, in particular for ΔT ∼ 1. Likewise, the intensity 99th percentiles indicate an increase of the probability of high intensities relative to the input field for large values of m and a decrease for low values of m.
Fig. 7. (a) Scintillation noise, (b) stretching exponent, (c) intensity 99th percentile, (d) ratio of output and input bandwidth, and (e) fluence gain for an incoherent pulse undergoing self-focusing. The red line on the color bars indicates the value of the plotted quantity for the input field.
The spatial period corresponding to the largest intensity gain between the input and output field is different for each realization of the incoherent process. The average of these spatial periods is between 2 and 3.5, with a standard deviation of the order of 0.5, for m = 0.01%, 0.1%, and 1% at all coherence times. These values correspond to the modulation periods pmax for coherent fields at higher nonlinear phases [Fig. 1(a)]. This trend is different at m = 10%, for which the average modulation period ranges from 4 with a standard deviation of 0.1 for broadband pulses to 6 with a standard deviation of 1.5 for narrowband pulses. Significant spatiotemporal coupling arises around the time and location where self-focusing occurs. For a laser system generating deterministic coherent pulses, a range of periods for which self-focusing occurs and a period for maximal gain can unambiguously be analytically determined (Sec. 2.3). Setting guidelines for spatial filtering to prevent the growth of small-scale fluctuations in a laser system generating incoherent pulses is more complex and requires further study.
Fields with coherence times smaller than 1 are not significantly broadened by the nonlinear propagation, but the rms bandwidth increases for fields with long coherence times. A similar behavior is observed for the spectral density of the field at the location of strongest self-focusing [peak of the spatial modulation in Eq. (11)], which is shown in Fig. 7(d), and for the spectral density spatially integrated over the whole beam, although the bandwidth enhancement observed in the latter case is smaller. The rms bandwidth only increases by a few percent for broadband fields. The increase is more significant for large values of m, for which self-focusing is significant, showing that it is induced by the joint effect of the nonlinearity in the time and spatial domains.
The scintillation noise, stretching exponent, and intensity 99th percentile pertain to the statistics of the intensity (function of time and space) within the optical pulse. The time-integrated fluence at a given position within the beam is of interest to further capture the self-focusing behavior of incoherent pulses. Self-focusing increases the local intensity, and such effect is expected to occur with higher probability at times when the intensity is high. Similar to the definition of the intensity gain Gmax in Sec. 2.3, a fluence gain gmax is defined as the ratio of the maximal output and input fluences at the peak of the sinusoidal modulation. For the monochromatic pulse assumed in the Bespalov–Talanov analysis, the intensity and fluence are proportional, and gmax = Gmax. The fluence gain depends on the spatial modulation amplitude but it does not depend strongly on the coherence time [Fig. 7(e)]. The modulation amplitude is expected to impact gmax because it does so for coherent pulses [Fig. 1(b)]. The coherence time leads to a weak mitigation of self-focusing in the time-integrated sense, with more bandwidth leading to a lower fluence gain. For narrowband pulses, self-focusing leads to much higher values of the intensity (as indicated by the intensity statistics) at times when it occurs, but does not impact the spatial shape at other times. For broadband pulses, self-focusing is mitigated by the significant changes in pulse shape occurring during propagation. It leads to lower values of the (instantaneous) intensity on the output pulse at any given time but it impacts the spatial beam shape over a range of times that spans multiple coherence times. Because of these effects, the modulation gain gmax tends to be similar at all coherence times.
4.3 Discussion
The simulation results presented in Sec. 4.2 show that nonlinear propagation leads to a range of behaviors depending on the spatial modulation amplitude and coherence time. Self-focusing mitigation for an incoherent field propagating within a medium depends on the normalized coherence time, defined in relation to the dispersion and length of the medium, and not on the physical bandwidth itself (e.g., in units of THz). A pulse with a given physical bandwidth has different normalized coherence times within the components of a laser system, hence different self-focusing behaviors.
In the presence of a small spatial modulation, the changes in the output field are dominated by the time-domain behavior studied in Sec. 3. In particular, fields with ΔT ∼ 1 exhibit the strongest mitigation of the high-intensity values. In this regime, strong mitigation of the intensity modulations, in the sense defined in Sec. 4.1, is observed, i.e., the output field has lower modulation than the input field. Furthermore, narrowband $(\Delta T \gg 1)$ and broadband pulses $(\Delta T \ll 1)$ have statistical properties similar to that of the input field. For larger modulation amplitudes, narrowband pulses result in output statistical properties that are consistent with predictions of the Bespalov–Talanov analysis independently applied at each intensity level, and there is no self-focusing mitigation in that case. Mitigation of high values of the intensity is observed for broadband fields, as compared to the Bespalov–Talanov analysis. This mild mitigation, in the sense defined in Sec. 4.1, still corresponds to an enhancement of the probability of high intensities relative to the input field because of self-focusing. In practical applications, beam modulation much smaller than 1% is technologically difficult to avoid. It is therefore likely that the self-focusing behavior is generally dictated by the coherence time, and in particular shorter coherence times result in self-focusing mitigation relatively to expectations derived from the Bespalov–Talanov analysis at identical instantaneous intensities.
The relative bandwidth increase via nonlinear propagation is only significant with relatively narrowband pulses. This is observed at all values of the modulation amplitude, indicating that the large changes of temporal profile occurring during propagation in the medium do not lead to accumulation of a temporal phase, which could lead to bandwidth increase by adding instantaneous frequencies. Narrowband pulses accumulate a nonlinear phase proportional to their intensity at a given time over the entire length of the medium, resulting in the creation of new instantaneous frequencies and an increase in relative bandwidth. Larger modulation amplitudes, for which more spatial focusing is observed, lead to a larger bandwidth increase. In this regime, the increase in bandwidth of the output pulse has a trend that indicates the relative strength of self-focusing, with mitigation being stronger for pulses having smaller changes in bandwidth. However, this is only a weak indicator of the absolute impact of self-focusing because the bandwidth does not significantly increase at short coherence times, for which self-focusing is only mildly mitigated and still results in a higher probability of output intensities. Therefore, comparison of the measured input and output optical spectrum might not clearly indicate the nonlinear propagation, particularly for broadband pulses.
Short coherence times do not lead to significant mitigation of spatial modulation growth on the time-integrated beam, whereas they clearly mitigate the likelihood of high intensities within the pulse, compared to a monochromatic pulse of similar intensity. In narrowband pulses, self-focusing occurs selectively at times when the input intensity is high (typically concentrated to less than the coherence time), because the intensity remains high during the propagation in the nonlinear medium. In broadband pulses, some amount of self-focusing occurs at a given longitudinal position within the medium and time within the pulse where the intensity is high. However, the pulse shape changes during propagation, leading to a lower overall self-focusing at a given time. Because of the redistribution of the intensity within the pulse during propagation, the self-focusing behavior impacts the beam over a range of times larger than its coherence time. Self-focusing can generally be identified from the beam profile of the incoherent pulse after propagation, as in the case of quasi-monochromatic coherent pulses. The observed change in time-integrated beam modulation is not directly linked to the actual changes in intensity within the pulse, but it is quite relevant to the operation of laser systems. In particular, a higher fluence can increase the probability of optical damage.
5. Conclusion
The nonlinear propagation of incoherent pulses has been analyzed using a normalized nonlinear Schrödinger equation and statistical analysis, revealing a complex interplay of different phenomena in which the coherence time plays a central role. When spatial modulations are small, pulses with moderate coherence times have the strongest mitigation of their intensity statistics in the common case of a medium with positive dispersion. With significant spatial modulations, the intensity increase caused by self-focusing is generally better mitigated for broadband pulses, for which the pulse shape changes significantly over the length of the medium. In this regime, the likelihood of high intensities is lower than predicted using the Bespalov–Talanov analysis at the same input intensity, but it is higher than in the input pulse. The nonlinear propagation analysis for incoherent pulses could be extended to take into account the two transverse spatial dimensions and non-sinusoidal perturbations [43,44].
Practical applications of this work include the shaping of the intensity statistics and bandwidth of incoherent pulses via nonlinear propagation, the design of laser systems producing high-energy incoherent pulses, and the associated metrology. Another important aspect for high-energy laser systems is the identification of safe operating conditions to mitigate the probability of damage resulting from nonlinear self-focusing and prevent the growth of small-scale modulations via filtering.
Appendix
The PDF of the intensity of an incoherent field undergoing nonlinear propagation in the absence of dispersion is calculated. Following Sec. 2.3, a coherent field with an intensity Iin and a spatial modulation of amplitude m results in the output intensity Iout given by
Funding
National Nuclear Security Administration (DE-NA0003856); Office of Science (DE-SC0021032); University of Rochester; New York State Energy Research and Development Authority.
Acknowledgment
This report was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof.
Disclosures
The author declares no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the author upon reasonable request.
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