Temporal optical besselon waves for high-repetition rate picosecond sources

We consider a continuous optical wave with amplitude ψ₀ and carrier angular frequency ω_c, $\Psi (t) = {\psi _0}\;\;\psi (t)\;\;{{\text{e}}^{i\;{\omega _{\text{c}}}\;t}}$, whose phase is temporally modulated by a sinusoidal wave:

$$\begin{equation}\psi (t) = {{\text{e}}^{i\;\;{A_{\text{m}}}\;\cos \left( {{\omega _{\text{m}}}\;t} \right)}}\end{equation} \tag{ 1 }$$

where A_m is the amplitude of the phase modulation and ω_m is its frequency. As a result of this phase modulation, the spectrum of frequency components of the wave envelope will consist of a series of discrete lines evenly spaced by ω_m, which can be obtained from the Jacobi–Anger expansion [18–20] of ψ(t):

$$\begin{equation}\psi (t) = \;\sum\limits_{n = - \infty }^\infty {{i^n}\;{J_n}} ({A_{\text{m}}})\;{{\text{e}}^{i\;n\;{\omega _{\text{m}}}t}}.\end{equation} \tag{ 2 }$$

Therefore, the nth spectral component will have an intensity proportional to $J_n^2({A_{\text{m}}})$, where J_n (x) is the Bessel function of the first kind and order n. An important characteristic of this spectrum is the existence of a π/2 phase shift between successive frequency components. The application of a quadratic spectral phase profile—such as produced by propagation through a purely dispersive element—to ψ(t) will enable partial compensation of the initial sinusoidal phase. This will in turn lead to the emergence of temporally localised, periodic pulse structures with the period T₀ = 2π/ω_m [12, 13], as shown in figure 1(a). There, we have used the value of accumulated dispersion that optimises the peak power of the resulting pulses for each amplitude value A_m of the initial modulation. This optimum accumulated dispersion decreases continuously with increasing A_m from 45 ps² (for A_m = 1 rad) to 3.4 ps² (for A_m = 12 rad). We have also checked that the highest peak power corresponds to the shortest pulse duration. We can see in figure 1(a) that increasing the modulation amplitude causes shortening of the central part of the pulses accompanied by a concomitant increase of the pulse peak power. However, since the initial spectral phase cannot be perfectly cancelled, the presence of a residual continuous background impairs the resulting pulse intensity profiles [14].

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On the other hand, the recent advances in linear pulse shaping technology make it now possible to manipulate optical frequency combs line by line. With our pulse shaping approach, imprinting a π/2 phase shift to each individual spectral component of the phase-modulated continuous wave enables the synthesis of a new optical pulse field, which we call 'besselon' because of the important role played by Bessel functions in the description of the waveform features. The besselon wave envelope, ψ_B, can be represented as

$$\begin{equation}{\psi _{\text{B}}}(t) = \;\sum\limits_{n = - \infty }^\infty {{J_n}} ({A_{\text{m}}})\;{{\text{e}}^{i\;n\;{\omega _{\text{m}}}\;t}} = {J_0}({A_{\text{m}}}) + 2\sum\limits_{n = 1}^\infty {{J_n}({A_{\text{m}}})\;\cos \left( {n\;{\omega _{\text{m}}}t} \right)} .\end{equation} \tag{ 3 }$$

In [15], we have emphasised that this periodic pulse structure exhibits an excellent extinction ratio combined with the absence of spurious pedestals for values of the initial phase-modulation amplitude A_m around 1.1 rad. Figure 1(b) shows the evolution of the wave intensity profile |ψ_B(t)|² with A_m across a wide range of modulation amplitude values, and features distinctly different from that of the pulse pattern resulting from spectral manipulation with a quadratic spectral phase. The besselon does not experience a continuous decrease of its temporal duration with increasing modulation amplitude, and its extinction ratio may also change significantly with varying amplitude. The central part of the pulses also evolves in a different manner: sidelobes develop and tend to merge with the central part that displays an increasingly large number of strong oscillations.

To gain better insight into the besselon pulse wave, we can obtain from (3) the maximum value of the field envelope at t = 0:

$$\begin{equation}{\psi _{\text{B}}}(t = 0) = {J_0}({A_{\text{m}}}) + 2\sum\limits_{n = 1}^\infty {{J_n}({A_{\text{m}}})} \end{equation} \tag{ 4 }$$

which, using the relations [18, 21]

$$\begin{equation}\begin{gathered} 2\sum\limits_{n = 1}^\infty {{J_{2n}}({A_{\text{m}}})} = 1 - {J_0}({A_{\text{m}}}) \hfill \\ 2\sum\limits_{n = 0}^\infty {{J_{2n + 1}}({A_{\text{m}}}) = \int\limits_0^{{A_m}} {{J_0}(x)\;{\text{d}}x} } , \hfill \\ \end{gathered} \end{equation} \tag{ 5 }$$

can be written in the form

$$\begin{equation}{\psi _{\text{B}}}(t = 0) = \;1 + {I_{J0}}({A_{\text{m}}})\end{equation} \tag{ 6 }$$

where I_{J 0} (figure 2(d), red curve) is defined by

$$\begin{equation}{I_{J0}}({A_{\text{m}}}) = \int\limits_0^{{A_{\text{m}}}} {{J_0}(x)\;{\text{d}}x} .\end{equation} \tag{ 7 }$$

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We have plotted in figure 2(a) the peak-power predictions given by (6). The Fourier-transform limited nature of the besselon for A_m⩽ 2.40 rad brings about a continuous increase of the peak power with A_m, which reaches higher values than the peak power of the pulse wave obtained after a dispersive element. With further increase of the modulation amplitude, however, the peak power of the besselon oscillates and remains below the value of 6.10 obtained at A_m = 2.40 rad. The amplitude values A_m,p of the phase modulation at which the peak power is maximal are given by dI_{J 0}/dA_m|_{A m,p}= 0, that is J₀(A_m,p) = 0. In other words, the maxima of the peak power occur at the zeros of the Bessel function of zero order. For these amplitudes (A_m,p = 2.4048, 5.5201, 8.6537, ...), the central component of the frequency spectrum vanishes and the pulse train becomes carrier-suppressed.

Using the following asymptotic form of the zeroth-order Bessel function for large arguments [18] (figure 2(d), black circles)

$$\begin{equation}{J_0}(x) \simeq \;\sqrt {\frac{2}{{\pi \;x}}} \cos \left( {x - \frac{\pi }{4}} \right)\end{equation} \tag{ 8 }$$

we can obtain an approximate expression for the location of the pth maximum:

$$\begin{equation}{A_{{\textrm{m,p}}}} \simeq {\mkern 1mu} \frac{{3{\mkern 1mu} \pi }}{4} + (p - 1){\mkern 1mu} \pi {\textrm{}}\:{\textrm{}},{\textrm{}}\:\:{\textrm{}}p \geqslant 1.\end{equation} \tag{ 9 }$$

Equation (9) indicates that the power of the besselon at t =0 experiences periodic growth and decay with a period of π rad. We also note that as I_{J 0}(x) asymptotically tends to 1 for large values of x [21], the amplitude of the oscillations decreases with increasing values of A_m.

The minimum power of the besselon (calculated at t = T₀/2) can also be obtained from (3):

$$\begin{equation}{\psi _{\textrm{B}}}\left( {t = \frac{{{T_0}}}{2}} \right) = \;1 - {I_{J0}}({A_{\textrm{m}}}).\end{equation} \tag{ 10 }$$

This quantity enables us to evaluate the extinction ratio ER of the pulses, which we define here as the ratio of the peak power at the pulse centre t = 0 to the minimum power of the pulse value at t = T₀/2:

$$\begin{equation}{\text{ER}}({A_{\text{m}}}) = \;\frac{{1 + {I_{J0}}({A_{\text{m}}})}}{{1 - {I_{J0}}({A_{\text{m}}})}}.\end{equation} \tag{ 11 }$$

We note that this metric does not quantify the level of potential sidelobes in the pulse profiles but provides an estimate of the ability of the pulse train to return to a null value between successive pulses. The minimum power of the besselon is shown in figure 2(b) and displays strong variations with varying amplitude of the initial phase modulation. This contrasts with the pulse wave obtained after a dispersive element, where the power level between successive pulses remains almost constant for A_m above 2 rad. We can deduce that for amplitudes of the sinusoidal modulation satisfying the equation I_{J 0} (A_m) = 1, the background level between pulses is zero, hence the extinction ratio is maximal. This condition is consistent with the condition derived in [15] based on numerical simulation data. In a similar manner to the peak and minimum powers, the extinction ratio of the pulses varies strongly with A_m (figure 2(c)), while remaining larger than the extinction ratio of the pulse wave synthesised by use of a quadratic spectral phase profile across the whole A_m variation range. Note that, for I_{J 0}(A_m) = 1, the maximum amplitude of the besselon is simply 2, yielding an intensity of 4.

We now provide an insight into the shape of the besselon at the points A_m of maximal extinction ratio (given by I_{J 0} (A_m) = 1). The temporal amplitude profiles ψ_B(t) obtained at A_m =1.1086, 4.0628, 7.15 and 10.2695 rad are shown in figure 3 (panel (a)). From (3), we can note that the envelope of the besselon is a real-valued function of time, contrary to the wave envelope that is obtained after quadratic phase compensation. Moreover, for phase-modulation amplitudes A_m less than the value A_m,1 at which the first peak power maximum occurs, the besselon envelope takes only positive values, whereas for higher values of A_m the amplitude profile features alternating positive- and negative-valued portions. The latter evidences a shift of π rad between successive wave oscillations. We also note that for the modulation amplitudes A_m,p of maximal peak power, the temporal intensity profile of the besselon features 2p + 1 peaks.

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The origin and properties of these intensity peaks can be readily understood if we express the besselon envelope in the form

$$\begin{equation}{\psi _{\text{B}}}(t) = \;{J_0}({A_{\text{m}}}) + 2\sum\limits_{n = 1}^\infty {{J_{2n}}({A_{\text{m}}})\;\cos \left( {2n\;{\omega _{\text{m}}}t} \right)} \\ + \;2\sum\limits_{n = 0}^\infty {{J_{2n + 1}}({A_{\text{m}}})\;\cos \left( {(2n + 1)\;{\omega _{\text{m}}}t} \right)} \\ \end{equation} \tag{ 12 }$$

and account for the relationship [18, 21]

$$\begin{equation}{J_0}({A_{\textrm{m}}}) + 2\sum\limits_{n = 1}^\infty {{J_{2n}}({A_{\textrm{m}}})\;\cos \left( {2n\;{\omega _{\textrm{m}}}t} \right)} = \cos \left( {{A_{\textrm{m}}}\;\sin ({\omega _{\textrm{m}}}t)} \right) = {\psi _{\textrm{D}}}(t){\textrm{}}\end{equation} \tag{ 13 }$$

where we have denoted by ψ_D(t) the resulting function. We can see in figure 3 that the central part of the besselon amplitude profile (solid black curves) is remarkably well described by 2ψ_D(t) (dashed blue curves). We can also note that the repetition rate of the pulse wave described by ψ_D(t) is twice the repetition rate of the besselon, and that 2ψ_D(t) = ψ_B(t) + ψ_B(t − T₀ /2). It is worth noting that the amplitude profile ψ_D(t) can be generated directly from a Mach–Zehnder device driven by a sinusoidal electrical wave.

We can then deduce several interesting features of the central part of the besselon from 2ψ_D(t). Firstly, the power of the central peaks equals 4, and their temporal locations t_M are given by

$$\begin{equation}{t_{\textrm{M}}} = {\textrm{a}}sin\left( {{{n\;\pi } \over {{A_{\textrm{m}}}}}} \right)/\left( {{{n\;\pi } \over {{A_{\textrm{m}}}}}} \right){\omega _{\textrm{m}}},\end{equation} \tag{ 14 }$$

where n ⩽ (N− 1)/2, and N is the number of peaks given by $\left\lfloor {2{A_{\text{m}}}/\pi } \right\rfloor + 1$. We can also obtain the FWHM width of the besselon (calculated as twice the largest time for which |ψ_B|² = 2; purple dashed lines in figure 3(b2)):

$$\begin{equation}{t_{\text{B}}} = \frac{2}{{{\omega _{\text{m}}}}}\operatorname{asin} \left( {\frac{\pi }{{{A_{\text{m}}}}}\left( {\frac{N}{2} - \frac{1}{4}} \right)} \right),\end{equation} \tag{ 15 }$$

and the temporal duration of its most central part (calculated as twice the shortest time for which |ψ_B|² = 2; purple dotted lines in figure 3(b2)):

$$\begin{equation}t_{\text{C}}^{} = \frac{2}{{{\omega _{\text{m}}}}}\operatorname{asin} \left( {\frac{\pi }{{4{A_{\text{m}}}}}} \right).\end{equation} \tag{ 16 }$$

The analytical predictions from (15) and (16) are shown in figure 4, and are in excellent agreement with the results of numerical simulations. The differences with the corresponding temporal characteristics of the waveform resulting from quadratic spectral phase compensation are also noticeable.

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In this section, we consider a simplified model that efficiently describes the central part of the besselon for low values of A_m. Indeed, from figures 3(a1)–(b1) it appears that approximating the central part of the pulse with the wave profile described by 2ψ_D(t) (equation (13)) does not provide sufficient accuracy for small amplitudes of the initial sinusoidal phase modulation (typically, for A_m < 1.5 rad). We can circumvent this difficulty by considering a simplified model for the besselon. In our previous work, we have shown that for A_m = 1.1 rad, the generated waveform is very close to a Gaussian profile [15]. We have also recently highlighted the similarity between the besselon profile obtained at small A_m and the typical profile of an Akhmediev breather [22], that is, a time-localised, space-periodic solution of the nonlinear Schrödinger equation on a constant background [23], which can be observed in nonlinear fibre optics. Here, we propose to interpret the besselon at small A_m from a different perspective. As shown in figure 5(a), in the limit of small A_m the frequency spectrum can be regarded as the result of the coherent superposition of three waves: a continuous wave ψ₁ with amplitude J₀ −2J₂, and two partially modulated waves, ψ₂ and ψ₃, which consist of a continuous background with amplitude J₁, modulated by a sinusoidal wave with amplitude 2J₂ and frequency ω_m, and are frequency shifted by −ω_m and ω_m, respectively. This leads to the following time-domain representation of the besselon:

$$\begin{equation}{\psi _{\text{B}}}(t) \simeq \;\left[ {{J_0} - 2{J_2}} \right]{\text{ }} + \left[ {\left( {{J_1} + 2{J_2}\cos \left( {{\omega _{\text{m}}}t} \right)} \right){\operatorname{e} ^{i{\omega _{\text{m}}}t}}} \right]{\text{ }} + \left[ {\left( {{J_1} + 2{J_2}\cos \left( {{\omega _{\text{m}}}t} \right)} \right){\operatorname{e} ^{ - i{\omega _{\text{m}}}t}}} \right]{\text{ }}\end{equation} \tag{ 17 }$$

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that is,

$$\begin{equation}{\psi _{\text{B}}}(t) \simeq \;\left[ {{J_0} - 2{J_2}} \right] + 2\left[ {{J_1} + 2{J_2}\cos \left( {{\omega _{\text{m}}}t} \right)} \right]\;\cos \left( {{\omega _{\text{m}}}t} \right).\end{equation} \tag{ 18 }$$

The amplitude and phase profiles of the three waves are shown in figures 5(b1) and (b2), which help understand how the besselon is formed. At t = 0, the three waves are in phase, so they interfere constructively and the amplitude of the besselon is maximal:

$$\begin{equation}{\psi _{\text{B}}}(t = 0) \simeq {J_0}({A_{\text{m}}}) + 2{J_1}({A_{\text{m}}}) + 2{J_2}({A_{\text{m}}}).{ }\end{equation} \tag{ 19 }$$

On the contrary, at t = T₀/2, the partially modulated waves are both π-shifted with respect to the continuous wave, hence they interfere destructively, yielding a minimum of the besselon amplitude:

$$\begin{equation}{\psi _{\text{B}}}(t = {T_0}/2) \simeq \;{J_0}({A_{\text{m}}}) - 2{J_1}({A_{\text{m}}}) + 2{J_2}({A_{\text{m}}}).\end{equation} \tag{ 20 }$$

From (20), zero amplitude is produced when

$$\begin{equation}{J_0}({A_{\text{m}}}) = \;2\left( {{J_1}({A_{\text{m}}}) - {J_2}({A_{\text{m}}})} \right),\end{equation} \tag{ 21 }$$

thus giving an optimum phase-modulation amplitude of A_m = 1.187 rad, which is consistent with the value of 1.109 rad obtained from the condition I_{J 0} (A_m) = 1 (cf section 2.1). Moreover, at t = T₀/4, the two sinusoidal waves have a phase difference of π, thereby cancelling each other and producing ${\psi _{\text{B}}}(t = {T_0}/4) \simeq \;{J_0}({A_{\text{m}}}) - 2{J_2}({A_{\text{m}}})$. This approximate equation shows that in the limit of small phase-modulation amplitudes, the minimum amplitude of the besselon, ${\psi _{\text{B}}}(t = {T_0}/4)$, continuously decreases with increasing A_m, thus the pulse duration becomes increasingly shorter. The besselon intensity profile obtained from the approximate equation (18) for A_m = 0.6 rad is shown in figure 5(b3). The deviation of this profile from the exact waveform calculated from (3) is almost indiscernible. As can be seen in figures 3(a1)–(b1), the agreement between the approximation to the wave profile based on (18) and the exact waveform is also rather good for A_m = 1.10 rad. The evolutions of the peak power and extinction ratio of the besselon with the phase-modulation amplitude predicted by this simplified three-wave model are plotted in figures 6(a) and (b) (purple circles), and they show excellent agreement with the exact evolutions provided that the level of the wave component J₃ is sufficiently low.

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We now focus on the use of the besselon in the context of high-repetition-rate, high-quality pulse sources. To this end, we limit our discussion to amplitudes of the initial sinusoidal phase modulation smaller than 4 rad. We can see in figure 1(b) and the magnified view of the central part of the besselon shown in figure 6(c1) that noticeable sidelobes develop in the temporal intensity profiles of the pulses for A_m > A_m,1. Because these sidelobes contain a non-negligible fraction of the total pulse energy, the peak power decreases concomitantly, as highlighted in figure 6(a). It is worth recalling here that the central component of the frequency spectrum decreases down to zero for A_m ⩽ A_m,1, and then it takes negative values up to A_m = A_m,2. We can therefore deduce from the simplified model developed in the previous section that at t = 0 and for A_m,1 < A_m < A_m,2, the continuous background has a phase offset of π relative to the sinusoidally modulated waves, which entails that the interference of the waves is now destructive. The resulting wave structure has a reduced peak power compared to the wave obtained for A_m,1 < A_m, and is no longer Fourier-transform limited. A solution to compensate this effect is to impress an additional π phase shift on the central component of the spectrum for A_m > A_m,1. We therefore obtain a modified besselon defined by $\psi {^{^{\prime}}}_{\text{B}}^{}(t) = \psi _{\text{B}}^{}(t)$ when A_m < A_m,1 and

$$\begin{equation}\psi {^{^{\prime}}}_{\text{B}}^{}(t) = \,{ } - {J_0}({A_{\text{m}}}) + 2\mathop \sum \limits_{n = 1}^\infty {J_n}({A_{\text{m}}})\,\cos \left( {n\,{\omega _{\text{m}}}t} \right) = { }\psi _{\text{B}}^{}(t)\, - 2\,{J_0}({A_{\text{m}}}){ }\end{equation} \tag{ 22 }$$

otherwise. The peak amplitude of the wave for A_m > A_m,1 then becomes

$$\begin{equation}{\psi {^{^{\prime}}}_{\text{B}}}(t = 0) = \,1 + {I_{J0}}({A_{\text{m}}}) - 2{J_0}({A_{\text{m}}})\end{equation} \tag{ 23 }$$

thus increasing by 2 |J₀(A_m)| relative to the peak amplitude of the unmodified besselon. The condition for maximal peak power is now J₀(A_m) = − 2 J₁(A_m). This brings about the occurrence at A'_m,1 = 3.40 rad (figure 6(a), cyan vertical line) of a peak power maximum of 8.93, which represents a 46% increase relative to the value obtained for ψ_B(t). The extinction ratio of the modified pulse wave is given by

$$\begin{equation}{{ER{^{^{\prime}}}}}({A_{\text{m}}}) = \;\frac{{1 + {I_{J0}}({A_{\text{m}}}) - 2{J_0}({A_{\text{m}}})}}{{1 - {I_{J0}}({A_{\text{m}}}) - 2{J_0}({A_{\text{m}}})}},\end{equation} \tag{ 24 }$$

hence it tends to infinity when the equation

$$\begin{equation}{I_{J0}}({A_{\text{m}}}) = 1 - 2{J_0}({A_{\text{m}}})\end{equation} \tag{ 25 }$$

is satisfied. This yields A_m = 2.86 rad (figure 6(b)). This value differs quite significantly from the one found for the unmodified besselon. We also note that extinction ratio values above 10² are attained for amplitudes of the initial phase modulation in the range 2.6 rad to 3.15 rad. The functional dependencies of the temporal intensity profiles of the modified and unmodified besselons on A_m are compared in figure 6(c). We can clearly see that at A_m = 2.86 rad the level of the sidelobes in the intensity profiles of the modified besselon is −13 dB, whereas the sidelobes are at only −7.22 dB of the peak power for the unmodified besselon. While the use of an additional π phase shift entails enhancement of the peak power of the besselon (figure 6(a)), no significant improvement is observed in terms of temporal duration (figure 4). Nonetheless, it is noteworthy that the modified besselon attains a FWHM duration of 0.1T₀ at A_m = 2.86 rad, which is significantly smaller than the duration of the unmodified besselon at the point A_m = 1.1086 rad of maximal extinction ratio. Finally, we note that for phase-modulation amplitudes larger than 3.83 rad, the first sidelobes in the besselon experience a π phase shift as J₁ changes of sign. Therefore, further enhancement of the peak power would require the insertion of additional phase shifts into the wave.

In this section, we demonstrate the possibility of interleaving modified besselons in the time domain to double the pulse repetition rate. Several solutions are available for repetition-rate multiplication, such as the use of the fractional Talbot effect [24, 25], temporal delay lines [26], or spectral pulse shaping involving spectral manipulation of the intensity [27] and/or phase [28] components of the pulses. Here, we achieve doubling of the repetition rate by suppressing the odd components of the frequency spectrum of the wave. The envelope of the resulting pulse wave is then given by

$$\begin{equation}\left\{ \begin{gathered} \hfill {\psi _{2{\text{B}}}}(t) = \;{J_0}({A_{\text{m}}}) + 2\sum\limits_{n = 1}^\infty {{J_{2n}}({A_{\text{m}}})\;\cos \left( {2n\;{\omega _{\text{m}}}t} \right)} \,\,\, \operatorname{for} \;{A_{\text{m}}} \le {A_{{\text{m}},1}} \\ \hfill \psi^\prime_{2{\textrm{B}}}(t) = \; - {J_0}({A_{\text{m}}}) + 2\sum\limits_{n = 1}^\infty {{J_{2n}}({A_{\text{m}}})\;\cos \left( {2n\;{\omega _{\text{m}}}t} \right)} \,\,\,\operatorname{for} \;{A_{\text{m}}} > {A_{{\text{m}},1}} \\ \end{gathered} \right.\end{equation} \tag{ 26 }$$

which, using (13), we can recast in the form

$$\begin{equation}\left\{ \begin{array}{l} {\psi _{2{\textrm{B}}}}(t) = \;{\psi _{\textrm{D}}}(t)\;\;\;{\mathop{\text{for}}\nolimits} \;\;{A_{\textrm{m}}} \le {A_{{\textrm{m}},1}}\\ \psi^\prime_{2{\textrm{B}}}(t) = \;{\psi _{\textrm{D}}}(t) - 2\;{J_0}({A_{\textrm{m}}})\;\;{\mathop{\text{for}}\nolimits} \;\;{A_{\textrm{m}}} > {A_{{\textrm{m}},1}} \end{array} \right.\!\! .\end{equation} \tag{ 27 }$$

The evolution of the temporal intensity profile of the repetition-rate-doubled pulse train with the amplitude of the initial phase modulation is shown in figure 7(a). We can infer from this figure that maximal extinction ratio of the pulses is achieved for A_m = 1.57 rad and A_m = 3.78 rad, rather than at A_m = 1.10 rad and A_m = 2.86 rad as one could expect from (11) and (25). Indeed, the minimum amplitude of the wave is given by

$$\begin{equation}\left\{ \begin{array}{l} {\psi _{2{\textrm{B}}}}(t = {T_0}/4) = \;\cos ({A_{\textrm{m}}})\;\;\;\;{\mathop{\text{for}}\nolimits} \;\;{A_{\textrm{m}}} \le {A_{{\textrm{m}},1}}\\ \psi^\prime_{2{\textrm{B}}}(t = {T_0}/4) = \;\cos ({A_{\textrm{m}}}) - 2\;{J_0}({A_{\textrm{m}}})\;\;{\mathop{\text{for}}\nolimits} \;\;{A_{\textrm{m}}} > {A_{{\textrm{m}},1}} \end{array} \right.\end{equation} \tag{ 28 }$$

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and vanishes when

$$\begin{equation}\left\{ \begin{array}{l} {A_{\textrm{m}}} = \pi /2\;\;\;{\mathop{\text{for}}\nolimits} \;\;{A_{\textrm{m}}} \le {A_{{\textrm{m}},1}}\\ 2\;{J_0}({A_{\textrm{m}}}) = \;\cos \left( {{A_{\textrm{m}}}} \right)\;\;{\mathop{\text{for}}\nolimits} \;\;{A_{\textrm{m}}} > {A_{{\textrm{m}},1}} \end{array} \right. .\end{equation} \tag{ 29 }$$

In figure 7(b), we compare the intensity profiles of the waves obtained before and after repetition-rate doubling. We can see that suppressing the odd spectral components enables cancellation of the residual background between successive pulses [26]. Neat pulse shapes are achieved especially at A_m = 3.78 rad (optimum operating point for A_m > A_m,1), where the spurious sidelobes are at a level of nearly −20 dB from the peak power. At this operating point, the FWHM duration of the pulses is 0.092 T₀, entailing a duty cycle of 0.18. However, the main drawback of this approach is a drop of the pulse peak power, which decreases by more than a factor of two relative to the pulse train at the fundamental repetition rate.

The experimental setup for generating and processing besselon waves is based on devices that are commercially available and typical of the telecommunication industry, as sketched in figure 8(a). A continuous-wave laser at 1550 nm is phase modulated in the time domain using a lithium niobate electro-optic device driven by a sinusoidal electrical signal at a frequency f_m = 14 GHz. The maximum amplitude of phase modulation that can be imparted by this low-voltage modulator is A_m = 3.75 rad, but modulations exceeding 10 rad are feasible by use of a resonant microwave modulator [29–31] or cross-phase modulation in a highly nonlinear fibre [8].

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A spectral pulse shaper (Finisar Waveshaper) based on liquid crystal on silicon technology [32] then imparts a set of discrete spectral phase shifts of π/2 to individual successive components of the resulting frequency comb to synthesise the besselon waves ψ_B and ψ'_B (figure 8, panels (b1) and (b2)). To generate the repetition-rate-doubled besselon pulse trains, ψ_2B and ψ'_2B, we replace in the spectral profile programmed on the filter two successive π/2 phase shifts with single π phase shifts which are then imprinted on the odd frequency components (figure 8, panels (b3) and (b4), red curves). Indeed, these discrete spectral π phase shifts create a notch filter [33] that dramatically attenuates the odd harmonics of the spectrum. Attenuation curves recorded using an amplified spontaneous emission source as the input signal are shown in figure 8(b) (black solid curves), and confirm that the desired spectral amplitude and phase can be achieved by simple phase-only spectral processing. Note that in the case of ψ'_2B, we do not filter out the harmonics at ±14 GHz. Indeed, at the optimum modulation depth (A_m = 3.72 rad, very close to the first zero of the Bessel function of order one) the intensity of these spectral sidebands is two orders of magnitude smaller than that of the neighboring peaks. It is worth mentioning that for operation at fixed wavelength and repetition rate, this linear shaping stage can also be fully realised by means of a cascaded uniform fibre Bragg grating [34]. We also note that increase of the pulse repetition rate by suppression of each of two frequency components of the signal can be achieved by other approaches, such as the use of a birefringent fibre [35] or Fabry–Perot interferometers [36].

The generated signal is recorded with a high-speed optical sampling oscilloscope (1 ps resolution) and a high-resolution optical spectrum analyser. A low-noise erbium-doped fibre amplifier is placed before the spectral shaper to achieve the optimum power level on the temporal detection device.

Figure 9 shows the temporal and spectral intensity profiles of the besselon that are generated at various modulation depths A_m. The agreement between the experimental results and the theoretical predictions is excellent in both the time and frequency domains. We can recognise all the various traits of the besselon that we have discussed in section 2. For low values of A_m (typically below 0.8 rad), the wave is only partially modulated and a continuous background remains (panel (a)). For values around 1.1 rad (panel (b)), the pulses feature a very high extinction ratio and a Gaussian-like shape. With further increase of A_m, the background impairs significantly the pulse train (A_m= 2.47 rad, panel (c)), and strong sidelobes subsequently develop on the pulses (panel (d1)). Note that the picosecond resolution of our optical sampling oscilloscope device may impair the measurements at large modulation depths and ultimately limit the agreement with the analytical solution. The pedestals can be suppressed by imparting an additional π phase shift to the central frequency component of the wave, hence creating a modified besselon (panel (d3)).

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The experimentally observed dependencies of the temporal intensity profiles of both besselon structure types ψ_B and ψ'_B on the amplitude of the initial phase modulation are illustrated in figure 10. Once again, these results are in remarkably good agreement with the predicted dependencies shown in figure 6(c). Indeed, the contour plots based on the analytical solution closely reproduce both the shape of the central part of the pulses and the evolution of the temporal sidelobes. We also show in figure 10 the intensity profiles recorded at the compression point where the extinction ratio is maximal (A_m = 2.83 rad), which further stresses the beneficial impact of imparting a π phase shift to the central spectral component.

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The excellent agreement between theory and experiment is also visible in figure 11, where we show the evolutions of the peak power, FWHM duration and extinction ratio of the besselon with A_m. The predicted behaviours are quantitatively reproduced in the experiment, thus validating our theoretical model. The key values of initial phase modulation amplitude leading to enhanced extinction ratio, i.e. A_m = 1.10 rad and A_m = 2.83 rad, are thereby confirmed.

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Figure 12 shows the temporal and spectral characteristics of the besselon pulse trains generated at a rate of 28 GHz for two different amplitudes of the initial phase modulation, A_m = π/2 and A_m = 3.72 rad. The odd harmonics of the spectra (panel 2) are reduced by more than 20 dB in intensity as a result of the π phase shifts imparted to them. Even though these frequency components are not completely suppressed, their very low levels no longer affect the overall stability of the resulting pulse trains, as confirmed by the eye diagrams. The experimental results are in excellent agreement with the theoretical predictions given by (27). The very low intensity of the pulse pedestals stemming from reduction of the odd frequency components entails very large extinction ratios. For A_m = 3.72 rad, the FWHM duration of the pulses is 6.9 ps, yielding a duty cycle of 0.19.

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