Propagation of temporal coherence gratings in dispersive medium with a chirper

C. Ding; C. Ding; C. Ding; O. Korotkova; O. Korotkova; D. Zhao; D. Li; Z. Zhao; L. Pan; L. Pan

doi:10.1364/OE.386598

1. Introduction

As we all know, the coherence state of a light source plays an important role in the spatio-temporal intensity distribution of the radiated optical field [1,2]. Hence, the modulation of source coherence is an efficient tool for controlling the far-field light intensity distribution [3–7]. After Gori and Santarsiero introduced a simple mathematical procedure for devising genuine spatial correlation functions of stochastic optical sources in 2007 [8] modeling the novel classes of partially coherent beams became a broad research subfield of statistical optics. Structuring of the source coherence state was shown to result in a variety of extraordinary evolution scenarios for the beam average intensity, including self-splitting, self-focusing, self-steering, and flat-topping; as well as formation as rectangular, ring-shaped, dark-hollow, gridded, or cusped profiles in the far field [9–16]. Recently, the outcomes of phase structuring of the complex coherence states have also been thoroughly investigated and were shown to break Cartesian symmetry in the propagating beam’s average intensity and control its tilt and radial acceleration [17–20].

On the other hand stochastic optical pulses with partial spectral or temporal coherence that can be radiated by a variety of currently available sources, such as free-electron lasers, excimer lasers, random lasers and supercontinuum light generated in microstructured fibers, have recently attracted widespread attention because of their possible applications in optical telecommunication, imaging, and fiber optics [21–36]. In the majority of these studies concerning the coherence properties of pulses, the traditional Gaussian Schell-model form is chosen to characterize the correlations between different frequency components or time instants. In 2013 Lajunen and Saastamoinen introduced the correlation functions of the stochastic pulse trains having non-Gaussian correlation Schell-model form and found that the partially coherent pulsed source with non-uniform correlation may lead to the temporal self-focusing phenomenon upon dispersive-medium propagation [37]. Our own recent investigations indicated that in the dispersive fibers the average pulse intensity profiles can be finely controlled by assigning various source temporal coherence profiles [38–42]. Just like in spatial domain, average intensity of stochastic pulses can self-split, form flat-top profiles, form multi-level signals with prescribed order, etc. In addition, Tang et al. pointed out that partially coherent pulsed source with the sinc Schell-model correlation could give rise to a double-layer flat-top intensity distribution and flat-top temporal degree of polarization distribution [43]. Moreover, if the partially coherent pulsed sources are described by frequency-dependent transverse scale parameters, the interesting spatiotemporal coupling is revealed in few-cycle or sub-cycle regimes [44]. A possible experimental procedure for generation of partially coherent pulses has been proposed in [45].

In this paper we will investigate the propagation properties of Temporal Coherence Grating (TCG) pulse trains introduced by Ma and Ponomarenko [46]. A TCG is statistically stationary and has a time-periodic DOC with a given characteristic period, being non-Gaussian correlation Schell-model. We first derive the analytical expression of the mean intensity distribution of the TCG pulse train by means of a temporal ABCD matrix approach. Then we explore the evolution of the mean intensity distribution of the TCG pulse train in dispersive medium with a chirper, where the influence of pulse parameters and medium parameters on is emphasized.

2. Theory

We begin by considering a partially coherent pulsed train source with a two-time Mutual Coherence Function (MCF) having the form of a temporal coherence grating:

(1)$${\varGamma _0}({{\tau_1},{\tau_2}} )= \sum\limits_{n = 0}^N {\frac{{{v_n}}}{{\sqrt \pi }}\exp \left( { - \frac{{\tau_1^2 + \tau_2^2}}{{2T_0^2}}} \right)} \exp \left[ { - \frac{{i\pi n}}{{a{T_0}}}({{\tau_1} - {\tau_2}} )} \right]\exp [{ - i{\omega_0}({{\tau_2} - {\tau_1}} )} ],$$

which corresponds to the pseudo-mode expansion proposed in [47]. Here n is a non-negative integer, T₀ denotes the pulse duration, a is a dimensionless grating constant, v_n is the power distribution of pseudo-modes, N indicates the number of grating lobes and ω₀ is the carrier frequency of pulse. Here we only consider the plane-wave pulses for simplicity, i.e., we assume that the transverse intensity distribution is uniform and the wave fronts are planar. It then follows from Eq. (1) that the MCF reduced to the fundamental Gaussian pulses when N=0. As can be seen from Eq. (1), each source has a Gaussian intensity profile,

(2)$$I(\tau )= {I_0}\exp \left( { - \frac{{\tau_{}^2}}{{T_0^2}}} \right), {I_0} = \frac{1}{{\sqrt \pi }}\sum\limits_{n = 0}^N {{v_n}} ,$$

and its temporal Degree of Coherence (DOC), defined as [1,2]

(3)$$\mu ({{\tau_1},{\tau_2}} )= \frac{{{\varGamma _0}({{\tau_1},{\tau_2}} )}}{{\sqrt {I({\tau _1})I({\tau _2})} }},$$

can be expressed as

(4)$$\mu ({{\tau_1},{\tau_2}} )= \frac{1}{{{I_0}}}\sum\limits_{n = 0}^N {{v_n}} \exp \left[ { - \frac{{i\pi n}}{{a{T_0}}}({{\tau_1} - {\tau_2}} )} \right]\exp [{ - i{\omega_0}({{\tau_2} - {\tau_1}} )} ].$$

In Fig. 1 the absolute value of the temporal DOC is given as a function of (τ₂-τ₁)/aT₀ for different values of N. We can see that the DOC, possessing the distribution of temporal coherence gratings modulated by the number of grating lobes N, shows the periodic properties with an adjustable grating constant a. Different power distributions of pseudo-modes v_n, results in different modulations of incident pulse source. As we will show in subsequent sections, the distributions of temporal DOC with different periodicity lead to the interesting propagation behavior of the partially coherent pulses generated by this kind of source. Hence, we may call this kind of pulse source the temporal coherent gratings (TCG) pulse train, which is essentially consistent with Ponomarenko’s model [46].

Fig. 1. Absolute value of the temporal DOC for different N. (a)-(c): v_n=const., (d)-(f): v_n=αⁿ/n!, α=5.

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In this paper, we consider the paraxial propagation of the MCF of the TCG pulse train through an ABCD optical system. According to extended Huygens-Fresnel integral [22,48], the MCF at the z plane from the source can be expressed as

(5)$$\begin{aligned} \varGamma ({{t_1},{t_2}, z} ) &= \frac{{{\omega _0}}}{{2\pi B}}\exp \left[ { - \frac{{i{\omega_0}D}}{{2B}}({t_1^2 - t_2^2} )} \right] \\ &\times \int {\int_{ - \infty }^\infty {{\varGamma _0}({{\tau_1},{\tau_2}} )} } \exp \left[ { - \frac{{i{\omega_0}A}}{{2B}}({\tau_1^2 - \tau_2^2} )} \right]\exp \left[ {\frac{{i{\omega_0}}}{B}({{\tau_1}{t_1} - {\tau_2}{t_2}} )} \right]d{\tau _1}d{\tau _2}, \end{aligned}$$

where A, B, and D are elements of the temporal ABCD matrix. Here we have assumed that the time coordinate is measured in the reference frame moving at the group velocity of the pulse train. For mathematical convenience we now introduce the following average and difference coordinates:

(6)$$\tau = {{({{\tau_1} + {\tau_2}} )} \mathord{\left/ {\vphantom {{({{\tau_1} + {\tau_2}} )} 2}} \right.} 2}, \Delta \tau = {\tau _2} - {\tau _1}, t = {{({{t_1} + {t_2}} )} \mathord{\left/ {\vphantom {{({{t_1} + {t_2}} )} 2}} \right.} 2}, \Delta t = {t_2} - {t_1}.$$

Inserting from Eq. (1) into Eq. (5) and changing coordinates according to Eq. (6), we obtain the following formula for the MCF:

(7)$$\begin{aligned}\varGamma ({{t_1},{t_2}, z}) &= \frac{{{\omega _0}}}{{2\pi B}}\exp \left[ {\frac{{i{\omega_0}D}}{B}t\Delta t} \right] \\ &\times \sum\limits_{n = 0}^N {\frac{{{v_n}}}{{\sqrt \pi }}} \int {\int_{ - \infty }^\infty {\exp \left[ { - \frac{{{\tau^2}}}{{T_0^2}}} \right]\exp \left[ {\frac{{i{\omega_0}}}{B}({A\Delta \tau - \Delta t} )\tau } \right]} } d\tau \\ &\times \exp \left[ { - \frac{{\Delta {\tau^2}}}{{4T_0^2}}} \right]\exp \left[ {\left( {\frac{{i\pi n}}{{a{T_0}}} - \frac{{i{\omega_0}t}}{B}} \right)\Delta \tau } \right]d\Delta \tau.\end{aligned}$$

First integrating over τ, then integrating over Δτ, we can express the Eq. (7) in the following form

(8)$$\begin{aligned}\varGamma ({t,\Delta t, z}) &= \sum\limits_{n = 0}^N {\frac{{{v_n}{T_0}}}{{\sqrt \pi T(z)}}} \exp \left[ {\frac{{i{\omega_0}D}}{B}t\Delta t} \right]\exp \left[ { - \frac{{\Delta {t^2}}}{{4{T^2}(z)}}} \right] \\ &\times \exp \left[ { - {{{{\left( {t - \frac{{\pi nB}}{{a{T_0}{\omega_0}}}} \right)}^2}} \mathord{\left/ {\vphantom {{{{\left( {t - \frac{{\pi nB}}{{a{T_0}{\omega_0}}}} \right)}^2}} {{T^2}(z)}}} \right.} {{T^2}(z)}}} \right]\exp \left[ { - \frac{{iT_0^2A{\omega_0}}}{{B{T^2}(z)}}\left( {t - \frac{{\pi nB}}{{a{T_0}{\omega_0}}}} \right)\Delta t} \right], \end{aligned}$$

where

(9)$$T(z) = {T_0}\sqrt {{A^2} + \frac{{{B^2}}}{{T_0^4\omega _0^2}}} .$$

Expression (8) shows the MCF at the output plane z of an arbitrary temporal optical ABCD system. Here, our aim is to investigate the evolution of the mean intensity profile and the temporal DOC of the TCG pulse train propagating in a linearly dispersive medium, where the ABCD matrix has form

(10)$$\left( {\begin{array}{cc} A &B\\ C &D \end{array}} \right) = \left( {\begin{array}{cc} {1 + s{\beta_2}z} &{{\omega_0}{\beta_2}z}\\ {{s \mathord{\left/ {\vphantom {s {{\omega_0}}}} \right.} {{\omega_0}}}} &1 \end{array}} \right),$$

with β₂ representing the group-velocity dispersion coefficient and s being the chirp coefficient of the ideal chirper. On substituting from Eq. (10) into Eq. (8), letting Δt=0, we obtain the mean intensity profile of the TCG pulse train in linearly dispersive medium:

(11)$$I({t, z} )= \sum\limits_{n = 0}^N {\frac{{{v_n}{T_0}}}{{\sqrt \pi T(z)}}} \exp \left[ { - {{{{\left( {t - \frac{{\pi n{\beta_2}z}}{{a{T_0}}}} \right)}^2}} \mathord{\left/ {\vphantom {{{{\left( {t - \frac{{\pi n{\beta_2}z}}{{a{T_0}}}} \right)}^2}} {{T^2}(z)}}} \right.} {{T^2}(z)}}} \right],$$

where

(12)$$T(z) = {T_0}\sqrt {{{(1 + s{\beta _2}z)}^2} + {{\beta _2^2{z^2}} \mathord{\left/ {\vphantom {{\beta_2^2{z^2}} {T_0^4}}} \right.} {T_0^4}}} .$$

And the temporal DOC of the TCG pulse train in linearly dispersive medium can be derived by the formula

(13)$$\gamma ({{t_1},{t_2}, z} )= \frac{{\varGamma ({{t_1},{t_2}, z} )}}{{\sqrt {I({t_1})I({t_2})} }}.$$

3. Numerical calculation results and analysis

In this section, we will illustrate the evolution of the temporal intensity profile and the temporal DOC of TCG pulse train in a linearly dispersive medium according to the analytical expressions given in Eqs. (11) and (13). The parameters used in numerical calculations are chosen to be T₀=20ps, a=0.25, s=−0.8 × 10⁻²ps⁻², β₂=50ps²/km and v_n=1 unless different values are specified.

Figure 2 presents the color-coded plot of normalized intensity distribution I(t, z) as a function of propagation distance z and time t for different numbers of grating lobes N with a=0.25. It is shown that the temporal coherence distribution of the incident pulse, characterized by the number of grating lobes N, has an important impact on the propagation properties of the TCG pulse train, especially, on its far-field temporal intensity profile. On the condition of uniform power distribution, i.e., v_n=const., as shown in Figs. 2(a)–2(d), for N=1 [Fig. 2(b)], the single pulse begins to split into double pulses with the same peak intensity upon dispersive-medium propagation at propagation distance z=1km. And the single pulse starts to split into three pulses and four pulses for N=2 and N=3, respectively. More generally, the single pulse splits into K+1 subpulses for N = K. For the case of the non-uniform power distribution, i.e., v_n=αⁿ/n! with α=3, the peak intensities of the subpulses are different from each other [see Figs. 2(f)–2(h)], strongly depending on the value of parameter α. For larger values, e.g., α=5, the intensity distribution becomes less uniform and some subpulses even disappear near the optical axis [see Figs. 2(j)–2(l)].

Fig. 2. Color-coded plot of normalized intensity distributionI(t, z) as a function of propagation distance z and time t. (a)-(d): v_n=const.,(e)-(h): v_n=αⁿ/n!, α=3.(i)-(l): v_n=αⁿ/n!, α=5.

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In addition, the self-focusing phenomenon occurs during the dispersive-medium propagation. We can give a physical interpretation of this as follows. It is evident from Eq. (12) that, the pulse duration T(z) at z plane can be rewritten as

(14)$${{{T^2}(z)} \mathord{\left/ {\vphantom {{{T^2}(z)} T}} \right.} T}_0^2 = ({s^2}\beta _2^2 + {{\beta _2^2} \mathord{\left/ {\vphantom {{\beta_2^2} {T_0^4}}} \right.} {T_0^4}}){z^2} + 2s{\beta _2}z + 1.$$

From Eq. (14) we see that ${{{T^2}(z)} \mathord{\left/ {\vphantom {{{T^2}(z)} T}} \right.} T}_0^2$ is a quadratic function of z and will reach a minimum when

(15)$${z_{\min }} ={-} \frac{s}{{{\beta _2}({{s^2} + {1 \mathord{\left/ {\vphantom {1 {T_0^4}}} \right.} {T_0^4}}} )}},$$

i.e., the intensity peak takes place at this position. Hence Eq. (15) implies that there must be a minimum when the chirp coefficient is much smaller than zero. Thus the self-focusing phenomenon occurs at z_min. On the other hand, we can also use the temporal ABCD matrix to give further physical explanation. The ABCD matrix in Eq. (10) can be rewritten as a product of two matrixes M₁×M₂

(16)$${M_1} = \left({\begin{array}{cc} 1 &{{\omega_0}{\beta_2}z}\\ 0 &1 \end{array}} \right),$$

and

(17)$${M_2} = \left( {\begin{array}{cc} 1 &0\\ {{s \mathord{\left/ {\vphantom {s {{\omega_0}}}} \right.} {{\omega_0}}}} &1 \end{array}} \right),$$

where matrix M₁ can be the free-space propagation of a Gaussian beam, where $z^{\prime} = {\omega _0}{\beta _2}z$ indicates the generalized propagation distance, while matrix M₁ corresponds to a lens with focal length f in Gaussian beam optics, with ${1 \mathord{\left/ {\vphantom {1 f}} \right.} f} \leftrightarrow - {s \mathord{\left/ {\vphantom {s {{\omega_0}}}} \right.} {{\omega _0}}}$. Thus the self-focusing phenomenon of the TCG pulse train upon dispersive-medium propagation is understandable. In Fig. 3 we present the color-coded plot of the normalized intensity distribution I(t, z) as a function of propagation distance z and time t for different values of chirp coefficient s with N=2, a=0.25 and v_n=const. As can be seen the self-focusing phenomenon is remarkable when the value of chirp coefficient s is much smaller than zero. With increasing s, the self-focusing phenomenon disappears gradually when s≥0. This self-focusing phenomenon of a partially coherent pulse is different from that one found by Lajunen and Saastamoinen [37], which is attributed to the non-uniform temporal coherent distribution of the incident pulse source. However, in our model, the pulse chirp leads to the self-focusing phenomenon upon propagation.

Fig. 3. Color-coded plot of normalized intensity distribution I(t, z) as a function of propagation distance z and time t for different chirp coefficient s.

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Figures 4(a)–4(c) shows the color-coded plot of normalized intensity distribution I(t, z) of the TCG pulse train as a function of grating constant a and time t for different N with v_n=const., and the corresponding 3D plot is given in Figs. 4(d)–4(f). Here the calculation parameters are z=2 km and T₀=20ps. It is shown that, at certain propagation distances (e.g., z=2 km), many subpulses merge into a single pulse possessing larger pulse width with increasing grating constant a. Figures 5(a)–5(c) show the color-coded plot of normalized intensity distribution I(t, z) of the TCG pulse train as a function of pulse duration T₀ and time t for different N, and corresponding 3D plot is given in Figs. 5(d)–5(f). The calculation parameters are z=2 km and a=0.25. As can be seen, the influence of pulse duration T₀ on intensity distribution I(t, z) has a similar behavior to that of grating constant a, i.e., many subpulses become a single pulse with increasing pulse duration T₀. Compared with the effect of grating constant a on intensity distribution I(t, z), the single pulse has much wider pulse width, which approximates a flat-top intensity profile. Hence, Figs. 4 and 5 imply that one can obtain much wider pulse width by adjusting grating constant a or pulse duration T₀ of incident pulse.

Fig. 4. Color-coded plot (a)-(c) of normalized intensity distribution I(t, z) and corresponding 3D plot (d)-(f) as a grating constant a and time t for different N.

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Fig. 5. Color-coded plot (a)-(c) of normalized intensity distribution I(t, z) and corresponding 3D plot (d)-(f) as a function of pulse duration T₀ and time t for different N.

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Figure 6 shows the effect of second-order dispersion coefficient β₂ on the normalized intensity distribution I(t, z) of the TCG pulse train with v_n=const. As can be seen, the position of the focal spots shifts toward the source plane with increasing β₂. And the distances among different focal spots broaden much larger with the increase of β₂. These results can be explained according to Eqs. (11) and (15). It follows from Eq. (15) that the position of focal spots is z_min, whose value will decrease with increasing β₂. It is then implied by Eq. (11) that the center position of subpulses is t_c=πβ₂z/aT₀, which is a monotonic increasing function of β₂ and z. That is why these subpulses diverge so fast during the dispersive-medium propagation.

Fig. 6. Color-coded plot of normalized intensity distribution I(t, z) as a function of propagation distance z and time t for different second-order dispersion coefficient β₂ with v_n=const.

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Figure 7 gives the absolute value of the temporal DOC γ (t₁, t₂, z) of a TCG pulse train as a function of time interval t_d=t₂-t₁, upper limit parameter N and propagation distance z. It is shown that for large t_d the peak values of the temporal DOC decrease gradually with increasing propagation distance z. Here, term Exp[-(t₂-t₁)²/4T²(z)] in Eq. (8) plays an important role for the magnitude of the DOC. Hence, this term results in the decrease of the peak values of temporal DOC at large time lags t_d. However, for short t_d the minima of the temporal DOC increase gradually with increasing z. Thus, the distribution of the temporal DOC changes from initial periodic comb-form distribution to Gaussian form, with increasing propagation distance z. Figure 8 shows the influence of incident pulse duration T₀, grating constant a and second-order dispersion coefficient β₂ on the distribution of temporal DOC γ (t₁, t₂, z) upon dispersive-medium propagation. The calculation parameters are N=2 and z=0.2km. Compared with Fig. 7(b2), we can see from Fig. 8 that the distribution of temporal DOC will degenerate into Gaussian form with decreasing incident pulse duration T₀ and grating constant a, or increasing second-order dispersion coefficient β₂.

Fig. 7. Absolute value of temporal DOC γ (t₁, t₂, z) of TCG pulse train versus different time interval t_d=t₂-t₁in dispersive medium for different N and propagation distance z.

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Fig. 8. Absolute value of temporal DOC γ (t₁, t₂, z) of TCG pulse train versus different time interval t_d=t₂-t₁in dispersive medium for different pulse duration T₀, grating constant a and second-order dispersion coefficient β₂. (a)-(b): a=0.25, β₂=50ps²/km; (c)-(d): T₀=20ps, β₂=50ps²/km; (e)-(f): T₀=20ps, a=0.25.

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4. Conclusion

In this paper, we have investigated the propagation of the TCG pulse train in dispersive medium with a chirper. The analytic expression for the two-time MCF of the TCG pulse train through an ABCD temporal matrix is derived, and used to explore the evolution of the mean temporal intensity and temporal DOC. We found that the number of grating lobes N plays an important role on the evolution of the mean temporal intensity. Upon dispersive-medium propagation, the single pulse splits into N+1 subpulses with the same or different peak intensities for the power distributions v_n=const. or v_n=αⁿ/n!, respectively. In addition, the self-focusing effect occurs during the propagation, which is chirp-induced non-linear phenomenon. The physical mechanism is different from the self-focusing phenomenon revealed in [37] (self-focusing being induced by the non-uniform temporal coherence distribution). The chirp of the medium is equivalent to a converging lens, which leads to the compression of the incident pulse duration. The distribution of the temporal DOC will degenerate into Gaussian form with increasing propagation distance or suitable choice of the incident pulse parameters and medium dispersion strength. The results obtained in this study would have potential application in pulse shaping or laser micromachining [49,50]. For example, if we would like to obtain a stochastic pulse with profile with a wide average pulse width, we can modulate the grating constant a or incident pulse duration T₀ to find desirable parameters. On the contrary, we can acquire a compressed pulse profile with short pulse duration by introducing a pulse chirper to realize a focusing function. And in the material micromachining, by the suitable choice of grating lobes N, the peak intensities of self-focusing spots can be modulated to ablate the appropriate material and improve fabrication resolution.

Funding

National Natural Science Foundation of China (NSFC) (61575091, 61675094, 11874321); Excellent Team of Spectrum Technology and Application of Henan province (18024123007).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Propagation of temporal coherence gratings in dispersive medium with a chirper

Abstract

1. Introduction

2. Theory

3. Numerical calculation results and analysis

4. Conclusion

Funding

Disclosures

References

Cited By

Figures (8)

Equations (17)

Optics Express