Position modulation with random pulses

Min Yao; Olga Korotkova; Chaoliang Ding; Liuzhan Pan

doi:10.1364/OE.22.016197

1. Introduction

The ability of the statistically stationary stochastic fields to modify their properties, such as the average intensity profile, the spectral composition, the degree of coherence, and the polarization state has been explored in details in spatial domain [1–5]. Such changes are frequently termed as “correlation-induced” to be distinguished from other effects, for instance, the Doppler-like shifts. It is well known [1] that in the spatial domain, the shape of the source degree of coherence determines the average intensity distribution in the far field, the two quantities being related by a Fourier-type law. In addition to the classic Gaussian-Schell-model sources and beams [1] in which the source degree of coherence and, hence, the far-field intensity are both Gaussian functions, other pairs have been recently explored: J0-Bessel- [6–8], J0-Bessel-Gaussian-, Laguerre-Gaussian- and cos-Gaussian-correlated [9–12] sources all leading to ring-shaped far-fields; non-uniformly correlated sources [13,14] capable of producing a shift in the intensity maximum on propagation, Multi-Gaussian-correlated sources producing flat-top far field intensity distribution with circular [15–17], rectangular [18] symmetries, and optical frames [19] being the linear superpositions of [15–18]. Both scalar and electromagnetic versions of such random sources can be practically realized with the help of the holograms/ground-glass plates [20,21] and the spatial light modulators [22].

Similar developments in the time domain have led to description of sources with prescribed temporal degree of coherence [1] generating random pulses with certain ensemble average intensity profiles [23–29]. The first class of random pulses, Gaussian-correlated pulses, was found to lead to the Gaussian intensity profiles (after sufficiently long propagation distance in dispersive media) with adjustable average duration that is fully controlled by the r.m.s. width of the temporal source degree of coherence [24]. Recently several other random pulse classes were shown to originate with the Gaussian intensity distribution but to acquire other profiles on propagation in dispersive media: non-uniformly-correlated pulse ensemble produces a temporal shift in their maximum intensity [30]; cos-Gaussian-correlated pulse ensemble [31] splits from a single average pulse distribution into two symmetric parts and Multi-Gaussian-correlated pulse ensemble acquires a flat-top intensity profile [32]. The possibility of experimental realization of the random pulses has been illustrated in [33].

The purpose of this article is to demonstrate that a linear superposition of several Multi-Gaussian-correlated pulse ensembles can be used for production of a new ensemble with average flat intensities with different heights, durations and the edge sharpness. Such superposition idea has been recently explored in the spatial domain resulting in a variety of optical frames [19]. We are to show that in the temporal domain a similar idea can be employed for quite arbitrary pulse average intensity modulation, which can be regarded as the extension of the pulse-position modulation [34] from deterministic to random domain. The important distinction must be made between the direct intensity modulation of the ensemble of random pulses and the modulation through the temporal degree of coherence: in the latter case the modulated ensemble average intensity profile, after being formed on propagation, remains shape invariant. This feature makes this type of modulation preferential in situations when the information must be sent securely at a distance, without the possibility of being intercepted close to the source. The superposition procedure of the source correlation functions, especially with some terms having negative signs, has been recently explored in details in [35] (see also [36]) where the conditions for its validity have been outlined.

The building blocks for our random pulse modulation method are the multi-Gaussian distributions, being the weighted sums of Gaussian functions with different heights and widths. Hence the method offers substantial tractability: even after propagation in the dispersive media the solution is expressed in the closed form. In addition to being used as correlation functions, multi-Gaussian distributions have been previously successfully employed for modeling of aperture edges [37], field and intensity distributions [38,39] and scattering potentials [40].

2. Source model for the simplest random pulse case: two flat average pulses

The second-order temporal correlation properties of the ensemble of random, statistically stationary pulses can be characterized by their mutual coherence function [1]

Γ^{(0)} (t_{1}, t_{2}) = < U^{(0) *} (t_{1}) U^{(0)} (t_{2}) >,

where U(t) is the complex analytic signal of a single pulse realization, star denotes complex conjugate and the angular brackets denote the statistical ensemble.

In this section we will establish the form of the mutual coherence function (1) of a source producing the far-field pulse ensemble with average intensity consisting of two parts each with arbitrary intensity distributions. We begin by recalling that any valid mutual coherence function has integral representation [41]

Γ^{(0)} (t_{1}, t_{2}) = \int p (v) H^{*} (t_{1}, v) H (t_{2}, v) d v,

where H is an arbitrary kernel but the weighting function p is non-negative, Fourier-transformable function. A broad class of the Gaussian Schell-model pulses can be established if H takes on the form:

H (t, v) = \exp (- \frac{t^{2}}{4 σ_{t 0}^{2}}) \exp (- i v t) .

Here $σ_{t 0}^{}$ is the r.m.s. width of the pulse ensemble. On substituting from Eq. (3) into Eq. (2) and after integration one finds that

Γ^{(0)} (t_{1}, t_{2}) = \exp (- \frac{t_{1}^{2} + t_{2}^{2}}{4 σ_{t 0}^{2}}) γ (t_{1}, t_{2}),

where the source temporal degree of coherence becomes:

γ^{(0)} (t_{1}, t_{2}) = \int p (v) \exp [- i v (t_{2} - t_{1})] d v .

In order to produce the simplest random pulse with two different flat-top parts we will choose function p in the form of a difference

p (v) = p_{b} (v) - p_{f} (v),

where both terms obey one-dimensional multi-Gaussian distributions [32]

p_{f} (v) = \frac{1}{\sqrt{2 π}} \frac{A_{t}}{C_{t}} \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \exp [- \frac{m}{2} δ_{f t}^{2} v^{2}],

p_{b} (v) = \frac{1}{\sqrt{2 π}} \frac{A_{t}}{C_{t}} \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \exp [- \frac{m}{2} δ_{b t}^{2} v^{2}],

and hence

p (v) = \frac{1}{\sqrt{2 π}} \frac{A_{t}}{C_{t}} \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) [\exp (- \frac{m}{2} δ_{b t}^{2} v^{2}) - \exp (- \frac{m}{2} δ_{f t}^{2} v^{2})] .

The normalization

C_{t} = \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}),

is used in Eqs. (7)–(9) in order to set the maximum height of the distribution independent of the upper summation index M.

In order for function p to be non-negative for any value of v we set

δ_{b t} < δ_{f t} .

In addition, by setting

A_{t} = {(\frac{1}{δ_{b t}} - \frac{1}{δ_{f t}})}^{- 1},

we confirm that the source degree of coherence which is just the Fourier transform of p, having form

γ^{(0)} (t_{1}, t_{2}) = \frac{A_{t}}{C_{t}} \sum_{m = 1}^{M} \frac{{(- 1)}^{m - 1}}{\sqrt{m}} (\begin{matrix} M \\ m \end{matrix}) {\frac{1}{δ_{b t}} \exp [- \frac{1}{2 m} \frac{{(t_{2} - t_{1})}^{2}}{δ_{b t}^{2}}] - \frac{1}{δ_{f t}} \exp [- \frac{1}{2 m} \frac{{(t_{2} - t_{1})}^{2}}{δ_{f t}^{2}}]},

takes on value 1 for the coinciding time instants.

Figures 1(a) and 1(b) shows the variation of the degree of coherence on the time lag $t_{2} - t_{1}$ for several selected values of upper summation index M and of the r.m.s. source correlation widths $δ_{b t}$ and $δ_{f t}$ . As we will see from the next section such form of the degree of coherence leads to the split of the pulse ensemble average on propagation in the dispersive medium into two replicas both having flat intensity profiles.

Fig. 1 The temporal degree of coherence of random optical pulses at the source plane for several different sources: (a) for two parts in the far field, $δ_{b t} = 1 p s, δ_{f t} = 2 p s$ , (b) for two parts in the far field, $δ_{b t} = 1 p s, δ_{f t} = 10 p s$ , (c) for four parts in the far field, $δ_{1 f t} = 10 p s, δ_{1 b t} = 1.5 p s, δ_{2 f t} = 0.8 p s, δ_{2 b t} = 0.4 p s$ , (d) for six parts in the far field, $δ_{1 f t} = 10 p s,$ $δ_{1 b t} = 5 p s, δ_{2 f t} = 2 p s, δ_{2 b t} = 1.2 p s, δ_{3 f t} = 0.4 p s, δ_{3 b t} = 0.3 p s$ . Several curves correspond to different values of M. Other pulse and media parameters are $σ_{t 0} = 15 p s$ , $β_{2} = 50 p s^{2} / k m$ .

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3. Propagation of the modulated pulse

The propagation law of the mutual coherence function in the second-order dispersive medium has the form [24]

\begin{array}{l} Γ (t_{1}, t_{2}, z) = \frac{1}{2 π β_{2} z} \iint Γ^{(0)} (t_{10}, t_{20}) \\ \times \exp {\frac{i}{2 β_{2} z} [(t_{10}^{2} - t_{20}^{2}) - 2 (t_{10} t_{1} - t_{20} t_{2}) + (t_{1}^{2} - t_{2}^{2})]} d t_{10} d t_{20}, \end{array}

β_{2}

being the group velocity dispersion coefficient. On substituting from Eqs. (4) and (13) into Eq. (14) and performing integration we find that

Γ (t_{1}, t_{2}, z) = \frac{A_{t}}{C_{t}} \sum_{m = 1}^{M} \frac{{(- 1)}^{m - 1}}{\sqrt{m}} (\begin{matrix} M \\ m \end{matrix}) (U_{b} - U_{f}),

where

U_{α} = \frac{1}{δ_{α t} Δ_{α m} (z)} \exp [- \frac{{(t_{2} + t_{1})}^{2}}{8 σ_{t 0}^{2} Δ_{α m}^{2} (z)}] \exp [- \frac{{(t_{2} - t_{1})}^{2}}{2 Q_{α m}^{2} Δ_{α m}^{2} (z)}] \exp [i \frac{(t_{2}^{2} - t_{1}^{2})}{2 β_{2} R_{α m} (z)}],

and

\begin{array}{l} Q_{α m}^{2} = {(\frac{1}{4 σ_{t 0}^{2}} + \frac{1}{m δ_{α t}^{2}})}^{- 1}, Δ_{α m}^{2} (z) = 1 + \frac{β_{2}^{2} z^{2}}{σ_{t 0}^{2} Q_{α m}^{2}}, \\ R_{α m} (z) = z (1 + \frac{σ_{t 0}^{2} Q_{α m}^{2}}{β_{2}^{2} z^{2}}), (α = b, f) . \end{array}

From Eqs. (16) and (17) the average intensity I of the pulse ensemble can be determined at the coinciding time instants:

\begin{array}{l} I (t, z) = Γ (t, t, z) = \frac{A_{t}}{C_{t}} \sum_{m = 1}^{M} \frac{{(- 1)}^{m - 1}}{\sqrt{m}} (\begin{matrix} M \\ m \end{matrix}) \\ \times {\frac{1}{δ_{b t} Δ_{b m} (z)} \exp [- \frac{t^{2}}{2 σ_{t 0}^{2} Δ_{b m}^{2} (z)}] - \frac{1}{δ_{f t} Δ_{f m} (z)} \exp [- \frac{t^{2}}{2 σ_{t 0}^{2} Δ_{f m}^{2} (z)}]} . \end{array}

In addition, the temporal degree of coherence of the propagating ensemble can be found from normalization

γ (t_{1}, t_{2}, z) = \frac{Γ (t_{1}, t_{2}, z)}{\sqrt{I (t_{1}, z) I (t_{2}, z)}} .

For $t_{d} = - 2 t_{1} = 2 t_{2}$ , $t_{2} - t_{1} = t_{d}$ one finds that

γ (t_{d}, - t_{d}, z) = \frac{\sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) {\frac{1}{δ_{b t} Δ_{b m} (z)} \exp [- \frac{t_{d}^{2}}{2 Q_{b m}^{2} Δ_{b m}^{2} (z)}] - \frac{1}{δ_{f t} Δ_{f m} (z)} \exp [- \frac{t_{d}^{2}}{2 Q_{f m}^{2} Δ_{f m}^{2} (z)}]}}{\sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) {\frac{1}{δ_{b t} Δ_{b m} (z)} \exp [- \frac{t_{d}^{2}}{8 σ_{t 0}^{2} Δ_{b m}^{2} (z)}] - \frac{1}{δ_{f t} Δ_{f m} (z)} \exp [- \frac{t_{d}^{2}}{8 σ_{t 0}^{2} Δ_{f m}^{2} (z)}]}} .

Figure 2 shows the average intensity of the ensemble of pulse realizations being split into two replicas. The duration of the replicas and the gap between them are controlled by the parameters $δ_{b t}$ and $δ_{f t}$ , while the intensity profile flatness is determined by the value of M. In Fig. 2(d) the profile is adjusted in such a way that pulse intensity averages are flat and separated by a time interval smaller than their average durations.

Fig. 2 Normalized intensity distribution $I (t, z)$ of random optical pulse ensembles (split to two parts at far field) versus distance z and time t for different values of parameters: (a), (c) for $δ_{f t} = 2 p s$ ; (b), (d) for $δ_{f t} = 10 p s$ ; (a), (b) for $M = 1$ ; (c), (d) for $M = 30$ . The parameter $δ_{b t} = 1 p s$ and the other parameters as in Fig. 1.

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4. Generalization to larger number of pulse modulations

The model of the previous section can be readily extended to the case when generation of two replicas of pulse trains having N modulations each is needed. We now set the temporal degree of coherence of the source to be a linear combination of those in Eq. (13), i.e.

γ^{(0)} (t_{1}, t_{2}) = B \sum_{n = 1}^{N} A_{n t} γ_{n}^{(0)} (t_{1}, t_{2}), B = {(\sum_{n = 1}^{N} A_{n t})}^{- 1},

where B is used for setting the maximum value of the degree to one and the weights A_nt are defined the same way as A_t in Eq. (12).

Figures 1(c) and 1(d) show the degree of coherence calculated from Eq. (21) for the cases of 4 and 6 pulses in the sequence, respectively.

If the initial average intensity of the pulse train is the same for all members, and, for instance, is taken as in Eq. (4) then each mutual coherence function in the superposition becomes

Γ_{n}^{(0)} (t_{10}, t_{20}) = \exp (- \frac{t_{1}^{2} + t_{2}^{2}}{4 σ_{t 0}^{2}}) γ_{n} (t_{1}, t_{2}) .

Hence, the propagation law, generalized to n-th term:

\begin{array}{l} Γ_{n} (t_{1}, t_{2}, z) = \frac{1}{2 π β_{2} z} \iint Γ_{n}^{(0)} (t_{10}, t_{20}) \\ \times \exp {\frac{i}{2 β_{2} z} [(t_{10}^{2} - t_{20}^{2}) - (t_{10} t_{1} - t_{20} t_{2}) + (t_{1}^{2} - t_{2}^{2})]} d t_{10} d t_{20}, \end{array}

yields the result

Γ_{n} (t_{1}, t_{2}, z) = \frac{A_{t}}{C_{t}} \sum_{m = 1}^{M} \frac{{(- 1)}^{m - 1}}{\sqrt{m}} (\begin{matrix} M \\ m \end{matrix}) (U_{n b m} - U_{n f m}),

where

U_{n α m} = \frac{1}{δ_{n α t} Δ_{n α m} (z)} \exp [- \frac{{(t_{2} + t_{1})}^{2}}{8 σ_{t 0}^{2} Δ_{n α m}^{2} (z)}] \exp [- \frac{Δ t^{2}}{2 Q_{n α m}^{2} Δ_{n α m}^{2} (z)}] \exp [i \frac{(t_{2}^{2} - t_{1}^{2})}{2 β_{2} R_{n α m} (z)}],

and

\begin{array}{l} Q_{n α m}^{2} = {(\frac{1}{4 σ_{t 0}^{2}} + \frac{1}{m δ_{n α t}^{2}})}^{- 1}, Δ_{n α m}^{2} (z) = 1 + \frac{β_{2}^{2} z^{2}}{σ_{t 0}^{2} Q_{n α m}^{2}}, \\ R_{n α m} (z) = z (1 + \frac{σ_{t 0}^{2} Q_{n α m}^{2}}{β_{2}^{2} z^{2}}), (α = b, f) . \end{array}

On summing up the results for N terms and calculating the result at the coinciding time arguments we find that the intensity of the ensemble average of pulses becomes

I = B \sum_{n = 1}^{N} a_{n} I_{n},

where

\begin{array}{l} I_{n} (t, z) = Γ_{n} (t, t, z) = \frac{A_{n}_{t}}{C_{n}_{t}} \sum_{m = 1}^{M} \frac{{(- 1)}^{m - 1}}{\sqrt{m}} (\begin{matrix} M \\ m \end{matrix}) \\ \times {\frac{1}{δ_{n b t} Δ_{n b m} (z)} \exp [- \frac{t^{2}}{2 σ_{t 0}^{2} Δ_{n b m}^{2} (z)}] - \frac{1}{δ_{n f t} Δ_{n f m} (z)} \exp [- \frac{t^{2}}{2 σ_{t 0}^{2} Δ_{n f m}^{2} (z)}]}, \end{array}

and

A_{n t} = {(\frac{1}{δ_{n b t}} - \frac{1}{δ_{n f t}})}^{- 1}, δ_{n b t} < δ_{n f t} .

Figure 3 illustrates the average intensity of the random pulse ensemble generated by the source with the degree of coherence in Figs. 1(c) and 1(d) that splits into two replicas each including two parts, i.e. four parts in total.

Fig. 3 Normalized average intensity distribution $I (t, z)$ of random optical pulse ensemble split to four parts versus distance z and time t. The source parameters are as in Figs. 1(c) and (d), respectively.

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Figure 4 shows the dependence of the gap between the adjacent pulses on the deference $δ_{(n - 1) b t} - δ_{n f t}$ that can be termed as the gap parameter. Depending on its value the gap between the adjacent parts can be made wider or narrower, as in Figs. 4(a) and 4(b), or can be eliminated, as in Fig. 4(c). If its value becomes negative then the gap transforms into the intensity spike, see Fig. 4(d).

Fig. 4 Normalized average intensity distribution $I (t, z)$ of random optical pulse ensemble (split to four parts in the far field) versus time t for different values of pulse coherent gap ( $δ_{1 b t} - δ_{2 f t}$ ): (a) $δ_{2 f t} = 0.8 p s$ , (b) $δ_{2 f t} = 1.2 p s$ , (c) $δ_{2 f t} = 1.5 p s$ , (d) $δ_{2 f t} = 1.8 p s$ . The parameter $δ_{1 b t} = 1.5 p s$ and the other parameters are as in Fig. 1(c).

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In Fig. 5 we illustrate the possibility of modulating the relative heights of the individual flat parts in the sequence.It can be done by adjusting the value of $a_{n} A_{n t} / (a_{1} A_{1 t})$ that can be viewed as the weight parameter. The inner parts can be made much taller [see Fig. 5(a)], slightly taller [see Fig. 5(b)], the same [see Fig. 5(c)] and shorter [see Fig. 5(d)] than the outer parts.

Fig. 5 Normalized average intensity distribution $I (t, z)$ of random optical pulse ensemble (split to four parts in the far field) versus time t for different values of pulse weight parameter $a_{2} A_{2 t} / (a_{1} A_{1 t})$ : (a) $0.5$ , (b) $0.8$ , (c) $1.0$ , (d) $1.2$ , The parameter $δ_{1 b t} = δ_{2 f t} = 1.5 p s$ and the other parameters are as in Fig. 1(c).

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Figure 6 presents the possible average intensity distributions in which the single pulse average distribution splits into two replicas, each having three parts. In Fig. 6(a) the sequence of the well-separated in time modulated parts is shown while Fig. 6(b) provides an example with adjacent boundaries between the modulated parts.

Fig. 6 Normalized average intensity distribution $I (t, z)$ of random optical pulse ensemble (split to six parts at far field) versus time t for different values of gap parameter: (a) with gaps $δ_{1 b t} = 5 p s,$ $δ_{2 f t} = 2 p s,$ $δ_{2 b t} = 1.2 p s,$ $δ_{3 f t} = 0.4 p s .$ (b) without gaps $δ_{1 b t} = δ_{2 f t} = 2 p s,$ $δ_{2 b t} = δ_{3 f t} = 0.7 p s$ . The values of the pulse weight parameter are $a_{2} A_{2 t} / (a_{1} A_{1 t}) = 1.2$ , $a_{3} A_{3 t} / (a_{1} A_{1 t}) = 0.8$ and other parameters as in Fig. 1(c).

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5. Conclusion

With the help of the linear superposition of several mutual coherence functions having the same average intensity profile, say Gaussian, but different Multi-Gaussian profiles for the degree of coherence we have illustrated the possibility of producing temporal modulation of the intensity of the pulse ensemble after its propagation at a sufficiently long distance in the dispersive medium. In particular, depending on the number, the weights and the r.m.s. correlation widths of the individual source degrees of coherence in the superposition it is possible to split the initial pulse intensity average into two replicas, appearing before and after the temporal pulse center, each replica to have N Gaussian-like or flat parts with individual heights and widths. The gaps between the parts in the sequence can also be controlled at will, eliminated or converted into spikes.

The application of the novel family of random pulses is envisioned in the domain of pulse position modulation for robust information transfer. The illustrated modulation method resulting in two or three distinctive pulses can be readily generalized to that for eight parts (or its multiple) if encoding of the digital information by a typical signal carrier is required.

Acknowledgments

This work is supported by the US ONR (N00189-14-T-0120), the US AFOSR (FA9550-12-1-0449), the National Natural Science Foundation of China under Grant Nos. 11304287, 61275150, and 61108090, special fund for high level project in Zhejiang International Studies University (090500452012), and the Program for Science & Technology Innovation Talents in Universities of Henan Province (13HASTIT048).

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Position modulation with random pulses

Abstract

1. Introduction

2. Source model for the simplest random pulse case: two flat average pulses

3. Propagation of the modulated pulse

4. Generalization to larger number of pulse modulations

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Equations (29)

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