About the Gaussian-Schell pulse train spectrum

B. Lacaze

doi:10.1364/OE.15.002803

1. The Gaussian-Shell pulse model

From [11] or [12], the Gaussian-Schell (G-S) pulse is defined as a complex zero-mean random process A={A(t),t ∈ ℝ} with correlation function

E [A (t) A^{*} (t^{'})] = \frac{a}{2 π σ^{' 2}} \exp [i ω_{0} (t - t') - \frac{{(t - m)}^{2}}{2 σ^{' 2}} - \frac{{(t' - m)}^{2}}{2 σ^{' 2}} - \frac{{(t - t')}^{2}}{{2 ρ}^{' 2}}]

where E [..] denotes mathematical expectation and the asterisk denotes the complex conjugate. Consequently, A expresses as follows:

A (t) = B (t) u (t)

where the deterministic function u(t) and the zero-mean process B = {B(t), t ∈ ℝ} are such that:

{\begin{array}{l} u (t) = \frac{1}{σ' \sqrt{2 π}} \exp [\frac{i ω_{0} t - {(t - m)}^{2}}{{2 σ}^{' 2}}] \\ E [B (t) B^{*} (t')] = a \exp [\frac{- {(t - t')}^{2}}{{2 ρ}^{' 2}}] \end{array}

Then, u(t) is a normalized Gaussian with mean m and standard deviation σ´, modulated by a monochromatic wave at the frequency ω ₀/2π. The process B is wide sense stationary. The autocorrelation function of B is Gaussian with correlation radius ρ'.

The bidimensional Fourier transform ℱ [..] of K _A(t,t´) = E[A(t)A ^*(t´)] is defined by:

ℱ [K_{A}] (ω, ω') = \iint_{ℝ^{2}} E [A (t) A^{*} (t')] e^{iωt - iω' t'} dtdt'

Classical derivations (developed in Appendix 1) lead to:

ℱ [K_{A}] (ω, ω') = \frac{aσ}{σ'} \exp [im (ω - ω') - \frac{σ^{2}}{2} ({(ω - ω_{0})}^{2} + {(ω' - ω_{0})}^{2}) - \frac{ρ^{2}}{2} {(ω - ω')}^{2}]

Eq. (5) is similar to the coherence function of a G-S beam. Consequently, the process A={A(t),t ∈ ℝ} is referred to as a G-S pulse. The parameters of K _A(t,t´) and of ℱ[K _A](ω), ω´) are related by

{\begin{array}{l} σ' = \sqrt{σ^{2} + 2 ρ^{2}} ρ' = \frac{σ}{ρ} \sqrt{σ^{2} + {2 ρ}^{2}} \\ σ = \frac{σ' ρ'}{\sqrt{{2 σ}^{' 2} + ρ^{' 2}}} ρ = \frac{σ^{' 2}}{\sqrt{{2 σ}^{' 2} + ρ^{' 2}}} \end{array}

In practical applications, a large number of G-S pulses is emitted. Let decompose the time into periods of duration T. If damages like jitter or width modulation can be neglected, each period contains one pulse at a given location Ref. [15]. Then, a G-S pulse sequence defines a continuous time process, referred to as a G-S pulse train. Generally, the duration of the pulse is very small with respect to the period T. Obviously, the process is not stationary, but exhibits a cyclo-stationary character Ref. [5], [1], [7]. Consequently, a one-dimensional power spectrum can be defined for this process.

This power spectrum is the power spectrum of one of the process stationarized versions. This paper mainly provides the power spectrum of G-S trains. Moreover, the properties of this power spectrum are studied as functions of the correlation between successive G-S pulses.

2. The Gaussian-Schell pulse train

2.1. Definition

A G-S pulse train is an infinite sequence of G-S pulses A _n,n ∈ ℤ. Each pulse location is given in the interval (nT,(n+1)T). Classically in signal processing, the resulting process Z = {Z(t),t ∈ ℝ} can be written as:

Z (t) = \sum_{n = - \infty}^{\infty} A_{n} (t - n T) .

In the case of optical lasers, the supports of the successive pulses are disjoined. For example, the pulse width, related to the parameter σ´, is in the range of ten picoseconds whereas T is in the range of ten nanoseconds. Consequently, at most one term differs from zero in the preceding infinite sum. Thus a more tractable expression is:

Z (t) = A_{\bar{t}} (\underline{t}), t = \bar{t} T + \underline{t}, \bar{t} \in ℤ, 0 \leq \underline{t} < T

where t̅ is the index of the period and t̲ = t - t̅T is the location inside the period. When T = 1,t̅ and t̲ are respectively the entire and fractional parts of t. Let consider that T = 1, without loss of generality.

Each process A _n is assumed to be a G-S pulse i.e

{\begin{array}{l} A_{n} (t) = B_{n} (t) u (t) \\ u (t) = \frac{1}{σ' \sqrt{2 π}} \exp [\frac{i ω_{0} t - {(t - m)}^{2}}{2 σ^{' 2}}] \\ E [B_{n} (t) B_{n}^{*} (t')] = a \exp [\frac{- {(t - t')}^{2}}{{2 ρ}^{' 2}}] \end{array}

where each B _n and u(t) verify (3). The B _n give a random character to the pulses. These processes can be dependent or not. For instance, slow drifts in a laser supply act on the pulse energy and relate the amplitudes of the successive pulses. However, the pulse variations may result from independent causes. In both cases, let assume that the following four conditions P ₁ to P ₄ hold:

{\begin{array}{l} P_{1} : A_{n} (t) = 0, t \notin (0, 1) \\ P_{2} : E [A_{n} (t)] = 0, t∈ (0, 1) \\ P_{3} : E [A_{m} (t) A_{m - n}^{*} (t')] = α_{n} (t, t') is independent of m \\ P_{4} : {lim}_{n \to \infty} α_{n} (t, t') = 0 with uniform convergence in (t, t') \end{array}

P ₁ assumes that the pulses are isolated. This approximation is justified by the very small value of σ´ with respect to the repetition period. P ₂ is part of the G-S pulse definition. The properties P ₃ and P ₄ relate to the dependence between pulses A _n. Both properties hold if the B _n’s verify

{\begin{array}{l} P_{3}^{'} : E [B_{m} (t) B_{m - n}^{*} (t')] = β_{n} (t, t') is a quantity independent of m \\ P_{4}^{'} : {lim}_{n \to \infty} β_{n} (t, t') = 0 with uniform convergence in (t, t') \end{array}

Note that these properties hold for the most commonly used telecommunication models Ref.[2], [7], [16].

2.2. The power spectrum in the uncorrelated case

Obviously, the process Z defined by (7) and (8) with properties P ₁ to P ₄ is not stationary, but rather cyclo-stationary. Consequently:

E [Z (t) Z^{*} (t')] = E [Z (t + 1) Z^{*} (t' + 1)]

The power spectrum s _Z(ω) of Z(t) can be defined by randomization of the origin time Ref. [7], [1]. Appendix 2 shows that:

s_{Z} (ω) = \frac{1}{2 π} \sum_{n = - \infty}^{\infty} [\iint_{{(0, 1)}^{2}} α_{n} (t, t') e^{i ω (t' - t)} dtdt'] e^{- i nω}

where the α_n(t,t´) are defined in (9). When Z is constructed from a G-S pulse train, equation (8) leads to

α_{0} (t, t') = E [A_{n} (t) A_{n}^{*} (t')] = a u (t) u^{*} (t') \exp [- \frac{{(t - t')}^{2}}{2 {ρ'}^{2}}]

The values of α_n(u,v), for n ≠ 0, depend on the pulse generator through $E [B_{m} (t) B_{m - n}^{*} (t')]$ . If these terms can be neglected, i.e if the processes B _n are assumed uncorrelated, s_z(ω) is reduced to (from Eq.(5), (6) and (10))

s_{Z} (ω) = \frac{aσ}{2 πσ'} \exp [- σ^{2} {(ω - ω_{0})}^{2}]

The spectrum is Gaussian with width $\frac{1}{σ \sqrt{2}} = \sqrt{\frac{1}{{2 σ}^{' 2}} + \frac{1}{ρ^{' 2}}}$ . This property is well-known, see for instance Ref [9], [10]. The parameter ρ´ defines the pulse autocorrelation, and the width $\frac{1}{σ \sqrt{2}}$ is a decreasing function of ρ´. When the pulse is deterministic (ρ´ = ∞), the spectral width is 1/σ´√2 . At the opposite (for small ρ´), the spectrum flattens out. The width is multiplied by 2 when ρ´ decreases from ρ´ = ∞ to ρ´ = σ´√3. The width is almost 1/ρ´ when ρ´ ≪ σ´. In this case, the power spectrum is independent of the pulse width σ´, and Z and B have spectral widths in the same range.

Furthermore, a unit power process is obtained when the constant a verifies:

\int_{- \infty}^{\infty} s_{Z} (ω) dω = 1 \Rightarrow a = 2 σ' \sqrt{π} .

In practical situations, ω ₀ = 10¹⁵s^-1 for a He-Ne laser, σ´ = 10^-11s, with a repetition frequency at f ₁ =10⁸s^-1 (note that shorter pulses of few 10^-14s can be emitted, see for example [13]). For normalized clock period, the previous quantities reduce to ω ₀ = 10⁷, σ; ₀ = 10^-3,f ₁ = 1.

For a clock period T and for the true values of the parameters (ω ₀,ρ´,σ ₀, Eq. (13) and (14) become respectively:

{\begin{array}{l} s_{Z} (ω) = \frac{aσ}{2 πσ' T} \exp [- σ^{2} {(ω - ω_{0})}^{2}] \\ \int_{- \infty}^{\infty} s_{Z} (ω) dω = 1 \Rightarrow a = 2 σ' T \sqrt{π} \end{array}

These formulas show that the bandwidth of Z is independent of the repetition rate 1/T. On the contrary, the mean energy depends on 1/T.

2.3. The power spectrum in the dependent case

The simplest correlation model for the processes B _n is, for T = 1:

B_{n} (t) = C (t + n), t \in [0, 1 [, n \in ℤ

where the underlying process C = {C(t),t ∈ ∝} is stationary Ref.[1]. The associated spectrum is given in Appendix 2. Note that the spectrum of the pulse train is no longer Gaussian. However, it is easy to show that such a model cannot explain a strong correlation between pulses combined with quick random variations inside pulses. A strong correlation between pulses A _n and A _n´ implies slow variations of the processes B _n and B _n´. These processes can then be assumed independent of t but different for n ≠ n´. Conversely, fast variations of the A _n (small ρ´) imply weak correlation between pulses. It is not the case in the following model.

For each n ∈ ℤ, let define a zero mean uncorrelated process C _n = {C _n(t),t ∈ ℝ} by:

{\begin{array}{l} E [C_{n} (t) C_{n}^{*} (t')] = \frac{a}{2} \exp [\frac{{- (t - t')}^{2}}{{2 ρ'}^{2}}] \\ B_{n} (t) = C_{n} (t) + C_{n - 1} (t) \end{array}

Then, B _n is like a moving average process of order 1 (MA₁) Ref. [6]. In this situation, the pulse A _n relates to the nearest pulses A _n-1 and A _n+1 but A _n is uncorrelated with the other A _n-m,m ≠ 0, ±1. Furthermore, the G-S pulse A _n verifies (8). Following the notations of (10)

{\begin{array}{l} β_{0} (t, t') = a \exp [\frac{{- (t - t')}^{2}}{{2 ρ'}^{2}}] \\ β_{\pm 1} (t, t') = \frac{a}{2} \exp [\frac{{- (t - t')}^{2}}{{2 ρ'}^{2}}] \\ β_{n} (t, t') = 0, n \neq 0, \pm 1 . \end{array}

The Appendix 2 allows to derive the power spectral density s_Z(ω) of the pulse train. In the case T = 1, we obtain:

s_{Z} (ω) = \frac{aσ}{2 πσ'} (1 + \cos ω) \exp [{- σ}^{2} {(ω - ω_{0})}^{2}]

Obviously, the spectrum is no longer Gaussian, but is a strong unit period modulation of a Gaussian. Furthermore, generalizing Eq. (17) to

B_{n} (t) = \sum_{k} b_{k} C_{n - k} (t)

when Σ_k∣b_k∣² = 2, the G-S pulses A _n verify Eq. (8), and the train of pulses Z is such as

{\begin{array}{l} s_{Z (ω) = P (ω) \exp [{- σ}^{2} {(ω - ω_{0})}^{2}]} \\ P (ω) = \frac{aσ}{4 π σ^{'}} {∣ \sum_{k} b_{k} e^{- ikω} ∣}^{2} \end{array}

P(ω) is constructed from any trigonometrical polynomial or from the associated limit. It is a periodic function by construction, with unit period corresponding to the laser repetition period. The internal coherence of the pulse (defined by β₀(t,t′)) is unchanged whatever P(ω), which takes into account the correlations between the pulses. Generally, 1/σ is large with respect to 1, and then a large number of periods of P(ω) will appear in s_Z(ω). Obviously, formulas for B _n different from Eq. (18) define other power spectra.

3. Conclusion

The frontier between stationary and non-stationary processes is not clear, when the time range is large enough. For instance, the monochromatic wave exp[i(ω ₀ t + ϕ)] is non-stationary if its phase ϕ is given, but stationary if ϕ is considered as a uniformly distributed random variable on the interval (0,2π/ω ₀). Measures generally involve time averaging. Consequently, the same results are obtained whatever the model. Therefore, the model choice is not critical. However, it is often easier to perform computations in the stationary than in the non-stationary context. It is the case in the context of G-S pulse trains, where the cyclo-stationary character is obvious. The measured spectrum is thus confused with the spectrum of any stationarized version. We have proved that correlation between pulses removes the pure Gaussian character of the spectrum. Furthermore, the correlation between pulses can be constructed independently of the internal coherence of the pulses. Finally, other models than the proposed moving average one would lead to other spectral shapes.

4. Appendices

4.1. Appendix 1

The characteristic function of the multi-dimensional Gaussian law verifies Ref. [4]:

\int_{ℝ^{n}} \frac{\sqrt{\det [Ω]}}{{(2 π)}^{\frac{n}{2}}} \exp [- \frac{1}{2} {(ω - ω_{0})}^{t} Ω (ω - ω_{0}) + {i u}^{t} ω] dω = \exp [{i u}^{t} ω_{0} - \frac{1}{2} u^{t} Ω^{- 1} u]

where ω, ω ₀, u are column matrices, x ^t the transpose of x, and Ω a symmetric strictly definite positive matrix. Particularly, we have n = 2 in Eq. (4),(5), and the quadratic forms are respectively defined by

{\begin{array}{l} σ^{2} ({(ω - ω_{0})}^{2} + {(ω' - ω_{0})}^{2}) + ρ^{2} {(ω - ω')}^{2} \\ \frac{1}{{σ'}^{2}} ({(t - m)}^{2} + {(t' - m)}^{2}) + \frac{1}{{ρ'}^{2}} {(t - t')}^{2} \end{array}

Then, with

Ω = [\begin{array}{l} σ^{2} + ρ^{2} & {- ρ}^{2} \\ {- ρ}^{2} & σ^{2} + ρ^{2} \end{array}]; Ω^{- 1} = \frac{1}{σ^{2 (σ^{2} + {2 ρ}^{2})}} [\begin{array}{l} σ^{2} + ρ^{2} & ρ^{2} \\ ρ^{2} & σ^{2} + ρ^{2} \end{array}]

The relations (6) also link (σ′,ρ′) with (σ,ρ).

4.2. Appendix 2

1) Let consider the process Z fulfilling the properties P ₁ to P ₄ of (9). First, E[Z(t)] = 0, whatever t. Second, from Eq. (7)
$E [Z (t) Z^{*} (t - τ)] = E [A_{\bar{t}} (\underset{—}{t}) A_{\bar{t - τ}}^{*} (\underline{t - τ})] = α_{\bar{t} - \bar{t - τ}} (\underset{—}{t}, \underline{t - τ})$
where t̄-t-τ̅ are the entire and fractional parts of t. Obviously, t̄-t-τ̅ is periodic in t, like t and t-τ;̲, with unit period. Thus, E[Z(t)Z ^*(t - τ)] has also the property, leading to a cyclo-stationary process Z. If Φ is a random variable independent of Z, with uniform distribution in (0,1), the new process U defined by
$U (t) = Z (t - Φ)$
is a stationarized version of Z. Both processes have the same power spectrum, because a time origin change has no influence on time averages. However, derivations are easier with the sta-tionarized version. Then, using conditional mathematical expectations
$E [U (t) U^{*} (t - τ)] = \int_{0}^{1} α_{\bar{t - ϕ} - \overset{}{\bar{t - τ - ϕ}}} (\underline{t - ϕ}, \underline{t - τ - ϕ}) d ϕ$
The integral on (0,1) is equal to the integral on (a,a+1) whatever a, because of the periodicity in ϕ of the integrand, and then
$E [U (t) U^{*} (t - τ)] = \int_{0}^{1} α_{- \bar{u - τ}} (u, \underline{u - τ}) du$
is a quantity independent oft. The process U is now stationary. The B _n are zero-mean processes, and the dependence between B ₀ and B _n is assumed negligible for large n. These hypotheses imply a spectral density s_Z (ω)) for U and Z defined by the inverse Fourier transform Ref. [3], [14]
$s_{Z} (ω) = \frac{1}{2 π} \int_{- \infty}^{\infty} [\int_{0}^{1} α_{- \bar{u - τ}} (u, \underline{u - τ}) du] e^{- iωτ} dτ$
This integral can be written as
$s_{Z} (ω) = \frac{1}{2 π} \sum_{n = - \infty}^{\infty} e^{- inω} [\iint_{{(0,1)}^{2}} α_{n} (t, t') e^{iω (t' - t)} dtdt']$
after a linear variable change (u = t,u-τ=t′) and a decomposition of ℝ in unit length intervals. Let consider that the functions α_n (t,t′) are negligible outside (0,1)². Consequently, the set (0,1)² in (20) can be replaced by the set (-∞,∞)² , which allows to use the bivariate Gaussian law formulae (see Appendix 1).
Note that, whatever the repetition frequency 1/T of the pulse generator, the process has no spectral lines. This property is mainly related to the mean value of the random variable A_n(t), which is assumed equal to 0, whatever t.
2) In the case where the B _n come from a unique B like in Eq. (16), we have:
$α_{n} (t, t') = au (t) u^{*} (t^{'}) \exp [- \frac{1}{{2 ρ}^{' 2}} {(t - t' + n)}^{2}]$
Equation (20) and the Appendix 1 leads to
$s_{Z} (ω) = \frac{aσ}{2 π σ^{'}} e^{- σ^{2} {(ω - ω_{0})}^{2}} [1 + 2 \sum_{n = 1}^{\infty} \exp [\frac{- n^{2}}{2 ({2 σ}^{' 2} + ρ^{' 2})}] \cos [\frac{2 n (ω - ω_{0})}{2 + (\frac{ρ'}{σ'})^{2}} - nω]]$
which can be transformed by the Poisson summation formula.
3) When the B _n are defined by (17), (18), we have (the C _n are uncorrelated)
$β_{n} (t, t') = [\sum_{k} b_{k} b_{k - n}^{*}] E [C_{0} (t) C_{0}^{*} (t')]$
Holding in (20), and using the identity
$\sum_{n, k} b_{k} b_{k - n}^{*} e^{- inω} = {∣ \sum_{k} b_{k} e^{- ikω} ∣}^{2}$
leads to (19). The chirp case studied notably in Ref. [9] is not very different, because the problem is equivalent to find the Fourier transform of a quadratic form exponential (see Appendix 1).

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