Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Tutorials Issues in Progress Current Issue All Issues Early Posting Feature Issues

Pulse-quality metric for nonstationary partially coherent fields

Milo W. Hyde, Olga Korotkova, and Mark F. Spencer

Open Access Open Access

Abstract

This paper generalizes a pulse-quality metric referred to as ${P^2}$, i.e., the time analogue of Siegman’s beam quality factor ${M^2}$, to include pulsed (nonstationary) random fields of any state of coherence. The analysis begins with the derivation of a general ${P^2}$ relation, which we then specialize to the important cases of coherent and Schell-model pulsed beams. As examples, we derive the ${P^2}$ for two stochastic sources: (1) a cosine Gaussian-correlated Schell-model pulsed beam and (2) a nonuniformly correlated pulsed beam. For both of these sources, we generate (in simulation) random instances of each and compare the simulated (Monte Carlo) ${P^2}$, i.e., computed directly from its definition, to the theoretical quantity. The agreement is excellent, thereby validating our ${P^2}$ analysis.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. INTRODUCTION

The traditional metric for gauging the quality of a pulsed laser beam is the time-bandwidth product (TBP). The TBP is the product of the beam’s pulse and power spectrum diameters, where the diameter is defined as the full width at half maximum (FWHM). There are several practical shortcomings with this FWHM definition. For instance, it can easily underestimate the width of a multimode laser, where the spectrum might contain features that fall below the half-maximum threshold. In addition, the FWHM generally cannot be obtained directly from pulse-shape measurements (e.g., the intensity autocorrelation in the case of femtosecond pulses) without assuming forms for the magnitude and phase of the pulse. These and other weaknesses of the FWHM definition are summarized in [1–4].

Naturally, these drawbacks have led to other pulse-quality metrics, most notably the pulse quality factor ${P^2}$, which is the time analogue of Siegman’s beam quality factor ${M^2}$ [5]. Like ${M^2}$, ${P^2}$ (more detail below) is the product of the pulse’s root-mean-square (RMS) widths in the time and frequency domains. Consequently, ${P^2}$ should provide more accurate estimates of the pulse duration and bandwidth than the TBP for sources with complex pulse shapes and spectra. ${M^2}$ is the laser industry standard for assessing beam quality and, since its introduction, has been generalized to include all manner of beam types: hard-apertured [6,7], vortex [8], and stochastic [9–15]. Interestingly, a similar generalization of ${P^2}$ has not occurred. Indeed, there is only a handful of references that discuss ${P^2}$ [1–4,16,17]. All of these references focus on the ${P^2}$ of coherent light sources, which generally excludes multimode lasers or lasers that have been artificially or unintentionally broadened. These cases include white-noise or pseudo-random binary sequence broadening in the former [18] and spontaneous emission from optical amplifiers in the latter [19–21].

In this paper, we generalize the ${P^2}$ metric to include pulsed (nonstationary) random fields of any state of coherence. We begin with the theoretical derivation of ${P^2}$ for such fields. Our analysis follows that of ${M^2}$ presented in [9]. We specialize our general ${P^2}$ relation to coherent and Schell-model beams and derive the ${P^2}$ for two example random fields: a cosine Gaussian-correlated Schell model and nonuniformly correlated pulsed beam. We then generate (in simulation) both of these sources and compare the simulated ${P^2}$, i.e., computed directly from the ${P^2}$ definition, to the theoretical quantity to validate our work. We conclude with a brief summary and discussions of future work and applications.

2. THEORY

We begin with our expression for the pulse quality factor ${P^2}$, such that

$${P^2} = 2{\sigma _t}{\sigma _\omega},$$
where ${\sigma _t}$ and ${\sigma _\omega}$ are the normalized second central moments, or RMS widths of the time-domain pulse intensity and frequency-domain spectral density, respectively. They are mathematically defined as
$$\begin{split}{\sigma _t^2}&= {\frac{{\int_{- \infty}^\infty {{\left({t - \langle t\rangle} \right)}^2}\Gamma \!\left({t,t} \right){\rm d}t}}{{\int_{- \infty}^\infty \Gamma \!\left({t,t} \right){\rm d}t}}}\\{\sigma _\omega ^2}&={\frac{{\int_{- \infty}^\infty {{\left({\omega - \langle \omega \rangle} \right)}^2}W\!\left({\omega ,\omega} \right){\rm d}\omega}}{{\int_{- \infty}^\infty W\!\left({\omega ,\omega} \right){\rm d}\omega}}.}\end{split}$$
In Eq. (2), $\langle t\rangle$ and $\langle \omega \rangle$ are the normalized first moments of intensity and spectral density, i.e.,
$$\begin{split}{\langle t\rangle} &= {\frac{{\int_{- \infty}^\infty t\Gamma \!\left({t,t} \right){\rm d}t}}{{\int_{- \infty}^\infty \Gamma \!\left({t,t} \right){\rm d}t}}}\\{\langle \omega \rangle}&= {\frac{{\int_{- \infty}^\infty \omega W\!\left({\omega ,\omega} \right){\rm d}\omega}}{{\int_{- \infty}^\infty W\!\left({\omega ,\omega} \right){\rm d}\omega}}.}\end{split}$$
Last, $\Gamma$ and $W$ are the mutual coherence function (MCF) and cross-spectral density (CSD) function of the stochastic pulsed source. Consequently, $\Gamma ({t,t}) = \left\langle {I(t)} \right\rangle$ is the ensemble-averaged intensity and $W({\omega ,\omega}) = S(\omega)$ is the spectral density. Using Wolf’s convention [22], the MCF and CSD function are related via the Fourier transform
$$\begin{split}{W\!\left({{\omega _1},{\omega _2}} \right)} &={ \frac{1}{{{{\left({2\pi} \right)}^2}}} \iint_{- \infty}^\infty \Gamma \!\left({{t_1},{t_2}} \right)\exp \!\left[{{\rm j}\!\left({{\omega _1}{t_1} - {\omega _2}{t_2}} \right)} \right]{\rm d}{t_1}{\rm d}{t_2}},\\{\Gamma \!\left({{t_1},{t_2}} \right)}&={ \iint_{- \infty}^\infty W\!\left({{\omega _1},{\omega _2}} \right)\exp \!\left[{- {\rm j}\!\left({{\omega _1}{t_1} - {\omega _2}{t_2}} \right)} \right]{\rm d}{\omega _1}{\rm d}{\omega _2}.}\end{split}$$

Traditionally, the moments in ${P^2}$, like those in ${M^2}$, are defined in the waist plane [3,4,16]. This makes ${P^2}$ a single invariant number that completely describes ${\sigma _t}$ at any distance from the waist plane in a second-order-dominant dispersive medium. Here, we are interested in pulse quality in the transmit (or source) plane, including any potential chirp and, therefore, proceed by computing ${\sigma _t}$, ${\sigma _\omega}$, and ultimately ${P^2}$ directly in the source plane. As a consequence, our ${P^2}$ does not have the same physical meaning as it does in [3,4,16]; however, it is more consistent with the TBP, which is our goal here. Nevertheless, the analysis to follow is applicable in either case.

Following the approach taken in [9,10], we can reformulate ${\sigma _\omega}$ in terms of the MCF or ${\sigma _t}$ in terms of the CSD function. Here, we choose the former. Proofs for the following are shown in the appendix:

$$\langle \omega \rangle = \frac{{\rm j}}{J}\int_{- \infty}^\infty {\left. {\frac{{\partial \Gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|_{t,t}}{\rm d}t,$$
$$\langle {\omega ^2}\rangle = \frac{1}{J}\int_{- \infty}^\infty {\left. {\frac{{{\partial ^2}\Gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}\partial {t_2}}}} \right|_{t,t}}{\rm d}t,$$
where ${\rm j} = \sqrt {- 1}$, $J$ is
$$J = \int_{- \infty}^\infty \Gamma \!\left({t,t} \right){\rm d}t,$$
and ${|_{t,t}}$ denotes that ${t_1} = {t_2} = t$ after computing the partial derivative.

A. General Stochastic Sources

Let us start by writing $\Gamma$ as

$$\Gamma \!\left({{t_1},{t_2}} \right) = {\left[{\left\langle {I\!\left({{t_1}} \right)} \right\rangle \left\langle {I\!\left({{t_2}} \right)} \right\rangle} \right]^{1/2}}\gamma \!\left({{t_1},{t_2}} \right) = i\!\left({{t_1}} \right)i\!\left({{t_2}} \right)\gamma \!\left({{t_1},{t_2}} \right),$$
where $\gamma$ is the normalized temporal correlation function (i.e., $\gamma ({t,t}) = 1$), also known as the complex degree of coherence (CDoC). In addition, because the MCF is Hermitian [22], $\gamma ({{t_1},{t_2}}) = {\gamma ^*}({{t_2},{t_1}})$.

Substituting Eq. (8) into Eqs. (5)–(7) and evaluating the straightforward yet tedious derivatives produces

$$\langle \omega \rangle = \frac{{\rm j}}{J}\int_{- \infty}^\infty i\!\left(t \right)i^\prime \!\left(t \right){\rm d}t + \frac{{\rm j}}{J}\int_{- \infty}^\infty {\left[{i\!\left(t \right)} \right]^2}{\left. {\frac{{\partial \gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|_{t,t}}{\rm d}t,$$
$$\begin{split}{\langle {\omega ^2}\rangle}&={ \frac{1}{J}\int_{- \infty}^\infty {{\left[{i^\prime \!\left(t \right)} \right]}^2}{\rm d}t + \frac{1}{J}\int_{- \infty}^\infty {{\left[{i\!\left(t \right)} \right]}^2}{{\left. {\frac{{{\partial ^2}\gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}\partial {t_2}}}} \right|}_{t,t}}{\rm d}t}\\&\quad+ {\frac{2}{J}{\rm Re}\!\left[{\int_{- \infty}^\infty i\!\left(t \right)i^\prime \!\left(t \right){{\left. {\frac{{\partial \gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|}_{t,t}}{\rm d}t} \right]},\end{split}$$
$$J = \int_{- \infty}^\infty {\left[{i\!\left(t \right)} \right]^2}{\rm d}t,$$
where $i^\prime (t) = {\rm d}[{i(t)}]/{\rm d}t$. The first integral in Eq. (9) can be evaluated in closed form using substitution or integration by parts:
$$\int_{- \infty}^\infty i\!\left(t \right)i^\prime \!\left(t \right){\rm d}t = \frac{1}{2}\mathop {\lim}\limits_{t \to \infty} \left\{{{{\left[{i\!\left(t \right)} \right]}^2} - {{\left[{i\!\left({- t} \right)} \right]}^2}} \right\}.$$
We can safely assume that $i(t) \to 0$ as $t \to \pm \infty$ (as it would be for any physical pulsed beam); therefore, the integral is zero. In addition, to arrive at Eq. (10), we used
$${\left. {\frac{{\partial \gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|_{t,t}} = {\left[{{{\left. {\frac{{\partial \gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_2}}}} \right|}_{t,t}}} \right]^*},$$
which we prove in Appendix C.

We can make further progress by expressing $\gamma$ as

$$\gamma \!\left({{t_1},{t_2}} \right) = a\!\left({{t_1},{t_2}} \right)\exp \!\left[{{\rm j}\psi \!\left({{t_1},{t_2}} \right)} \right],$$
where $a({t,t}) = 1$, $\psi ({t,t}) = 0$, $a({{t_1},{t_2}}) = a({{t_2},{t_1}})$, and $\psi ({{t_1},{t_2}}) = - \psi ({{t_2},{t_1}})$ as a consequence of the CDoC being normalized and Hermitian. The above derivatives are
$$\begin{split}{{{\left. {\frac{{\partial \gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|}_{t,t}}} &={ {{\left. {\frac{{\partial a\!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|}_{t,t}} + {\rm j}{{\left. {\frac{{\partial \psi \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|}_{t,t}}}\\{{{\left. {\frac{{{\partial ^2}\gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}\partial {t_2}}}} \right|}_{t,t}}}&= {{{\left. {\frac{{{\partial ^2}a\!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}\partial {t_2}}}} \right|}_{t,t}} + {{\left[{{{\left. {\frac{{\partial \psi \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|}_{t,t}}} \right]}^2}} \\&\quad+ {{\rm j}{{\left. {\frac{{{\partial ^2}\psi \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}\partial {t_2}}}} \right|}_{t,t}}}.\end{split}$$
Inherent in these relations are the identities
$$\begin{split}{{{\left. {\frac{{\partial a\!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|}_{t,t}}}= {{{\left. {\frac{{\partial a\!\left({{t_1},{t_2}} \right)}}{{\partial {t_2}}}} \right|}_{t,t}}}\\[-4pt]{{{\left. {\frac{{\partial \psi \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|}_{t,t}}}= {- {{\left. {\frac{{\partial \psi \!\left({{t_1},{t_2}} \right)}}{{\partial {t_2}}}} \right|}_{t,t}},}\end{split}$$
which are easy to prove from the limit definition of the derivation and the symmetry relations of $a$ and $\psi$ or Eq. (13). Substituting Eq. (15) into Eqs. (9) and (10) produces
$$\begin{split}\langle \omega \rangle &= \frac{{\rm j}}{J}\int_{- \infty}^\infty {\left[{i\!\left(t \right)} \right]^2}{\left. {\frac{{\partial a\!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|_{t,t}}{\rm d}t\\[-4pt]&\quad - \frac{1}{J}\int_{- \infty}^\infty {\left[{i\!\left(t \right)} \right]^2}{\left. {\frac{{\partial \psi \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|_{t,t}}{\rm d}t,\end{split}$$
$$\begin{split}{\langle {\omega ^2}\rangle} &={\frac{1}{J}\int_{- \infty}^\infty {{\left[{i^\prime \!\left(t \right)} \right]}^2}{\rm d}t + \frac{2}{J}\int_{- \infty}^\infty i\!\left(t \right)i^\prime \!\left(t \right){{\left. {\frac{{\partial a\!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|}_{t,t}}{\rm d}t}\\[-4pt]&\quad+{ \frac{1}{J}\int_{- \infty}^\infty {{\left[{i\!\left(t \right)} \right]}^2}{{\left. {\frac{{{\partial ^2}a\!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}\partial {t_2}}}} \right|}_{t,t}}{\rm d}t} \\[-4pt]&\quad+ {\frac{1}{J}\int_{- \infty}^\infty {{\left[{i\!\left(t \right)} \right]}^2}{{\left[{{{\left. {\frac{{\partial \psi \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|}_{t,t}}} \right]}^2}{\rm d}t}\\[-4pt]&\quad+{ \frac{{\rm j}}{J}\int_{- \infty}^\infty {{\left[{i\!\left(t \right)} \right]}^2}{{\left. {\frac{{{\partial ^2}\psi \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}\partial {t_2}}}} \right|}_{t,t}}{\rm d}t.}\end{split}$$
Since $\langle \omega \rangle$ and $\langle {\omega ^2}\rangle$ must be real,
$$\begin{split}{\int_{- \infty}^\infty {{\left[{i\!\left(t \right)} \right]}^2}{{\left. {\frac{{\partial a\!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|}_{t,t}}{\rm d}t} = {0},\\[-4pt]{\int_{- \infty}^\infty {{\left[{i\!\left(t \right)} \right]}^2}{{\left. {\frac{{{\partial ^2}\psi \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}\partial {t_2}}}} \right|}_{t,t}}{\rm d}t}= {0}.\end{split}$$
In actuality,
$$\begin{split}{{{\left. {\frac{{\partial a\!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|}_{t,t}}} = {0},\\[-4pt]{{{\left. {\frac{{{\partial ^2}\psi \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}\partial {t_2}}}} \right|}_{t,t}}}= {0},\end{split}$$
which we prove in Appendix D using the coherent-modes representation of $\Gamma$, and Eqs. (17) and (18) simplify accordingly.

We are now in a position to derive an expression for $\sigma _\omega ^2 = \langle {\omega ^2}\rangle - {\langle \omega \rangle ^2}$. Using Eqs. (17), (18), and (20), we find

$$\begin{split}{\sigma _\omega ^2} &={ \frac{1}{J}\int_{- \infty}^\infty {{\left[{i^\prime \!\left(t \right)} \right]}^2}{\rm d}t + \frac{1}{J}\int_{- \infty}^\infty {{\left[{i\!\left(t \right)} \right]}^2}{{\left. {\frac{{{\partial ^2}a\!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}\partial {t_2}}}} \right|}_{t,t}}{\rm d}t}\\[-4pt]&\quad+{ \frac{1}{J}\int_{- \infty}^\infty {{\left[{i\!\left(t \right)} \right]}^2}{{\left[{{{\left. {\frac{{\partial \psi \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|}_{t,t}}} \right]}^2}{\rm d}t} \\[-4pt]&\quad- {{{\left\{{\frac{1}{J}\int_{- \infty}^\infty {{\left[{i\!\left(t \right)} \right]}^2}{{\left. {\frac{{\partial \psi \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|}_{t,t}}{\rm d}t} \right\}}^2}.}\end{split}$$
Last, the ${P^2}$ of the partially coherent pulsed beam can be found by applying Eq. (1).

B. Coherent Sources

For a coherent source, the MCF factors, such that

$$\Gamma \!\left({{t_1},{t_2}} \right) = f\!\left({{t_1}} \right){f^*}\!\left({{t_2}} \right)\exp \!\left[{- {\rm j}{\omega _c}\!\left({{t_1} - {t_2}} \right)} \right],$$
where ${\omega _c}$ is the radian optical frequency (also known as the carrier frequency) and $f$ is the complex envelope of the pulse. In this case,
$$\begin{split}{i\!\left(t \right) }&= {\left| {f\!\left(t \right)} \right| = \sqrt {f\!\left(t \right){f^*}\!\left(t \right)}}\\{a\!\left({{t_1},{t_2}} \right) }&= 1\\{\exp \!\left[{{\rm j}\psi \!\left({{t_1},{t_2}} \right)} \right] }&= {\frac{{f\!\left({{t_1}} \right)f\!\left({{t_2}} \right)}}{{\left| {f\!\left({{t_1}} \right)} \right|\left| {f\!\left({{t_2}} \right)} \right|}}\exp \!\left[{- {\rm j}{\omega _c}\!\left({{t_1} - {t_2}} \right)} \right].}\end{split}$$
The derivatives in Eq. (21) can be computed from these relations, resulting in
$$\begin{split}{i^\prime \!\left(t \right)}&= \frac{{{\rm Re}\!\left[{f^\prime \!\left(t \right){f^*}\!\left(t \right)} \right]}}{{\left| {f\!\left(t \right)} \right|}}\\{{{\left. {\frac{{\partial \psi \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|}_{t,t}}}&= \frac{{{\rm Im}\!\left[{f^\prime \!\left(t \right){f^*}\!\left(t \right)} \right]}}{{{{\left| {f\!\left(t \right)} \right|}^2}}} - {\omega _c}.\end{split}$$
Clearly, the derivative of the CDoC amplitude $a$ is zero. Substituting these into Eq. (21) and simplifying yields
$$\sigma _\omega ^2 = \frac{1}{J}\int_{- \infty}^\infty {\left| {f^\prime \!\left(t \right)} \right|^2}{\rm d}t - {\left\{{{\rm Im}\!\left[{\frac{1}{J}\int_{- \infty}^\infty f^\prime \!\left(t \right){f^*}\!\left(t \right){\rm d}t} \right]} \right\}^2}.$$

C. Schell-Model Sources

The general $\sigma _\omega ^2$ expression derived above simplifies considerably if the MCF of the source takes a Schell-model form [22–24], i.e.,

$$\begin{split}\Gamma \!\left({{t_1},{t_2}} \right) &= i\!\left({{t_1}} \right)i\!\left({{t_2}} \right)\gamma \!\left({{t_1} - {t_2}} \right) \\&= i\!\left({{t_1}} \right)i\!\left({{t_2}} \right)a\!\left({{t_1} - {t_2}} \right)\exp \!\left[{{\rm j}\psi \!\left({{t_1} - {t_2}} \right)} \right].\end{split}$$
In this case, the derivatives in Eq. (21) are independent of $t$, and only the first two terms survive:
$$\sigma _\omega ^2 = \frac{1}{J}\int_{- \infty}^\infty {\left[{i^\prime \!\left(t \right)} \right]^2}{\rm d}t + {\left. {\frac{{{\partial ^2}a\!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}\partial {t_2}}}} \right|_{t,t}}.$$
The derivative in Eq. (25) can be recast as a derivative with respect to $\tau = {t_1} - {t_2}$ using the chain rule. The final expression is
$$\sigma _\omega ^2 = \frac{1}{J}\int_{- \infty}^\infty {\left[{i^\prime \!\left(t \right)} \right]^2}{\rm d}t - {\left. {\frac{{{{\rm d}^2}a\!\left(\tau \right)}}{{{\rm d}{\tau ^2}}}} \right|_{\tau = 0}}.$$

As discussed above, for a coherent source the derivative of $a$ is zero, and Eq. (28) simplifies to Eq. (25) [note that the second term in Eq. (25) is zero because $i$ is real]. Consequently, the first term in Eq. (28) must be the “coherent contribution” to $\sigma _\omega ^2$. This observation permits a physical form for ${P^2}$, namely,

$${P^2} = \sqrt {{{\left({{P^{{\rm coh}}}} \right)}^4} - 4\sigma _t^2{{\left. {\frac{{{{\rm d}^2}a\!\left(\tau \right)}}{{{\rm d}{\tau ^2}}}} \right|}_{\tau = 0}}} ,$$
where ${({{P^{{\rm coh}}}})^2} = 2{\sigma _t}\sigma _\omega ^{{\rm coh}}$ and $\sigma _\omega ^{{\rm coh}} = \sqrt {{{({\sigma _\omega ^{{\rm coh}}})}^2}}$ are given in Eq. (25). This result is the time analog of the Schell-model ${M^2}$ relation derived in [10].

D. Examples

In what follows, we present examples for (1) a cosine-Gaussian correlated Schell-model pulsed beam and (2) a nonuniformly correlated pulsed beam.

1. Cosine Gaussian-Correlated Schell-Model Pulsed Beam

The MCF of a cosine Gaussian-correlated Schell-model (CGSM) pulsed beam takes the form

$$\begin{split}{\Gamma \!\left({{t_1},{t_2}} \right)} &= {\exp \!\left({- \frac{{t_1^2}}{{4W_t^2}}} \right)\exp \!\left({- \frac{{t_2^2}}{{4W_t^2}}} \right)}\\&\quad \times{\cos \!\left[{\frac{{n\sqrt {2\pi} \!\left({{t_1} - {t_2}} \right)}}{\delta}} \right]\exp \!\left[{- \frac{{{{\left({{t_1} - {t_2}} \right)}^2}}}{{2{\delta ^2}}}} \right]}\\&\quad\times{\exp \!\left[{- {\rm j}{\omega _c}\!\left({{t_1} - {t_2}} \right)} \right],}\end{split}$$
where ${W_t}$ is the pulse width, $n$ is a positive constant, and $\delta$ is the temporal correlation (coherence) length [25]. The functions $i$ and $\gamma$ are clear in the above MCF:
$$\begin{split}{i\!\left(t \right)} &= \exp {\left({- \frac{{{t^2}}}{{4W_t^2}}} \right)}\\{\gamma \!\left(\tau \right)} &= a{\left(\tau \right)\exp \!\left[{{\rm j}\psi \!\left(\tau \right)} \right]} \\&= {\cos \!\left({\frac{{n\sqrt {2\pi} \tau}}{\delta}} \right)\exp \!\left({- \frac{{{\tau ^2}}}{{2{\delta ^2}}}} \right)\exp \!\left({- {\rm j}{\omega _c}\tau} \right).}\end{split}$$
Note that, when $n = 0$, the above MCF simplifies to that of a Gaussian–Schell-model pulsed beam [23,24].

A CGSM beam has the interesting characteristic in that it splits into two Gaussian-shaped pulses after propagating a certain distance (depends on $\delta$, $n$, ${W_t}$, and the group velocity dispersion coefficient) in a second-order-dominant dispersive medium. In addition, the form of the MCF makes it analytically tractable in many situations of practical interest. CGSM beams can easily be synthesized by exploiting the temporal version of the van Cittert–Zernike theorem [19,26–28].

Applying Eq. (29), one finds, after simple calculations, that

$${P^2} = \sqrt {1 + 4\frac{{W_t^2}}{{{\delta ^2}}}\!\left({1 + 2\pi {n^2}} \right)} .$$
Not surprisingly, when $n = 0$, this result is identical (time swapped for space) to the Gaussian Schell-model ${M^2}$ result in [10,13,29].

2. Nonuniformly Correlated Pulsed Beam

We now proceed to a more complicated example. The MCF of this stochastic pulsed beam is

$$\begin{split}{\Gamma \!\left({{t_1},{t_2}} \right)}& = {\exp \!\left({- \frac{{t_1^2}}{{4W_t^2}}} \right)\exp \!\left({- \frac{{t_2^2}}{{4W_t^2}}} \right)}\\&\quad\times {\exp \left\{{- \frac{{{{\left[{{{\left({{t_1} - {t_0}} \right)}^2} - {{\left({{t_2} - {t_0}} \right)}^2}} \right]}^2}}}{{{\delta ^4}}}} \right\}\exp \!\left[{{\rm j}\alpha \!\left({t_1^3 - t_2^3} \right)} \right]}\\&\quad \times{\exp \left\{{{\rm j}\beta \!\left[{{{\left({{t_1} - {t_0}} \right)}^2} - {{\left({{t_2} - {t_0}} \right)}^2}} \right]} \right\}\exp \!\left[{- {\rm j}{\omega _c}\!\left({{t_1} - {t_2}} \right)} \right].}\end{split}$$
Again, $i$ and $\gamma$—more importantly, $a$ and $\psi$—are easy to identify, namely,
$$\begin{split}{i\!\left(t \right) }&= {\exp \!\left({- \frac{{{t^2}}}{{4W_t^2}}} \right)},\\{a\!\left({{t_1},{t_2}} \right)} &= {\exp \left\{{- \frac{{{{\left[{{{\left({{t_1} - {t_0}} \right)}^2} - {{\left({{t_2} - {t_0}} \right)}^2}} \right]}^2}}}{{{\delta ^4}}}} \right\}},\\{\psi \!\left({{t_1},{t_2}} \right) }&= {\alpha \!\left({t_1^3 - t_2^3} \right) + \beta \!\left[{{{\left({{t_1} - {t_0}} \right)}^2} - {{\left({{t_2} - {t_0}} \right)}^2}} \right] - {\omega _c}\!\left({{t_1} - {t_2}} \right),}\end{split}$$
where $\alpha$ is the third-order chirp coefficient, $\beta$ is the second-order chirp coefficient, and ${t_0}$ is a time delay that shifts $\gamma$ in the ${t_1} - {t_2}$ plane.

This source is a more general version of the nonuniformly correlated (NUC) pulsed beam introduced in [30]. NUC beams of this type “self-focus” after propagating a near-zone distance in a second-order dispersive medium, such as an optical fiber and, therefore, provide a level of beam/pulse control beyond that of coherent and even Schell-model sources. NUC beams can be synthesized using a device known as a Fourier transform pulse shaper [31–33] as was recently done in [34].

To derive an expression for ${P^2}$, we start with $\sigma _\omega ^2$. Using Eq. (21), and after tedious yet straightforward calculations, we arrive at

$$\sigma _\omega ^2 = \frac{1}{{4W_t^2}}\left\{{1 + 8W_t^4\!\left[{9{\alpha ^2}W_t^2 + 2{\beta ^2} + \frac{4}{{{\delta ^4}}}\!\left({1 + \frac{{t_0^2}}{{W_t^2}}} \right)} \right]} \right\}.$$
Substituting Eq. (35) and ${\sigma _t} = {W_t}$ into Eq. (1) yields the final result:
$${P^2} = \sqrt {1 + 8W_t^4\!\left[{9{\alpha ^2}W_t^2 + 2{\beta ^2} + 4\frac{1}{{{\delta ^4}}}\!\left({1 + \frac{{t_0^2}}{{W_t^2}}} \right)} \right]} .$$
The first two terms in the brackets quantify how higher-order phase modulation affects pulse quality, while the last term accounts for temporal coherence. If $\alpha = \beta = 0$ and $\delta \to \infty$, the MCF in Eq. (33) simplifies to that of a fully coherent pulsed Gaussian beam and the ${P^2}$ in Eq. (36) equals 1.
 figure: Fig. 1.

Fig. 1. CGSM pulsed beam results: (a) theory and (b) simulation ${\rm Re}(\Gamma)$; (c) theory and (d) simulation ${\rm Im}(\Gamma)$; (e) $\Gamma ({t,t}) = \left\langle {I(t)} \right\rangle$ theory and simulation; and (f) ${P^2}$ versus CGSM realization number.

Download Full Size | PDF

In the next section, we validate the above results by computing ${P^2}$, using the definitions in Eqs. (1)–(4), of simulated realizations of CGSM and NUC pulsed beams.

3. SIMULATION

Before presenting the simulation results, we discuss the details of the simulation setup.

A. Setup

We computed the MCF and ${P^2}$ from 10,000 statistically independent CGSM and NUC beam realizations. The CGSM beam parameters were ${W_t} = 30\;{\rm ps}$, $\delta = 15\;{\rm ps}$, and $n = 1.5$, and the NUC parameters were ${W_t} = 30\;{\rm ps}$, $\delta = 30\;{\rm ps}$, $\alpha = 5 \times {10^{- 5}}\;{{\rm ps}^{- 3}}$, $\beta = 5 \times {10^{- 4}}\;{{\rm ps}^{- 2}}$, and ${t_0} = - 5 \;{\rm ps}$.

We generated the CGSM and NUC beam realizations using the method described in [35]. This approach uses the integral form of the MCF, known colloquially as the superposition rule [36–38], and requires the numerical evaluation of the following superposition integral:

$$U\!\left(t \right) = \int_{- \infty}^\infty r\!\left(v \right)\sqrt {\frac{1}{2}p\!\left(v \right)} H\!\left({t,v} \right){\rm d}v,$$
where $r$ is a zero-mean, unit-variance, delta-correlated, complex Gaussian random function. The $p$ and $H$ are source dependent. For the CGSM beam,
$$\begin{split}{p\!\left(v \right)}& = {\frac{\delta}{{\sqrt {2\pi}}}\cosh \!\left({n\sqrt {2\pi} \delta v} \right)\exp \!\left({- \frac{{{\delta ^2}{v^2} + 2\pi {n^2}}}{2}} \right)}\\{H\!\left({t,v} \right) }&= {\exp \!\left({- \frac{{{t^2}}}{{4W_t^2}}} \right)\exp \!\left({{\rm j}vt} \right),}\end{split}$$
and for the NUC beam,
$$\begin{split}{p\!\left(v \right)} &= {\frac{{{\delta ^2}}}{{2\sqrt \pi}}\exp \!\left[{- \frac{{{\delta ^4}}}{4}{{\left({v - \beta} \right)}^2}} \right]}\\{H\!\left({t,v} \right)}& ={ \exp \!\left({- \frac{{{t^2}}}{{4W_t^2}}} \right)\exp \!\left({{\rm j}\alpha {t^3}} \right)\exp \!\left[{{\rm j}{{\left({t - {t_0}} \right)}^2}v} \right].}\end{split}$$

Note that the complex exponential in the CGSM $H$ is the Fourier kernel; therefore, we generated CGSM realizations using fast Fourier transforms (FFTs) [39,40]. On the other hand, the NUC $H$ is a Fourier-like kernel, and, although NUC beams can be generated using FFTs and clever substitutions [35,41], here we evaluated Eq. (37) as a matrix-vector product.

For the CGSM and NUC beam realizations, we discretized the $t$ axis of $U$ using $N = 1000$ points. The sampling time was $\Delta t = \min ({{W_t},\delta})/50$, i.e., 0.3 and 0.6 ps for the CGSM and NUC simulations, respectively. These $\Delta t$ ensured that there were at least 50 points across the CGSM and NUC MCFs. Since we produced CGSM realizations using FFTs, the $v$-axis spacing, ${\rm d}v$ in Eq. (37), was set by $N$ and $\Delta t$, i.e., ${\rm d}v = \Delta v = 2\pi /({N\Delta t})$. We chose the $\Delta v$ for the NUC realizations so that 50 points spanned the width of $p$ in Eq. (39).

Last, for each CGSM and NUC realization, we computed $\sigma _t^2$ and $\sigma _\omega ^2$ using Eqs. (2) and (3) and evaluated the integrals using the trapezoid rule. In the case of $\sigma _\omega ^2$, we Fourier-transformed (using an FFT) $U(t)$, took the magnitude square of the result, and used the CSD expressions in Eqs. (2) and (3). ${P^2}$ was computed using Eq. (1).

B. Results

Figures 1 and 2 report the CGSM and NUC results, respectively. The figures are organized in the same manner: (a) and (b) show the theoretical and simulated ${\rm Re}[{\Gamma ({{t_1},{t_2}})}]$; (c) and (d) show the corresponding ${\rm Im}[{\Gamma ({{t_1},{t_2}})}]$; (e) plots the theoretical and simulated $\Gamma ({t,t}) = \left\langle {I(t)} \right\rangle$; and (f) shows ${P^2}$ versus stochastic field realization number. Images (a) and (b) are encoded using the same false color scale defined by the color bar immediately to the right of (b), likewise for images (c) and (d).

 figure: Fig. 2.

Fig. 2. NUC pulsed beam results: (a) theory and (b) simulation ${\rm Re}(\Gamma)$; (c) theory and (d) simulation ${\rm Im}(\Gamma)$; (e) $\Gamma ({t,t}) = \left\langle {I(t)} \right\rangle$ theory and simulation; and (f) ${P^2}$ versus NUC realization number.

Download Full Size | PDF

The agreement between the theoretical and simulated results is excellent. The simulated ${P^2}$ for both beams converges to the theoretical result within approximately 1000 realizations. The quality of the results displayed in Figs. 1 and 2 validates the analysis of the previous section.

4. CONCLUSION

In summary, the pulse quality factor ${P^2}$, defined in terms of RMS widths, has inherent advantages over the time-bandwidth product (TBP), i.e., the traditional measure of pulse quality. This last point is especially true for fields with complex pulse shapes and spectra. Our work generalized ${P^2}$, originally presented for coherent fields, to nonstationary random fields of any state of coherence. We began by deriving a general expression for ${P^2}$ in terms of the shape and temporal correlation function of the pulsed stochastic field. We then specialized that result to the important cases of coherent and Schell-model (uniformly correlated) sources before presenting two examples: a cosine Gaussian-correlated Schell model and nonuniformly correlated pulsed beam. Last, we validated our work by generating both of these sources in simulation and comparing sample ${P^2}$ to the derived theoretical quantities. The agreement between simulation and theory was excellent.

It is important to note that interferometric techniques could be used to experimentally measure ${P^2}$. For example, [42] recently showed that reduced pulse quality has a measurable effect on the signal-to-noise ratio of pulsed digital-holography systems. Therefore, we believe that the generalized theory presented in this paper will enable experimental validation in the near future. Analogously, the experimental measurement of ${P^2}$ could also be used to improve the performance of coherent-detection systems at large, which have numerous applications in unconventional imaging, as well as laser-based sensing and communications. These systems often involve the interference of highly sophisticated pulsed waveforms (in terms of magnitude and phase), which inevitably include cases where the TBP (in particular, the FWHM definition) has limited utility as a pulse-quality metric. This last point serves to further motivate the generalization of ${P^2}$ contained in this paper.

APPENDIX A: PROOF OF EQ. (5)

Starting with

$$\Gamma \!\left({{t_1},{t_2}} \right) = \iint_{- \infty}^\infty W\!\left({{\omega _1},{\omega _2}} \right)\exp \!\left[{- {\rm j}\!\left({{\omega _1}{t_1} - {\omega _2}{t_2}} \right)} \right]{\rm d}{\omega _1}{\rm d}{\omega _2},$$
we take the partial derivative with respect to ${t_1}$, such that
$$\begin{split}\frac{{\partial \Gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}&= - {\rm j}\iint_{- \infty}^\infty {\omega _1}W\!\left({{\omega _1},{\omega _2}} \right)\\&\quad\times\exp \!\left[{- {\rm j}\!\left({{\omega _1}{t_1} - {\omega _2}{t_2}} \right)} \right]{\rm d}{\omega _1}{\rm d}{\omega _2}.\end{split}$$
We now Fourier-transform both sides of Eq. (A2) and, after changing the order of the integrals, obtain
$$\begin{split}&{\frac{1}{{{{\left({2\pi} \right)}^2}}} \iint_{- \infty}^\infty \frac{{\partial \Gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}\exp \!\left[{{\rm j}\!\left({{{\omega ^\prime_1}}{t_1} - {{\omega ^\prime_2}}{t_2}} \right)} \right]{\rm d}{t_1}{\rm d}{t_2}}\\[-4pt]&\quad ={ \frac{{- {\rm j}}}{{{{\left({2\pi} \right)}^2}}}\iint_{- \infty}^\infty {\omega _1}W\!\left({{\omega _1},{\omega _2}} \right) \iint_{- \infty}^\infty \exp \!\left[{- {\rm j}\!\left({{\omega _1} - {{\omega ^\prime_1}}} \right){t_1}} \right]}\\[-4pt]&\qquad\times{\exp \!\left[{{\rm j}\!\left({{\omega _2} - {{\omega ^\prime_2}}} \right){t_2}} \right]{\rm d}{t_1}{\rm d}{t_2}{\rm d}{\omega _1}{\rm d}{\omega _2}.}\end{split}$$
The ${t_1},{t_2}$ integrals on the right-hand side of Eq. (A3) evaluate to ${({2\pi})^2}\delta ({{\omega _1} - {{\omega ^\prime_1}}})\delta ({{\omega _2} - {{\omega ^\prime_2}}})$, where $\delta$ is the Dirac delta function. The remaining integrals over ${\omega _1},{\omega _2}$ are now trivial. The simplified result is
$$\begin{split}&\frac{1}{{{{\left({2\pi} \right)}^2}}}\iint_{- \infty}^\infty \frac{{\partial \Gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}\exp \!\left[{{\rm j}\!\left({{\omega _1}{t_1} - {\omega _2}{t_2}} \right)} \right]{\rm d}{t_1}{\rm d}{t_2}\\[-4pt]&\quad = - {\rm j}{\omega _1}W\!\left({{\omega _1},{\omega _2}} \right).\end{split}$$
Last, we set ${\omega _1} = {\omega _2} = \omega$ and integrate both sides of Eq. (A4) over all $\omega$ resulting in
$$\begin{split}\int_{- \infty}^\infty \omega W\!\left({\omega ,\omega} \right){\rm d}\omega &= \frac{{\rm j}}{{{{\left({2\pi} \right)}^2}}} \iint_{- \infty}^\infty \frac{{\partial \Gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}\int_{- \infty}^\infty \\[-4pt]&\quad\exp \!\left[{{\rm j}\omega \!\left({{t_1} - {t_2}} \right)} \right]{\rm d}\omega {\rm d}{t_1}{\rm d}{t_2}.\end{split}$$
The $\omega$ integral on the right-hand side of Eq. (A5) is $2\pi \delta ({{t_1} - {t_2}})$, making one of the remaining two integrals trivial:
$$\int_{- \infty}^\infty \omega W\!\left({\omega ,\omega} \right){\rm d}\omega = \frac{{\rm j}}{{2\pi}}\int_{- \infty}^\infty {\left. {\frac{{\partial \Gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|_{t,t}}{\rm d}t.$$

Having derived the numerator of $\langle \omega \rangle$ [see Eq. (3)], we now proceed to the denominator. Starting with

$$W\!\left({{\omega _1},{\omega _2}} \right) = \frac{1}{{{{\left({2\pi} \right)}^2}}}\iint_{- \infty}^\infty \Gamma \!\left({{t_1},{t_2}} \right)\exp \!\left[{{\rm j}\!\left({{\omega _1}{t_1} - {\omega _2}{t_2}} \right)} \right]{\rm d}{t_1}{\rm d}{t_2},$$
we set ${\omega _1} = {\omega _2} = \omega$ and integrate both sides over all $\omega$, such that
$$\begin{split}\int_{- \infty}^\infty W\!\left({\omega ,\omega} \right){\rm d}\omega &= \frac{1}{{{{\left({2\pi} \right)}^2}}}\iint_{- \infty}^\infty \Gamma \!\left({{t_1},{t_2}} \right)\\[-4pt]&\quad\times\int_{- \infty}^\infty \exp \!\left[{{\rm j}\omega \!\left({{t_1} - {t_2}} \right)} \right]{\rm d}\omega {\rm d}{t_1}{\rm d}{t_2}.\end{split}$$
Again, the $\omega$ integral on the right-hand side of the above expression is a Dirac delta function. Evaluating one of the remaining two integrals simplifies Eq. (A8) to
$$\int_{- \infty}^\infty W\!\left({\omega ,\omega} \right){\rm d}\omega = \frac{1}{{2\pi}}\int_{- \infty}^\infty \Gamma \!\left({t,t} \right){\rm d}t.$$
Dividing Eq. (A6) by (A9) yields Eq. (5).

APPENDIX B: PROOF OF EQ. (6)

Following a similar approach as that described in Appendix A, we begin by taking the mixed partial of Eq. (A1), i.e.,

$$\begin{split}\frac{{{\partial ^2}\Gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}\partial {t_2}}} &= \iint_{- \infty}^\infty {\omega _1}{\omega _2}W\!\left({{\omega _1},{\omega _2}} \right)\\[-4pt]&\quad\times\exp \!\left[{- {\rm j}\!\left({{\omega _1}{t_1} - {\omega _2}{t_2}} \right)} \right]{\rm d}{\omega _1}{\rm d}{\omega _2}.\end{split}$$
We then Fourier-transform both sides of the above expression, producing
$$\begin{split}{\omega _1}{\omega _2}W\!\left({{\omega _1},{\omega _2}} \right) &= \frac{1}{{{{\left({2\pi} \right)}^2}}}\iint_{- \infty}^\infty \frac{{{\partial ^2}\Gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}\partial {t_2}}}\\[-4pt]&\quad\times\exp \!\left[{{\rm j}\!\left({{\omega _1}{t_1} - {\omega _2}{t_2}} \right)} \right]{\rm d}{t_1}{\rm d}{t_2}.\end{split}$$
Last, we set ${\omega _1} = {\omega _2} = \omega$ and integrate both sides over all $\omega$ resulting in
$$\int_{- \infty}^\infty {\omega ^2}W\!\left({\omega ,\omega} \right){\rm d}\omega = \frac{1}{{2\pi}}\int_{- \infty}^\infty {\left. {\frac{{{\partial ^2}\Gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}\partial {t_2}}}} \right|_{t,t}}{\rm d}t.$$
Dividing Eq. (B3) by (A9) yields Eq. (6).

APPENDIX C: PROOF OF EQ. (13)

Returning to Eq. (A2), we set ${t_1} = {t_2} = t$ and obtain

$$\begin{split}{\left. {\frac{{\partial \Gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|_{t,t}}& = - {\rm j} \iint_{- \infty}^\infty {\omega _1}W\!\left({{\omega _1},{\omega _2}} \right)\\[-4pt]&\quad\times\exp \!\left[{- {\rm j}\!\left({{\omega _1} - {\omega _2}} \right)t} \right]{\rm d}{\omega _1}{\rm d}{\omega _2}.\end{split}$$
Likewise, taking the partial derivative of Eq. (A1) with respect to ${t_2}$ and setting ${t_1} = {t_2} = t$ yields
$${\left. {\frac{{\partial \Gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_2}}}} \right|_{t,t}} = {\rm j} \iint_{- \infty}^\infty {\omega _2}W\!\left({{\omega _1},{\omega _2}} \right)\exp \!\left[{{\rm j}\!\left({{\omega _1} - {\omega _2}} \right)t} \right]{\rm d}{\omega _1}{\rm d}{\omega _2}.$$
We now take the conjugate of Eq. (C2) and make use of the fact that the CSD function is Hermitian, i.e., ${W^*}({{\omega _1},{\omega _2}}) = W({{\omega _2},{\omega _1}})$:
$$\begin{split}{\left[{{{\left. {\frac{{\partial \Gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_2}}}} \right|}_{t,t}}} \right]^*} &= - {\rm j}\iint_{- \infty}^\infty {\omega _2}W\!\left({{\omega _2},{\omega _1}} \right)\\[-4pt]&\quad\times\exp \!\left[{- {\rm j}\!\left({{\omega _2} - {\omega _1}} \right)t} \right]{\rm d}{\omega _2}{\rm d}{\omega _1}.\end{split}$$
The integrals in Eqs. (C1) and (C3) are equal; therefore,
$${\left. {\frac{{\partial \Gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|_{t,t}} = {\left[{{{\left. {\frac{{\partial \Gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_2}}}} \right|}_{t,t}}} \right]^*}.$$
Substituting in the expression for $\Gamma$ given in Eq. (8) and evaluating the derivatives yields Eq. (13).

APPENDIX D: PROOF OF EQ. (20)

A genuine MCF (square integrable, Hermitian, and non-negative definite) can be expanded in a set of orthogonal modes, such that

$$\Gamma \!\left({{t_1},{t_2}} \right) = \sum\limits_n {\lambda _n}{g_n}\!\left({{t_1}} \right)g_n^*\!\left({{t_2}} \right),$$
where ${\lambda _n} \geq 0$ and ${g_n}$ are solutions to the integral equation
$$\int_{- \infty}^\infty \Gamma \!\left({{t_1},{t_2}} \right){g_n}\!\left({{t_2}} \right){\rm d}{t_2} = {\lambda _n}{g_n}\!\left({{t_1}} \right).$$
Equation (D1) is known as the coherent-modes representation of $\Gamma$ [22–24,43–45]. Using Eq. (8), the CDoC $\gamma$ in terms of coherent modes is
$$\begin{split}{\gamma \!\left({{t_1},{t_2}} \right)} ={ a\!\left({{t_1},{t_2}} \right)\exp \!\left[{{\rm j}\psi \!\left({{t_1},{t_2}} \right)} \right]}\\ {= \sum\limits_n {\lambda _n}\frac{{{g_n}\!\left({{t_1}} \right)}}{{\sqrt {\sum\limits_n {\lambda _n}{{\left| {{g_n}\!\left({{t_1}} \right)} \right|}^2}}}}\frac{{g_n^*\!\left({{t_2}} \right)}}{{\sqrt {\sum\limits_n {\lambda _n}{{\left| {{g_n}\!\left({{t_2}} \right)} \right|}^2}}}}.}\end{split}$$
We now compute the partial derivative of $\gamma$ with respect to ${t_1}$ and evaluate the result at ${t_1} = {t_2} = t$:
$$\begin{split}{{{\left. {\frac{{\partial \gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|}_{t,t}}} = {{{\left. {\frac{{\partial a\!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|}_{t,t}} + {\rm j}{{\left. {\frac{{\partial \psi \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|}_{t,t}}}\\ = {\sum\limits_n {\lambda _n}\frac{{g_n^*\!\left(t \right)}}{{\sqrt {\sum\limits_n {\lambda _n}{{\left| {{g_n}\!\left(t \right)} \right|}^2}}}}\!\left[{\frac{{\rm d}}{{{\rm d}t}}\frac{{{g_n}\!\left(t \right)}}{{\sqrt {\sum\limits_n {\lambda _n}{{\left| {{g_n}\!\left(t \right)} \right|}^2}}}}} \right].}\end{split}$$
The derivative in the brackets, after some lengthy calculations, evaluates to
$$\begin{split}\frac{{\rm d}}{{{\rm d}t}}\frac{{{g_n}\!\left(t \right)}}{{\sqrt {\sum\nolimits_n {\lambda _n}{{\left| {{g_n}\!\left(t \right)} \right|}^2}}}}& = \frac{{{{g^\prime_n}}\!\left(t \right)}}{{\sqrt {\sum\nolimits_n {\lambda _n}{{\left| {{g_n}\!\left(t \right)} \right|}^2}}}} \\&\quad- \frac{{{g_n}\!\left(t \right)}}{{\sqrt {\sum\nolimits_n {\lambda _n}{{\left| {{g_n}\!\left(t \right)} \right|}^2}}}}\frac{{\sum\nolimits_n {\lambda _n}{\rm Re}\!\left[{{{g^\prime_n}}\!\left(t \right)g_n^*\!\left(t \right)} \right]}}{{\sum\nolimits_n {\lambda _n}{{\left| {{g_n}\!\left(t \right)} \right|}^2}}}.\end{split}$$
Substituting Eq. (D5) into Eq. (D4) and expanding out the terms yields
$$\begin{split}{{{\left. {\frac{{\partial \gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|}_{t,t}}}&= {\frac{{\sum\nolimits_n {\lambda _n}\left\{{{{g^\prime_n}}\!\left(t \right)g_n^*\!\left(t \right) - {\rm Re}\!\left[{{{g^\prime_n}}\!\left(t \right)g_n^*\!\left(t \right)} \right]} \right\}}}{{\sum\nolimits_n {\lambda _n}{{\left| {{g_n}\!\left(t \right)} \right|}^2}}}}\\&={ {\rm j}\frac{{\sum\nolimits_n {\lambda _n}{\rm Im}\!\left[{{{g^\prime_n}}\!\left(t \right)g_n^*\!\left(t \right)} \right]}}{{\sum\nolimits_n {\lambda _n}{{\left| {{g_n}\!\left(t \right)} \right|}^2}}}.}\end{split}$$
Since Eq. (D6) is purely imaginary,
$${\left. {\frac{{\partial a\!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|_{t,t}} = 0.$$

This proves the first identity in Eq. (20). To prove the second, we now compute the mixed partial of $\gamma$ and evaluate the result at ${t_1} = {t_2} = t$:

$$\begin{split}{{{\left. {\frac{{{\partial ^2}\gamma \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}\partial {t_2}}}} \right|}_{t,t}}}&= {{{\left. {\frac{{{\partial ^2}a\!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}\partial {t_2}}}} \right|}_{t,t}} + {{\left[{{{\left. {\frac{{\partial \psi \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}}}} \right|}_{t,t}}} \right]}^2} }\\&\quad+ {{\rm j}{{\left. {\frac{{{\partial ^2}\psi \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}\partial {t_2}}}} \right|}_{t,t}}}\\ &= {\sum\nolimits_n {\lambda _n}\!\left[{\frac{{\rm d}}{{{\rm d}t}}\frac{{g_n^*\!\left(t \right)}}{{\sqrt {\sum\nolimits_n {\lambda _n}{{\left| {{g_n}\!\left(t \right)} \right|}^2}}}}} \right]}\\&\quad\times{\left[{\frac{{\rm d}}{{{\rm d}t}}\frac{{{g_n}\!\left(t \right)}}{{\sqrt {\sum\nolimits_n {\lambda _n}{{\left| {{g_n}\!\left(t \right)} \right|}^2}}}}} \right]}\\& = {\sum\nolimits_n {\lambda _n}{{\left| {\frac{{\rm d}}{{{\rm d}t}}\frac{{{g_n}\!\left(t \right)}}{{\sqrt {\sum\nolimits_n {\lambda _n}{{\left| {{g_n}\!\left(t \right)} \right|}^2}}}}} \right|}^2}.}\end{split}$$
Because Eq. (D8) is real,
$${\left. {\frac{{{\partial ^2}\psi \!\left({{t_1},{t_2}} \right)}}{{\partial {t_1}\partial {t_2}}}} \right|_{t,t}} = 0.$$

Acknowledgment

M. Hyde would like to thank the Air Force Office of Scientific Research (AFOSR) Physical and Biological Sciences Branch (RTB) for supporting this work. O. Korotkova acknowledges the support from the University of Miami under the Cooper Fellowship program. M. Spencer would like to thank the AFOSR for sponsoring this research under the auspices of an Air Force Research Laboratory Science and Engineering Early Career Award. The views expressed in this paper are those of the authors and do not reflect the official policy or position of the US Air Force, the Department of Defense, or the US government.

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

REFERENCES

1. E. Sorokin, G. Tempea, and T. Brabec, “Measurement of the root-mean-square width and the root-mean-square chirp in ultrafast optics,” J. Opt. Soc. Am. B 17, 146–150 (2000). [CrossRef]  

2. J. Zhou, C. Zhu, and J. Kuhl, “Generation of high-quality 10-femtosecond laser pulses with intracavity spatial and spectral control,” Appl. Phys. B 73, 119–123 (2001). [CrossRef]  

3. G. Rousseau, N. McCarthy, and M. Piché, “Description of pulse propagation in a dispersive medium by use of a pulse quality factor,” Opt. Lett. 27, 1649–1651 (2002). [CrossRef]  

4. G. Rousseau, N. McCarthy, and M. Piche, “Generation and characterization of sub-10-fs laser pulses,” Proc. SPIE 4087, 910–920 (2000). [CrossRef]  

5. A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990). [CrossRef]  

6. R. Martínez-Herrero and P. M. Mejías, “Second-order spatial characterization of hard-edge diffracted beams,” Opt. Lett. 18, 1669–1671 (1993). [CrossRef]  

7. R. Martínez-Herrero, P. M. Mejías, and M. Arias, “Parametric characterization of coherent, lowest-order Gaussian beams propagating through hard-edged apertures,” Opt. Lett. 20, 124–126 (1995). [CrossRef]  

8. S. Ramee and R. Simon, “Effect of holes and vortices on beam quality,” J. Opt. Soc. Am. A 17, 84–94 (2000). [CrossRef]  

9. F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991). [CrossRef]  

10. M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16, 106–112 (1999). [CrossRef]  

11. B. Zhang, X. Chu, and Q. Li, “Generalized beam-propagation factor of partially coherent beams propagating through hard-edged apertures,” J. Opt. Soc. Am. A 19, 1370–1375 (2002). [CrossRef]  

12. X. Chu, B. Zhang, and Q. Wen, “Generalized M2 factor of a partially coherent beam propagating through a circular hard-edged aperture,” Appl. Opt. 42, 4280–4284 (2003). [CrossRef]  

13. S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18, 12587–12598 (2010). [CrossRef]  

14. M. W. Hyde IV and M. F. Spencer, “Modeling the effects of high-energy-laser beam quality using scalar Schell-model sources,” Proc. SPIE 10981, 31–41 (2019). [CrossRef]  

15. M. W. Hyde and M. F. Spencer, “M2 factor of a vector Schell-model beam,” Opt. Eng. 58, 074101 (2019). [CrossRef]  

16. G. Lin, C. Sui, and Q. Lin, “Non-Gaussian pulse propagation and pulse quality factor using intensity moment method,” Chin. Phys. Lett. 16, 415–417 (1999). [CrossRef]  

17. M. Alonso, T. Setälä, and A. Friberg, “Optimal pulses for arbitrary dispersive media,” J. Eur. Opt. Soc. 6, 11035 (2011). [CrossRef]  

18. B. Anderson, A. Flores, R. Holten, and I. Dajani, “Comparison of phase modulation schemes for coherently combined fiber amplifiers,” Opt. Express 23, 27046–27060 (2015). [CrossRef]  

19. V. Torres-Company, J. Lancis, and P. Andrés, “Space-time analogies in optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2011), Chap. 1, pp. 1–80.

20. M. E. Marhic, P. A. Andrekson, P. Petropoulos, S. Radic, C. Peucheret, and M. Jazayerifar, “Fiber optical parametric amplifiers in optical communication systems,” Laser Photon. Rev. 9, 50–74 (2015). [CrossRef]  

21. I. V. Doronin, E. S. Andrianov, A. A. Zyablovsky, A. A. Pukhov, Y. E. Lozovik, A. P. Vinogradov, and A. A. Lisyansky, “Second-order coherence properties of amplified spontaneous emission,” Opt. Express 27, 10991–11005 (2019). [CrossRef]  

22. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

23. H. Lajunen, J. Tervo, and P. Vahimaa, “Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses,” J. Opt. Soc. Am. A 21, 2117–2123 (2004). [CrossRef]  

24. H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22, 1536–1545 (2005). [CrossRef]  

25. C. Ding, O. Korotkova, Y. Zhang, and L. Pan, “Cosine-Gaussian correlated Schell-model pulsed beams,” Opt. Express 22, 931–942 (2014). [CrossRef]  

26. C. Dorrer, “Temporal van Cittert-Zernike theorem and its application to the measurement of chromatic dispersion,” J. Opt. Soc. Am. B 21, 1417–1423 (2004). [CrossRef]  

27. H. Lajunen, A. T. Friberg, and P. Östlund, “Quasi-stationary plane-wave optical pulses and the van Cittert–Zernike theorem in time,” J. Opt. Soc. Am. A 23, 2530–2537 (2006). [CrossRef]  

28. Y. Zhang, C. Ding, M. W. Hyde IV, and O. Korotkova, “Non-stationary pulses with complex-valued temporal degree of coherence,” J. Opt. 22, 105607 (2020). [CrossRef]  

29. Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16, 15563–15575 (2008). [CrossRef]  

30. H. Lajunen and T. Saastamoinen, “Non-uniformly correlated partially coherent pulses,” Opt. Express 21, 190–195 (2013). [CrossRef]  

31. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

32. A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929–1960 (2000). [CrossRef]  

33. C. Ding, M. Koivurova, J. Turunen, T. Setälä, and A. T. Friberg, “Coherence control of pulse trains by spectral phase modulation,” J. Opt. 19, 095501 (2017). [CrossRef]  

34. R. Talukder, A. Halder, M. Koivurova, C. Ding, T. Setälä, J. Turunen, and A. T. Friberg, “Generation of pulse trains with nonconventional temporal correlation properties,” J. Opt. 24, 055502 (2022). [CrossRef]  

35. M. W. Hyde, “Stochastic complex transmittance screens for synthesizing general partially coherent sources,” J. Opt. Soc. Am. A 37, 257–264 (2020). [CrossRef]  

36. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007). [CrossRef]  

37. R. Martínez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34, 1399–1401 (2009). [CrossRef]  

38. F. Gori and R. Martínez-Herrero, “Reproducing kernel Hilbert spaces for wave optics: tutorial,” J. Opt. Soc. Am. A 38, 737–748 (2021). [CrossRef]  

39. D. G. Voelz, Computational Fourier Optics: A MATLAB Tutorial (SPIE, 2011).

40. D. Voelz, X. Xiao, and O. Korotkova, “Numerical modeling of Schell-model beams with arbitrary far-field patterns,” Opt. Lett. 40, 352–355 (2015). [CrossRef]  

41. M. W. Hyde, X. Xiao, and D. G. Voelz, “Generating electromagnetic nonuniformly correlated beams,” Opt. Lett. 44, 5719–5722 (2019). [CrossRef]  

42. S. A. Owens, M. F. Spencer, D. E. Thornton, and G. P. Perram, “Pulsed laser source digital holography efficiency measurements,” Appl. Opt. 61, 4823–4832 (2022). [CrossRef]  

43. E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982). [CrossRef]  

44. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part II: Steady-state fields and higher-order correlations,” J. Opt. Soc. Am. A 3, 76–85 (1986). [CrossRef]  

45. A. S. Ostrovsky, Coherent-Mode Representations in Optics (SPIE, 2006).

Data availability

No data were generated or analyzed in the presented research.

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)
Fig. 1.
Fig. 1. CGSM pulsed beam results: (a) theory and (b) simulation ${\rm Re}(\Gamma)$; (c) theory and (d) simulation ${\rm Im}(\Gamma)$; (e) $\Gamma ({t,t}) = \left\langle {I(t)} \right\rangle$ theory and simulation; and (f) ${P^2}$ versus CGSM realization number.

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. NUC pulsed beam results: (a) theory and (b) simulation ${\rm Re}(\Gamma)$; (c) theory and (d) simulation ${\rm Im}(\Gamma)$; (e) $\Gamma ({t,t}) = \left\langle {I(t)} \right\rangle$ theory and simulation; and (f) ${P^2}$ versus NUC realization number.

View in Article | Download Full Size | PDF

Equations (64)

Equations on this page are rendered with MathJax. Learn more.

P 2 = 2 σ t σ ω ,
σ t 2 = ∫ − ∞ ∞ ( t − ⟨ t ⟩ ) 2 Γ ( t , t ) d t ∫ − ∞ ∞ Γ ( t , t ) d t σ ω 2 = ∫ − ∞ ∞ ( ω − ⟨ ω ⟩ ) 2 W ( ω , ω ) d ω ∫ − ∞ ∞ W ( ω , ω ) d ω .
⟨ t ⟩ = ∫ − ∞ ∞ t Γ ( t , t ) d t ∫ − ∞ ∞ Γ ( t , t ) d t ⟨ ω ⟩ = ∫ − ∞ ∞ ω W ( ω , ω ) d ω ∫ − ∞ ∞ W ( ω , ω ) d ω .
W ( ω 1 , ω 2 ) = 1 ( 2 π ) 2 ∬ − ∞ ∞ Γ ( t 1 , t 2 ) exp [ j ( ω 1 t 1 − ω 2 t 2 ) ] d t 1 d t 2 , Γ ( t 1 , t 2 ) = ∬ − ∞ ∞ W ( ω 1 , ω 2 ) exp [ − j ( ω 1 t 1 − ω 2 t 2 ) ] d ω 1 d ω 2 .
⟨ ω ⟩ = j J ∫ − ∞ ∞ ∂ Γ ( t 1 , t 2 ) ∂ t 1 | t , t d t ,
⟨ ω 2 ⟩ = 1 J ∫ − ∞ ∞ ∂ 2 Γ ( t 1 , t 2 ) ∂ t 1 ∂ t 2 | t , t d t ,
J = ∫ − ∞ ∞ Γ ( t , t ) d t ,
Γ ( t 1 , t 2 ) = [ ⟨ I ( t 1 ) ⟩ ⟨ I ( t 2 ) ⟩ ] 1 / 2 γ ( t 1 , t 2 ) = i ( t 1 ) i ( t 2 ) γ ( t 1 , t 2 ) ,
⟨ ω ⟩ = j J ∫ − ∞ ∞ i ( t ) i ′ ( t ) d t + j J ∫ − ∞ ∞ [ i ( t ) ] 2 ∂ γ ( t 1 , t 2 ) ∂ t 1 | t , t d t ,
⟨ ω 2 ⟩ = 1 J ∫ − ∞ ∞ [ i ′ ( t ) ] 2 d t + 1 J ∫ − ∞ ∞ [ i ( t ) ] 2 ∂ 2 γ ( t 1 , t 2 ) ∂ t 1 ∂ t 2 | t , t d t + 2 J R e [ ∫ − ∞ ∞ i ( t ) i ′ ( t ) ∂ γ ( t 1 , t 2 ) ∂ t 1 | t , t d t ] ,
J = ∫ − ∞ ∞ [ i ( t ) ] 2 d t ,
∫ − ∞ ∞ i ( t ) i ′ ( t ) d t = 1 2 lim t → ∞ ⁡ { [ i ( t ) ] 2 − [ i ( − t ) ] 2 } .
∂ γ ( t 1 , t 2 ) ∂ t 1 | t , t = [ ∂ γ ( t 1 , t 2 ) ∂ t 2 | t , t ] ∗ ,
γ ( t 1 , t 2 ) = a ( t 1 , t 2 ) exp [ j ψ ( t 1 , t 2 ) ] ,
∂ γ ( t 1 , t 2 ) ∂ t 1 | t , t = ∂ a ( t 1 , t 2 ) ∂ t 1 | t , t + j ∂ ψ ( t 1 , t 2 ) ∂ t 1 | t , t ∂ 2 γ ( t 1 , t 2 ) ∂ t 1 ∂ t 2 | t , t = ∂ 2 a ( t 1 , t 2 ) ∂ t 1 ∂ t 2 | t , t + [ ∂ ψ ( t 1 , t 2 ) ∂ t 1 | t , t ] 2 + j ∂ 2 ψ ( t 1 , t 2 ) ∂ t 1 ∂ t 2 | t , t .
∂ a ( t 1 , t 2 ) ∂ t 1 | t , t = ∂ a ( t 1 , t 2 ) ∂ t 2 | t , t ∂ ψ ( t 1 , t 2 ) ∂ t 1 | t , t = − ∂ ψ ( t 1 , t 2 ) ∂ t 2 | t , t ,
⟨ ω ⟩ = j J ∫ − ∞ ∞ [ i ( t ) ] 2 ∂ a ( t 1 , t 2 ) ∂ t 1 | t , t d t − 1 J ∫ − ∞ ∞ [ i ( t ) ] 2 ∂ ψ ( t 1 , t 2 ) ∂ t 1 | t , t d t ,
⟨ ω 2 ⟩ = 1 J ∫ − ∞ ∞ [ i ′ ( t ) ] 2 d t + 2 J ∫ − ∞ ∞ i ( t ) i ′ ( t ) ∂ a ( t 1 , t 2 ) ∂ t 1 | t , t d t + 1 J ∫ − ∞ ∞ [ i ( t ) ] 2 ∂ 2 a ( t 1 , t 2 ) ∂ t 1 ∂ t 2 | t , t d t + 1 J ∫ − ∞ ∞ [ i ( t ) ] 2 [ ∂ ψ ( t 1 , t 2 ) ∂ t 1 | t , t ] 2 d t + j J ∫ − ∞ ∞ [ i ( t ) ] 2 ∂ 2 ψ ( t 1 , t 2 ) ∂ t 1 ∂ t 2 | t , t d t .
∫ − ∞ ∞ [ i ( t ) ] 2 ∂ a ( t 1 , t 2 ) ∂ t 1 | t , t d t = 0 , ∫ − ∞ ∞ [ i ( t ) ] 2 ∂ 2 ψ ( t 1 , t 2 ) ∂ t 1 ∂ t 2 | t , t d t = 0 .
∂ a ( t 1 , t 2 ) ∂ t 1 | t , t = 0 , ∂ 2 ψ ( t 1 , t 2 ) ∂ t 1 ∂ t 2 | t , t = 0 ,
σ ω 2 = 1 J ∫ − ∞ ∞ [ i ′ ( t ) ] 2 d t + 1 J ∫ − ∞ ∞ [ i ( t ) ] 2 ∂ 2 a ( t 1 , t 2 ) ∂ t 1 ∂ t 2 | t , t d t + 1 J ∫ − ∞ ∞ [ i ( t ) ] 2 [ ∂ ψ ( t 1 , t 2 ) ∂ t 1 | t , t ] 2 d t − { 1 J ∫ − ∞ ∞ [ i ( t ) ] 2 ∂ ψ ( t 1 , t 2 ) ∂ t 1 | t , t d t } 2 .
Γ ( t 1 , t 2 ) = f ( t 1 ) f ∗ ( t 2 ) exp [ − j ω c ( t 1 − t 2 ) ] ,
i ( t ) = | f ( t ) | = f ( t ) f ∗ ( t ) a ( t 1 , t 2 ) = 1 exp [ j ψ ( t 1 , t 2 ) ] = f ( t 1 ) f ( t 2 ) | f ( t 1 ) | | f ( t 2 ) | exp [ − j ω c ( t 1 − t 2 ) ] .
i ′ ( t ) = R e [ f ′ ( t ) f ∗ ( t ) ] | f ( t ) | ∂ ψ ( t 1 , t 2 ) ∂ t 1 | t , t = I m [ f ′ ( t ) f ∗ ( t ) ] | f ( t ) | 2 − ω c .
σ ω 2 = 1 J ∫ − ∞ ∞ | f ′ ( t ) | 2 d t − { I m [ 1 J ∫ − ∞ ∞ f ′ ( t ) f ∗ ( t ) d t ] } 2 .
Γ ( t 1 , t 2 ) = i ( t 1 ) i ( t 2 ) γ ( t 1 − t 2 ) = i ( t 1 ) i ( t 2 ) a ( t 1 − t 2 ) exp [ j ψ ( t 1 − t 2 ) ] .
σ ω 2 = 1 J ∫ − ∞ ∞ [ i ′ ( t ) ] 2 d t + ∂ 2 a ( t 1 , t 2 ) ∂ t 1 ∂ t 2 | t , t .
σ ω 2 = 1 J ∫ − ∞ ∞ [ i ′ ( t ) ] 2 d t − d 2 a ( τ ) d τ 2 | τ = 0 .
P 2 = ( P c o h ) 4 − 4 σ t 2 d 2 a ( τ ) d τ 2 | τ = 0 ,
Γ ( t 1 , t 2 ) = exp ( − t 1 2 4 W t 2 ) exp ( − t 2 2 4 W t 2 ) × cos [ n 2 π ( t 1 − t 2 ) δ ] exp [ − ( t 1 − t 2 ) 2 2 δ 2 ] × exp [ − j ω c ( t 1 − t 2 ) ] ,
i ( t ) = exp ⁡ ( − t 2 4 W t 2 ) γ ( τ ) = a ( τ ) exp [ j ψ ( τ ) ] = cos ( n 2 π τ δ ) exp ( − τ 2 2 δ 2 ) exp ( − j ω c τ ) .
P 2 = 1 + 4 W t 2 δ 2 ( 1 + 2 π n 2 ) .
Γ ( t 1 , t 2 ) = exp ( − t 1 2 4 W t 2 ) exp ( − t 2 2 4 W t 2 ) × exp ⁡ { − [ ( t 1 − t 0 ) 2 − ( t 2 − t 0 ) 2 ] 2 δ 4 } exp [ j α ( t 1 3 − t 2 3 ) ] × exp ⁡ { j β [ ( t 1 − t 0 ) 2 − ( t 2 − t 0 ) 2 ] } exp [ − j ω c ( t 1 − t 2 ) ] .
i ( t ) = exp ( − t 2 4 W t 2 ) , a ( t 1 , t 2 ) = exp ⁡ { − [ ( t 1 − t 0 ) 2 − ( t 2 − t 0 ) 2 ] 2 δ 4 } , ψ ( t 1 , t 2 ) = α ( t 1 3 − t 2 3 ) + β [ ( t 1 − t 0 ) 2 − ( t 2 − t 0 ) 2 ] − ω c ( t 1 − t 2 ) ,
σ ω 2 = 1 4 W t 2 { 1 + 8 W t 4 [ 9 α 2 W t 2 + 2 β 2 + 4 δ 4 ( 1 + t 0 2 W t 2 ) ] } .
P 2 = 1 + 8 W t 4 [ 9 α 2 W t 2 + 2 β 2 + 4 1 δ 4 ( 1 + t 0 2 W t 2 ) ] .
U ( t ) = ∫ − ∞ ∞ r ( v ) 1 2 p ( v ) H ( t , v ) d v ,
p ( v ) = δ 2 π cosh ( n 2 π δ v ) exp ( − δ 2 v 2 + 2 π n 2 2 ) H ( t , v ) = exp ( − t 2 4 W t 2 ) exp ( j v t ) ,
p ( v ) = δ 2 2 π exp [ − δ 4 4 ( v − β ) 2 ] H ( t , v ) = exp ( − t 2 4 W t 2 ) exp ( j α t 3 ) exp [ j ( t − t 0 ) 2 v ] .
Γ ( t 1 , t 2 ) = ∬ − ∞ ∞ W ( ω 1 , ω 2 ) exp [ − j ( ω 1 t 1 − ω 2 t 2 ) ] d ω 1 d ω 2 ,
∂ Γ ( t 1 , t 2 ) ∂ t 1 = − j ∬ − ∞ ∞ ω 1 W ( ω 1 , ω 2 ) × exp [ − j ( ω 1 t 1 − ω 2 t 2 ) ] d ω 1 d ω 2 .
1 ( 2 π ) 2 ∬ − ∞ ∞ ∂ Γ ( t 1 , t 2 ) ∂ t 1 exp [ j ( ω 1 ′ t 1 − ω 2 ′ t 2 ) ] d t 1 d t 2 = − j ( 2 π ) 2 ∬ − ∞ ∞ ω 1 W ( ω 1 , ω 2 ) ∬ − ∞ ∞ exp [ − j ( ω 1 − ω 1 ′ ) t 1 ] × exp [ j ( ω 2 − ω 2 ′ ) t 2 ] d t 1 d t 2 d ω 1 d ω 2 .
1 ( 2 π ) 2 ∬ − ∞ ∞ ∂ Γ ( t 1 , t 2 ) ∂ t 1 exp [ j ( ω 1 t 1 − ω 2 t 2 ) ] d t 1 d t 2 = − j ω 1 W ( ω 1 , ω 2 ) .
∫ − ∞ ∞ ω W ( ω , ω ) d ω = j ( 2 π ) 2 ∬ − ∞ ∞ ∂ Γ ( t 1 , t 2 ) ∂ t 1 ∫ − ∞ ∞ exp [ j ω ( t 1 − t 2 ) ] d ω d t 1 d t 2 .
∫ − ∞ ∞ ω W ( ω , ω ) d ω = j 2 π ∫ − ∞ ∞ ∂ Γ ( t 1 , t 2 ) ∂ t 1 | t , t d t .
W ( ω 1 , ω 2 ) = 1 ( 2 π ) 2 ∬ − ∞ ∞ Γ ( t 1 , t 2 ) exp [ j ( ω 1 t 1 − ω 2 t 2 ) ] d t 1 d t 2 ,
∫ − ∞ ∞ W ( ω , ω ) d ω = 1 ( 2 π ) 2 ∬ − ∞ ∞ Γ ( t 1 , t 2 ) × ∫ − ∞ ∞ exp [ j ω ( t 1 − t 2 ) ] d ω d t 1 d t 2 .
∫ − ∞ ∞ W ( ω , ω ) d ω = 1 2 π ∫ − ∞ ∞ Γ ( t , t ) d t .
∂ 2 Γ ( t 1 , t 2 ) ∂ t 1 ∂ t 2 = ∬ − ∞ ∞ ω 1 ω 2 W ( ω 1 , ω 2 ) × exp [ − j ( ω 1 t 1 − ω 2 t 2 ) ] d ω 1 d ω 2 .
ω 1 ω 2 W ( ω 1 , ω 2 ) = 1 ( 2 π ) 2 ∬ − ∞ ∞ ∂ 2 Γ ( t 1 , t 2 ) ∂ t 1 ∂ t 2 × exp [ j ( ω 1 t 1 − ω 2 t 2 ) ] d t 1 d t 2 .
∫ − ∞ ∞ ω 2 W ( ω , ω ) d ω = 1 2 π ∫ − ∞ ∞ ∂ 2 Γ ( t 1 , t 2 ) ∂ t 1 ∂ t 2 | t , t d t .
∂ Γ ( t 1 , t 2 ) ∂ t 1 | t , t = − j ∬ − ∞ ∞ ω 1 W ( ω 1 , ω 2 ) × exp [ − j ( ω 1 − ω 2 ) t ] d ω 1 d ω 2 .
∂ Γ ( t 1 , t 2 ) ∂ t 2 | t , t = j ∬ − ∞ ∞ ω 2 W ( ω 1 , ω 2 ) exp [ j ( ω 1 − ω 2 ) t ] d ω 1 d ω 2 .
[ ∂ Γ ( t 1 , t 2 ) ∂ t 2 | t , t ] ∗ = − j ∬ − ∞ ∞ ω 2 W ( ω 2 , ω 1 ) × exp [ − j ( ω 2 − ω 1 ) t ] d ω 2 d ω 1 .
∂ Γ ( t 1 , t 2 ) ∂ t 1 | t , t = [ ∂ Γ ( t 1 , t 2 ) ∂ t 2 | t , t ] ∗ .
Γ ( t 1 , t 2 ) = ∑ n λ n g n ( t 1 ) g n ∗ ( t 2 ) ,
∫ − ∞ ∞ Γ ( t 1 , t 2 ) g n ( t 2 ) d t 2 = λ n g n ( t 1 ) .
γ ( t 1 , t 2 ) = a ( t 1 , t 2 ) exp [ j ψ ( t 1 , t 2 ) ] = ∑ n λ n g n ( t 1 ) ∑ n λ n | g n ( t 1 ) | 2 g n ∗ ( t 2 ) ∑ n λ n | g n ( t 2 ) | 2 .
∂ γ ( t 1 , t 2 ) ∂ t 1 | t , t = ∂ a ( t 1 , t 2 ) ∂ t 1 | t , t + j ∂ ψ ( t 1 , t 2 ) ∂ t 1 | t , t = ∑ n λ n g n ∗ ( t ) ∑ n λ n | g n ( t ) | 2 [ d d t g n ( t ) ∑ n λ n | g n ( t ) | 2 ] .
d d t g n ( t ) ∑ n λ n | g n ( t ) | 2 = g n ′ ( t ) ∑ n λ n | g n ( t ) | 2 − g n ( t ) ∑ n λ n | g n ( t ) | 2 ∑ n λ n R e [ g n ′ ( t ) g n ∗ ( t ) ] ∑ n λ n | g n ( t ) | 2 .
∂ γ ( t 1 , t 2 ) ∂ t 1 | t , t = ∑ n λ n { g n ′ ( t ) g n ∗ ( t ) − R e [ g n ′ ( t ) g n ∗ ( t ) ] } ∑ n λ n | g n ( t ) | 2 = j ∑ n λ n I m [ g n ′ ( t ) g n ∗ ( t ) ] ∑ n λ n | g n ( t ) | 2 .
∂ a ( t 1 , t 2 ) ∂ t 1 | t , t = 0.
∂ 2 γ ( t 1 , t 2 ) ∂ t 1 ∂ t 2 | t , t = ∂ 2 a ( t 1 , t 2 ) ∂ t 1 ∂ t 2 | t , t + [ ∂ ψ ( t 1 , t 2 ) ∂ t 1 | t , t ] 2 + j ∂ 2 ψ ( t 1 , t 2 ) ∂ t 1 ∂ t 2 | t , t = ∑ n λ n [ d d t g n ∗ ( t ) ∑ n λ n | g n ( t ) | 2 ] × [ d d t g n ( t ) ∑ n λ n | g n ( t ) | 2 ] = ∑ n λ n | d d t g n ( t ) ∑ n λ n | g n ( t ) | 2 | 2 .
∂ 2 ψ ( t 1 , t 2 ) ∂ t 1 ∂ t 2 | t , t = 0.
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use