Theoretical analysis of the stretched optical pulse ripple and novel chirped pulse retrieving algorithm

Ivan Ulyanov; Ivan Ulyanov

doi:10.1364/OE.27.028166

1. Introduction

Nowadays a chirped pulse amplification (CPA) technique is the most straightforward way for the generation of ultrafast high peak power optical pulses [1,2]. A pulse stretcher is the key component in CPA schemes. It is used for broadening pulses in time domain prior to amplification by inducing a group delay dispersion (GDD) and therefore reducing its peak power. Thus, an adverse influence of various nonlinear effects is considerably reduced. However, it is not possible to eliminate absolutely certain nonlinear effects, such as e.g. self-phase modulation (SPM), which have no threshold and therefore affect pulses with any energy. A SPM effect is crucial in the fiber based CPA scheme implementations, where optical pulses propagate in elongated medium with small mode field diameters, and consequently pulses with rather moderate peak powers considerably suffer from SPM distortions [2]. Both output optical pulse quality and its peak power are dramatically reduced due to a group delay ripple (GDR) and an amplitude ripple conditioned by SPM [3,4]. Nevertheless, for many industrial applications it is very promising to develop ripple-free ultrafast high peak power fiber lasers, because such sources are very efficient, robust and easily assembled [5–8]. Eventually, the investigation of the origin of the ripples and its evolution inside CPA lasers is an important task.

The evolution of rippled optical pulses due to SPM inside a CPA laser can be computed by solving numerically a Nonlinear Schrödinger Equation (NLSE). This method works as a black-box computation, a certain rippled pulse is considered as the input data and the resulting pulse is simulated. This procedure cannot provide a physical insight into the phenomenon of the ripple increase due to SPM. Some meaningful questions can be answered only by mathematical analysis: why the spectrum amplitude ripple is changing with power scaling? Are the ripple on different frequencies sense the SPM effect the same way or not? These problems will be addressed in detail in the next sections. Several attempts were made before to explain the SPM effect from the point of view of spectral phase and amplitude ripples [9–11]. In a current paper the approach for the ripple description and analysis is different. We will start to investigate both amplitude and phase ripples of the pulse complex envelope in time domain. Later it will be demonstrated that such approach is natural because these two kinds of the ripple play the same role in the pulse quality degradation at low energies and the approach is fruitful for pulse retrieving problem.

First of all, the formula that establish the relation between the time domain and spectral domain ripples of a chirped pulse is derived. This formula can be interpreted as an application of the spectral filter with a particular shape that acts not in an optical spectrum domain, but in the domain of Fourier harmonics of the pulse spectrum amplitude functions. In detail this is described in the next section. After that a simple quantitative explanation of the evolution of the ripples in Kerr nonlinear medium is provided, influence of group velocity dispersion effect (GVD) is discussed. Moreover, such knowledge of the interrelation between amplitude and phase ripples in time and spectral domains together with the evolution of the ripples in a time domain reveals the solution of a pulse retrieving problem, relying only onto two spectra of the chirped pulse measured at the input of a Kerr nonlinear medium and at its output. According to my knowledge, such chirped pulse retrieving algorithm is so far unknown.

2. Time and spectral domains

A linearly polarized optical pulse propagating in a single mode fiber can be completely described by the carrier frequency value ${\omega _c}$ and the complex envelope function $A\left ( t \right )$. Equivalently in a frequency domain the pulse is characterized by the complex spectrum function $S\left ( \omega \right )$. Further we will use the words “shape“ or “amplitude“ for denoting the absolute values either of these functions. As a rule, the spectrum shape $\left | {S\left ( \omega \right )} \right |$ can be easily measured using conventional laboratory equipment, e.g. optical spectrum analyzer. Typically when measured the spectrum shape function is centered at a carrier frequency. Further we will consider the spectrum function $S\left ( \omega \right )$ of the pulsed laser radiation in reference to an optical carrier frequency, i.e. oscillations at the carrier frequency are excluded from analysis. Then the value $S\left ( 0 \right )$ stands for the carrier harmonic, and for the transform-limited pulse the complex envelope $A\left ( t \right )$ is a pure real function.

The current paper is motivated by the investigation of fiber chirped pulse amplification laser. There stretcher typically is implemented with the diffraction gratings pair, linearly chirped fiber Bragg grating, chirped volume Bragg grating or even a fiber coil. Nevertheless for major variety of implementations stretcher can be mathematically represented as on operator, inducing only a second order dispersion on an optical pulse. The reason is simple — second order dispersion most effectively stretches the pulse in time domain. Third and high order dispersion are most commonly a next order perturbation of the spectrum phase, but sometimes it is possible to obtain a fruitful trade-off between high order dispersion and nonlinear effects [12]. It is well known, that the spectrum amplitude of the laser pulse with a significant accumulated group delay dispersion matches its complex envelope amplitude in a time domain [13]. However, this approximation is correct only to a certain extent and mathematical analysis should be carried out in order to verify the limits of its validity.

Three parameters that characterize optical pulses $A\left ( t \right ),\;S\left ( \omega \right ),\;{D_2}$, where the last one is an accumulated GDD of the laser pulse (measured in ${ps^2}$), are interconnected by the inverse Fourier transform:

(1)$$\begin{aligned} A(t) &= {\cal F}_{\omega }^{ - 1}\left\{ {\left| {{S}\left( {\omega } \right)} \right| \exp \left( {\frac{{{i}{{D}_2}}}{2}{{\omega }^2}} \right)} \right\}\left( {t} \right) = \nonumber\\ &= \frac{1}{{\sqrt {2{\pi }} }}\mathop \int _{ - \infty }^{ + \infty } \left| {{S}\left( {\omega } \right)} \right|\exp\left( {\frac{{{i}{{D}_2}}}{2}\left( {{{\omega }^2} - \frac{{2{\omega t}}}{{{{D}_2}}}} \right)} \right)\mathrm{d}\omega = \nonumber\\ &= \frac{1}{{\sqrt {2{\pi }} }}\exp\left( { - \frac{{{i}{{t}^2}}}{{2{{D}_2}}}} \right)\mathop \int _{ - \infty }^{ + \infty } \left| {{S}\left( {\omega } \right)} \right|\exp\left( {\frac{{{i}{{D}_2}}}{2}{{\left( {{\omega } - \frac{{t}}{{{{D}_2}}}} \right)}^2}} \right)\mathrm{d}\omega . \end{aligned}$$

The formula is exact for pulses with flat spectral phase $\arg {S (\omega )} \equiv 0$ at $D_2 = 0$, what is rarely observed for real CPA lasers. Also Eq. (1) is asymptotically true for $D_2 \to \infty$, as spectral phase plays negligible role comparing to $\exp \left ( {{{{i}{{D}_2}}}{{\omega }^2}/2} \right )$ term. Quantitative criterion for validity of the Eq. (1) could be formulated using [14]:

(2)$$\left|{D_2}\right| \gg \frac {P^2}{{8 \pi^2 {\sigma_{\nu}}^2}},$$

where $P^2$ is the pulse quality factor (equals $1$ for ideal Gaussian pulse and about several units for typical observed pulses) and ${\sigma _{\nu }}^2$ is the spectrum shape second moment. The Eq. (2) is derived by analogy with optical beams, and means that pulse should be in the ”far-field”. For typical CPA laser estimation of $P^2 \sim 2$, $\sigma _{\nu } \sim 1\;THz$ gives $\left |{D_2}\right | \gg 0.025\;{ps}^2$. In real lasers typical value of $D_2$ used for sufficient pulse stretching is an order of tens of ${ps}^2$, which means that the approximation used for Eq. (1) is reasonable.

In Eq. (1) there is a Fresnel type integral, which can be approximated using the stationary phase method [15]. The main idea of this approximation is that due to a large absolute value of ${D_2}$ the exponent term under the integral rapidly oscillates for every frequency $\omega$ except the small interval near the stationary point ${\omega _s} = t/{D_2}$. An overall contribution of the oscillating intervals into the integration result is zero. Thus, the spectrum shape $\left | {S\left ( \omega \right )} \right |$ can be approximated by its value in the stationary point $\left | {S\left ( {{\omega _s}} \right )} \right |$ and the Eq. (1) can be rewritten as:

(3)$$\begin{aligned} A\left( {t} \right) &\approx \frac{1}{{\sqrt {2{\pi }} }}{\exp}\left( { - \frac{{{i}{{t}^2}}}{{2{{D}_2}}}} \right)\left| {{S}\left( {\frac{{t}}{{{{D}_2}}}} \right)} \right|\mathop {\int}\limits _{ - \infty }^{ + \infty } {\exp}\left( {\frac{{{i}{{D}_2}}}{2}{{\left( {{\omega } - \frac{{t}}{{{{D}_2}}}} \right)}^2}} \right){\mathrm{d}\omega } = \nonumber\\ &= \sqrt {\frac{{i}}{{{{D}_2}}}} {\exp}\left( { - \frac{{{i}{{t}^2}}}{{2{{D}_2}}}} \right)\left| {{S}\left( {\frac{{t}}{{{{D}_2}}}} \right)} \right|. \end{aligned}$$

As follows, the complex envelope $A\left ( t \right )$ should have a linear chirp and its amplitude should be proportional to the spectrum amplitude $\left | {S\left ( \omega \right )} \right |$ with the substituted argument $\omega \to t/{D_2}$.

Due to the fact that it is impossible to obtain infinitely large group delay dispersion parameter ${D_2}$ the stationary phase approximation is valid only for slowly varying spectrum amplitudes $\left | {S\left ( \omega \right )} \right |$ and, as a consequence, $A\left ( t \right )$ functions. For rapidly varying components of the complex envelope, which are called ripples, this approximation is very rough. Obviously, in a general case the ripples of the spectrum amplitude and the pulse complex envelope amplitude will not coincide. In the next sections the criterion of validity of the Eq. (3) will be introduced and the connection formula between the ripple shapes in time and frequency domains will be derived.

3. Amplitude ripples in a time domain

In this section long chirped pulses with a flat top will be considered in order to avoid dealing with edge effects. However, later it will be shown that the results obtained here are actually valid for the pulses with arbitrary shapes. It is also assumed that optical pulses have only an amplitude ripple of the complex envelope $A\left ( t \right )$ and the chirp of the pulse is strictly linear. It means that the phase-frequency characteristic of the pulse complex envelope is parabolic. In this case the formula that interconnects the pulse spectrum and its shape in a time domain can be written as follows:

(4)$${S}\left( {\omega } \right) = {{\cal F}_{t}}\left\{ {{{A}_0}\left( {1 + {a}\left( {t} \right)} \right)\exp \left( { - \frac{{{i}{{t}^2}}}{{2{{D}_2}}}} \right)} \right\}\left( {\omega } \right).$$

Here ${A_0}$ — an average amplitude of the pulse, $a\left ( t \right )$ — an amplitude ripple real function with zero mean value, $\left | {a\left ( t \right )} \right | \ll 1$. This function can be expanded in a Fourier series: the progression $\left \{ {{k_n},\;{a_n},\;{\phi _n}} \right \}$ represents spectral harmonics of the ripple with amplitudes ${a_n}$ and frequencies ${k_n}/\left ( {2\pi } \right )$ shifted by ${\phi _n}$ radians:

(5)$$\begin{aligned} {S}\left( {\omega } \right) &= {{\cal F}_{t}}\left\{ {{{A}_0}\left( {1 + \mathop \sum _{{n} = 1}^\infty {{a}_{n}}\sin \left( {{{k}_{n}}{t} + {\phi _{n}}} \right)} \right)\exp \left( { - \frac{{{i}{{t}^2}}}{{2{{D}_2}}}} \right)} \right\}\left( {\omega } \right) = \nonumber\\ &= \frac{{{{A}_0}}}{{\sqrt {{i}/{{D}_2}} }}{\exp \left({\frac{1}{2}{i}{{\omega }^2}{{D}_2}}\right)}{\bigg(}{ 1 + \mathop \sum _{{n} = 1}^\infty {i}\frac{{{{a}_{n}}}}{2}\bigg[ {{\exp}\left( { - {i}{\phi _{n}} + \frac{1}{2}{i}{{k}_{n}}\left({{{k}_{n}} - 2{\omega }} \right){{D}_2}} \right) -}} \nonumber\\ &\quad -{{{\exp}\left( {{i}{\phi _{n}} + \frac{1}{2}{i}{{k}_{n}}\left( {{{k}_{n}} + 2{\omega }} \right){{D}_2}} \right)} ] }\bigg) = \nonumber\\ &= \frac{{{{A}_0}}}{{\sqrt {{i}/{{D}_2}} }}{\exp \left({\frac{1}{2}{i}{{\omega }^2}{{D}_2}}\right)}\left( {1 + \mathop \sum _{{n} = 1}^\infty {{a}_{n}}\left( {{\cos}\frac{{{k}_{n}^2{{D}_2}}}{2} + {i\sin}\frac{{{k}_{n}^2{{D}_2}}}{2}} \right)\sin \left( {{{k}_{n}}{\omega }{{D}_2} + {\phi _{n}}} \right)} \right). \end{aligned}$$

In fact it is more correct to make a Fourier transform of the function $a\left ( t \right )$. However, as it does not affect the final result, for the sake of simplicity we use a Fourier series decomposition, which is also more convenient for numerical simulations. Utilizing the smallness of the coefficients ${a_n}$, the spectrum amplitude of the pulse can be expressed as:

(6)$$\left| {{S}\left( {\omega } \right)} \right| \approx {{A}_0}\sqrt {\left| {{{D}_2}} \right|} \sqrt { {1 + \mathop \sum _{{n} = 1}^\infty 2{{a}_{n}}\cos \frac{{{k}_{n}^2{{D}_2}}}{2}\sin \left( {{{k}_{n}}{\omega }{{D}_2} + {\phi _{n}}} \right)} } .$$

Once again, the square root can be expanded in a Taylor series leaving only a first order of ${a_n}$ coefficients due to its smallness.

(7)$$\left| {{S}\left( {\omega } \right)} \right| \approx {{A}_0}\sqrt {\left| {{{D}_2}} \right|} \left[ {1 + \mathop \sum _{{n} = 1}^\infty \left( {{{a}_{n}}\cos \frac{{{k}_{n}^2{{D}_2}}}{2}} \right)\sin \left( {{{k}_{n}}{\omega }{{D}_2} + {\phi _{n}}} \right)} \right].$$

Similarly an argument of the spectrum can be expressed as

(8)$$\arg S\left( \omega \right) \approx{-} \frac{\pi }{2} \operatorname{sign} {{D_2}} + \frac{1}{2}{\omega ^2}{D_2} + \mathop \sum _{n = 1}^\infty \left( {{a_n}\sin \frac{{k_n^2{D_2}}}{2}} \right)\sin \left( {{k_n}\omega {D_2} + {\phi _n}} \right).$$

Substituting $t/{D_2}$ for ${\omega }$ we shall obtain from the Eq. (7):

(9)$$\left| {{S}\left( {\frac{{t}}{{{{D}_2}}}} \right)} \right| \approx {{A}_0}\sqrt {\left| {{{D}_2}} \right|} \left[ {1 + \mathop \sum _{{n} = 1}^\infty \left( {{{a}_{n}}\cos \frac{{{k}_{n}^2{{D}_2}}}{2}} \right)\sin \left( {{{k}_{n}}{t} + {\phi _{n}}} \right)} \right].$$

A thorough insight of the Eq. (9) reveals that the sum is nothing else than a ripple function with some filter applied in a Fourier space. Each Fourier harmonic of the ripple is multiplied by the filter function ${f_{filter}}\left ( k \right ) = \cos \left ( {{k^2}{D_2}/2} \right )$.

Generalizing the result in Eq. (9) for the Fourier transform case, one can obtain that for the linearly chirped pulse with a rippled flat top its spectrum shape $\left | {S\left ( \omega \right )} \right |$ is connected with the pulse shape $\left | {A\left ( t \right )} \right |$ by the formula

(10)$${{\cal F}_{t}}\left\{ {\left| {{S}\left( {\frac{{t}}{{{{D}_2}}}} \right)} \right|} \right\}\left( {\omega } \right) = \sqrt {\left| {{{D}_2}} \right|} {{\cal F}_{t}}\left\{ {\left| {{A}\left( {t} \right)} \right|} \right\}\left( {\omega } \right)\cos \frac{{{{\omega }^2}{{D}_2}}}{2}.$$

Similarly for an argument of the spectrum

(11)$${{\cal F}_t}\left\{ {\arg S\left( {\frac{t}{{{D_2}}}} \right) + \frac{\pi }{2}\operatorname{sign}{{D_2}} - \frac{{{t^2}}}{{2{D_2}}}} \right\}\left( \omega \right) = {{\cal F}_t}\left\{ {\left| {A\left( t \right)} \right|} \right\}\left( \omega \right)\sin \frac{{{\omega ^2}{D_2}}}{2}.$$

The similar formulas were introduced earlier in the papers [10,11], although they were obtained using some other mathematical apparatus and assuming a pure amplitude or pure phase ripple of the pulse spectrum, not the pulse complex envelope ripple as presented in current paper.

The Eq. (10) can be interpreted in such a way that the spectral shape of the rippled pulse coincides with its complex envelope shape modified by the special spectral filter. Here the term “spectral“ does not imply an optical spectral domain, but the domain of the mathematical decomposition of the pulse complex envelope amplitude into harmonic functions.

From the Eq. (10) it also follows that the ripple harmonic components of the complex envelope amplitude with the time period $T \gg 2\sqrt {\pi {D_2}}$ coincide with the ripple harmonics of the spectrum amplitude. It means that the Eq. (10) is applicable for arbitrary shaped pulses, not only flat top ones, because the pulse shape itself can be interpreted as a low frequency amplitude ripple.

Using numerical simulations it is easy to verify the Eqs. (3) and (10), for this purpose the ripple function $a\left ( t \right )$ is needed. Here and elsewhere in this paper ripple function is numerically generated by the following way: random variable with uniform distribution $U\left ( -l, l \right )$ is sampled for every $0.24\;ps$, then this array is low-pass filtered (first-order FIR filter) with cut-off frequency $f_{co}\;[ps^{-1}]$. Thus a random ripple function has two generating parameters: $l$ and $f_{co}$. Note, that ripple function root mean square value or peak-to-peak value linearly depends on $l$ parameter.

The complex envelope shape and the spectrum shape of the rippled pulse are plotted in Fig. 1. Tanh-shaped pulse with pulse duration ${\Delta }{t_{FWHM}} = 500\;ps$, top amplitude ${A_0} = 1\;{W^{1/2}}$, group delay dispersion ${D_2} = - 40\;{ps^2}$, ripple generating parameters $l = 0.1,\; f_{co} = 0.033\;ps^{-1}$ was used for the simulation. It is clear that slowly varying parts of these two functions coincide, Fig. 1(a), according to the Eq. (3). On the contrary the ripple parts of these functions do not match each other, Fig. 1(b), which points that simple statement ”chirped pulse shape matches its spectrum shape” is not valid for rippled pulses.

Fig. 1. Comparison of the complex envelope shape and spectral shape of the tanh-shaped pulse with an amplitude ripple (arbitrary assigned). The yellow curve represents the result of the product $\sqrt {\left | {{{D}_2}} \right |} \left | {{A}\left ( {t} \right )} \right |$, and the blue curve is the spectrum shape with a substituted argument $\left | {S\left ( {t/{D_2}} \right )} \right |$ that was numerically calculated via Fast Fourier Transform. (a) — full scale plot, (b) — a magnified top section.

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The Eq. (10) is a more general relation then (3) that connects the rippled complex envelope shape and rippled spectrum shape of the pulse. The result of its application is demonstrated in Fig. 2. The thick yellow curve represents the complex envelope shape with the spectral filter applied $\sqrt {\left | {{D_2}} \right |} {\cal F}_\omega ^{ - 1}\left \{ {\cal F}\left \{ {\left | {A\left ( t \right )} \right |} \right \}\left ( \omega \right )\cos \left ( {{{\omega ^2}{D_2}}}/{2} \right ) \right \}\left ( t \right )$, and the blue one represents the spectrum shape with substituted argument $\left | {S\left ( {t/{D_2}} \right )} \right |$ that was numerically calculated via Fast Fourier Transform of $A\left ( t \right )$ similar to that shown in Fig. 1. It can be clearly seen that in contrast to Fig. 1(b) at this time two curves are in very good agreement, so that it confirms the validity of the relation (10).

Fig. 2. Modeling results obtained for the tanh-shaped pulse with an amplitude ripple (arbitrary assigned). The thick yellow curve is the pulse shape with the spectral filter applied, and the blue curve is the spectrum shape with substituted argument $\left | {S\left ( {t/{D_2}} \right )} \right |$ that is numerically calculated via Fast Fourier Transform. (a) — a full scale plot of the pulse shape, (b) — a magnified top of the pulse shape.

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For producing Figs. 1 and 2 the ripple generating parameters were $l = 0.1,\; f_{co} = 0.033\;ps^{-1}$, which stands for a rather small ripple of $\sim 3\%$ in respect to the average amplitude ${A_0}$. In this case the assumptions made for the derivation of the Eq. (10) were satisfied, as an excellent match is observed between yellow and blue curves. However, this formula still works well for much greater ripples. For instance, on Figs. 3(a) and 3(b) the same ripple function as for the Fig. 2(b) is scaled to $\sim 60\%$ and $\sim 150\%$ in respect to the average amplitude respectively. It is clear that Eq. (10) continues to provide good qualitative agreement between the pulse shape with the spectral filter applied, and the spectrum shape with substituted argument. If not presented by a thick line, one could only find a difference between this two curves on Fig. 3 at their peaks and depths.

Fig. 3. Validation of the Eq. (10) for the greater ripple function. (a) — ripple function is $\sim 60\%$ in respect to the average amplitude, (b) — $\sim 150\%$.

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In order to estimate the errors of approximation, used for derivation of the Eq. (10), it should be investigated for the variety of random ripple functions. Then let us define a dimensionless error functional as

(12)$$err = \sqrt {\frac{1}{ {\left| {{D_2}} \right|}{A_0^2{T_0}}} \mathop{\int}\limits _{ - {T_0}/2}^{{T_0}/2} {{\left( {{f_1}\left( t \right) - {f_2}\left( t \right)} \right)}^2}\mathrm{d}t}.$$

Substituting ${f_1}\left ( t \right )$ for $\sqrt {\left | {{D_2}} \right |} {\cal F}_\omega ^{ - 1}\left \{ {\cal F}\left \{ {\left | {A\left ( t \right )} \right |} \right \}\left ( \omega \right )\cos \left ( {{{\omega ^2}{D_2}}}/{2} \right ) \right \}\left ( t \right )$ and ${f_2}\left ( t \right )$ for $\left | {S\left ( {t/{D_2}} \right )} \right |$, the $err$ functional obtains the physical meaning of the relative root mean square error between the complex envelope shape with the spectral filter applied and the spectrum shape with substituted argument.

The Monte Carlo method was used to find the expected value and the variance of the $err$ functional for various ripple parameters $l \in [0,5]$, $f_{co} = \{ 0.033\;ps^{-1}, 0.1\;ps^{-1}\}$ ($l$ parameters larger than $5$ makes no sense, as the ripple becomes greater then $A_0$). The pulse was again tanh-shaped with pulse duration ${\Delta }{t_{FWHM}} = 500\;ps$, top amplitude ${A_0} = 1\;{W^{1/2}}$, group delay dispersion ${D_2} = - 40\;{ps^2}$. The results are presented on a Fig. 4.

Fig. 4. Monte Carlo testing of the Eq. (10). $err$ functional dependence on $l$ parameter is plotted for $f_{co} = 0.033\;ps^{-1}$ — blue curve and $f_{co} = 0.1\;ps^{-1}$ — yellow curve.

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For the $f_{co} = 0.033\;ps^{-1}$ blue curve lies below $0.05$, which means that we could expect maximum $5\%$ root mean square error when applying the Eq. (10) for pulses with such low frequency ripple. For the $f_{co} = 0.1\;ps^{-1}$ the error is worse, but taking in account that increasing $x$ times $f_{co}$ also translates in ripple function RMS growth in $\sqrt {x}$ times, as if it is generated by $\sqrt {x} l$ uniform distribution parameter, one can find, that high frequency ripple practically makes no income to the $err$. In other words frequency of the ripple does not influence the accuracy of the Eq. (10), only root mean square value does.

Obviously, the relation (10) can be also used for the solution of an inverse problem. For example, if it is known that there is only an amplitude ripple of the pulse complex envelope function, then this function can be characterized simply by measuring the spectrum amplitude of the pulse.

Eventually, the following criterion of validity of the Eq. (3) can be formulated: if the Fourier image of the spectrum amplitude of the pulse ${{\cal F}_t}\left \{ {\left | {S\left ( {t/{D_2}} \right )} \right |} \right \}\left ( \omega \right )$ only comprises low frequency harmonics that satisfy the condition $\left | \omega \right | \ll \sqrt {\pi /{D_2}}$, then its complex envelope amplitude $\left | {{A}\left ( {t} \right )} \right |$ can be obtained using the Eq. (3) and the pulse chirp is quite accurately linear.

4. Phase ripples in a time domain

The considerations similar to those introduced in the previous section can be carried out for the pulses with the strictly flat top amplitude ${A_0}$, but with a perturbed linear chirp of the complex envelope, i.e. with the phase ripple of the complex envelope function:

(13)$${A}\left( {t} \right) = {{A}_0}\exp \left( { - \frac{{{i}{{t}^2}}}{{2{{D}_2}}} + {i}{{\varphi }_{{ripple}}}\left( {t} \right)} \right),$$

where ${\varphi _{ripple}}\left ( t \right )$ — is the ripple of the parabolic phase function of the complex envelope argument. An influence of such phase ripple on the pulse spectrum amplitude can be expressed by the following formula

(14)$${{\cal F}_{t}}\left\{ {{{\left| {S} \right|}_{{ripple}}}\left( {\frac{{t}}{{{{D}_2}}}} \right)} \right\}\left( {\omega } \right) ={-} {{\cal F}_{t}}\left\{ {{{\varphi }_{{ripple}}}\left( {t} \right)} \right\}\left( {\omega } \right)\sin \frac{{{{\omega }^2}{{D}_2}}}{2},$$

It was assumed that the spectrum amplitude $\left | {S\left ( \omega \right )} \right |$ can be represented as a sum of the slowly varying part ${\left | S \right |_{slow}}\left ( \omega \right )$ and the fast one ${\left | S \right |_{ripple}}\left ( \omega \right )$ conditioned by the high frequency ripple:

(15)$$\left| {{S}\left( {\omega } \right)} \right| \equiv {\left| {S} \right|_{{ripple}}}\left( {\omega } \right) + {\left| {S} \right|_{{slow}}}\left( {\omega } \right).$$

In a similar fashion to the previously considered case of an amplitude ripple, Fig. 2, we can compare the spectrally filtered phase ripple function ${\cal F}_\omega ^{ - 1}\left \{ {{\cal F}_t}\left \{ {{\varphi _{ripple}}\left ( t \right )} \right \}\left ( \omega \right )\sin \left ({{\omega ^2}{D_2}}/{2}\right ) \right \}\left ( t \right )$ with the spectrum amplitude function ${\left | S \right |_{ripple}}\left ( {{t}/{{{D_2}}}} \right )$. Both functions are plotted in Fig. 5. Once again, the tanh-shaped pulse was used for the simulations, the pulse duration ${\Delta }{t_{FWHM}} = 500\;ps,\;{A_0} = 1\;{W^{1/2}},\;{D_2} = - 40\;p{s^2}$, the phase ripple was arbitrary assigned.

Fig. 5. Modeling results obtained for the tanh-shaped pulse with the phase ripple (arbitrary assigned). The thick yellow curve is the phase ripple function with the spectral filter applied, and the blue one is the ripple part of the spectrum amplitude function with the substituted argument ${\left | S \right |_{ripple}}\left ( {t/{D_2}} \right )$.

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Consequently, the Eq. (14) can be used for the characterization of the pulse complex envelope function simply relying on the measured spectrum amplitude, providing that the pulse complex envelope has only a phase ripple.

It should be noticed, that there is an ambiguity in the definition of the function ${\left | S \right |_{ripple}}\left ( \omega \right )$. There are numerous ways for splitting the pulse spectrum amplitude into slowly and rapidly varying parts. Here the following recommendation can be introduced. As it is known from the previous chapter, the spectrum amplitude harmonics with frequencies $\left | \omega \right | \ll \sqrt {\pi /{D_2}}$ are directly transformed into the complex envelope amplitude harmonics of the pulse. According to this fact we can conventionally refer harmonics with frequencies $\left | \omega \right | \le {1}/{2}\sqrt {\pi /{D_2}}$ to the slowly varying part ${\left | S \right |_{slow}}\left ( \omega \right )$ and, consequently, those with frequencies $\left | \omega \right | > {1}/{2}\sqrt {\pi /{D_2}}$ to the ripple part ${\left | S \right |_{ripple}}\left ( \omega \right )$. Therefore a natural way to extract the rippled part of the spectrum amplitude $\left | {S\left ( {t/{D_2}} \right )} \right |$ is to apply a high-pass filter with the cut-off frequency that is equal to ${1}/{2}\sqrt {\pi /{D_2}}$.

Provided recommendations do not completely eliminate ambiguity, as there are multiple implementations of high pass filters and a cut-off frequency itself was chosen rather frivolously. In practice this ambiguity leads to the discrepancy between the spectrally filtered phase ripple function and the spectrum amplitude ripple function ${\left | S \right |_{ripple}}\left ( {{t}/{{{D_2}}}} \right )$ in those regions, where the spectrum shape changes drastically, e.g. the edges of a tanh-shaped pulse. There is no obvious solution for this problem, because it originates from the fundamental question of how to extract an original signal from the distorted (rippled) measurement data.

5. Evolution of the ripple

In the current section we will investigate ripple evolution in a Kerr nonlinear medium. In order to obtain the propagation equation for the ripple, we shall start from the propagation equation for an optical pulse — Nonlinear Schrödinger Equation. The ripple evolution, discussed in this section, is interesting only as basement for development of the pulse retrieving problem algorithm, where group velocity dispersion effect is unlikely and adverse. Later in this section we will discuss the quantitative parameter showing the influence of the GVD on the ripple, and the approach for the minimization of GVD effect will be presented. Thus in the current section we will consider only a self-phase modulation effect occurring during the propagation of the linearly polarized optical pulse with the complex envelope $A\left ( {t,z} \right )$ along the z-axis. Then the NLSE is simplified to the following propagation equation

(16)$$i\frac{{\partial A\left( {t,z} \right)}}{{\partial z}} ={-} \gamma {\left| {A\left( {t,z} \right)} \right|^2}A\left( {t,z} \right).$$

Here $\gamma$ is the nonlinear parameter of the fiber. It is obvious, that the function $A\left ( {t,z} \right ) = A\left ( {t,0} \right )\exp \left ( {i\gamma {{\left | {A\left ( {t,0} \right )} \right |}^2}z} \right )$ is the solution of the Eq. (16). Let us represent the pulse complex envelope using two complex functions: slowly varying in a time domain ${A_{slow}}\left ( t \right )$ and rapidly varying (ripple)$r\left ( {t,z} \right )$. Then the ansatz for the pulse complex envelope can be written as:

(17)$$A\left( {t,z} \right) = {A_{slow}}\left( t \right)\left( {1 + r\left( {t,z} \right)} \right)\exp \left( {i\gamma {{\left| {{A_{slow}}\left( t \right)} \right|}^2}z} \right).$$

According to the previous section of this paper the complex function ${A_{slow}}\left ( t \right )$ can be obtained from $A\left ( {t,z} \right )$ by its low-pass filtering with the cut-off frequency $\omega = {1}/{2}\sqrt {\pi /{D_2}}$. It should be stressed that ${A_{slow}}\left ( t \right )$ is assumed to be independent of $z$ coordinate. The exponential term is responsible for the spectrum shape broadening, i.e. due to a self-phase modulation effect the edges of ${A_{slow}}\left ( t \right )$ obtain a nonlinear chirp with distance. The arguments of the functions $A\left ( {t,z} \right ),\;{A_{slow}}\left ( t \right ),\;r\left ( {t,z} \right )$ will be omitted further for the sake of brevity. In order to understand a physical meaning of the function $r$, the complex envelope of the pulse should be rewritten in an exponential form:

(18)$$\begin{aligned} A &= {A_{slow}}\sqrt {{{\left( {1 + \operatorname{Re} r} \right)}^2} + {{\left( \operatorname{Im} r \right)}^2}} \exp \left( {i\arctan \frac{{\operatorname{Im}r}}{{1 + \operatorname{Re}r}} + i\gamma {{\left| {{A_{slow}}} \right|}^2}z} \right) \approx \nonumber\\ &\approx {A_{slow}}\left( {1 + \operatorname{Re} r} \right)\exp \left( {i\operatorname{Im}r + i\gamma {{\left| {{A_{slow}}} \right|}^2}z} \right),\;\;\forall t,\forall z:\left| r \right| \ll 1 \end{aligned}$$

In the first order approximation, when $\left | r \right | \ll 1$, the real part of the function $\operatorname {Re}r$ is responsible for an amplitude ripple, and its imaginary part $\operatorname {Im}r$ is connected with a phase ripple. The evolution equation for the $r$ function can be obtained by substituting the ansatz (17) into the propagation Eq. (16) and linearizing it with respect to $r$:

(19)$${{i}}\frac{{\partial {{r}}}}{{\partial {{z}}}} ={-} {{\gamma }}{\left| {{{{A}}_{{{slow}}}}} \right|^2}\left( {{{r}} + {{{r}}^{{*}}}} \right).$$

An Eq. (19) is just a reduced version of the Eq. (5.1.4) in [16] without GVD term. Basing on the theory described in [16] we can estimate effect of the group velocity dispersion on the ripple evolution. There is derived, that harmonic wave ripple function $r\left ( {t,z} \right )$ starts to propagate with phase velocity according to dispersion relation (5.1.7) [16] in the presence of GVD. Thus after propagating the length $L$ of Kerr nonlinear and dispersive medium the harmonic ripple function with angular frequency $\omega$ obtains a phase shift

(20)$$\theta = \frac{1}{2}\left| {{\beta _2}} \right|\omega L\sqrt {{\omega ^2} + \frac{{4\gamma A_0^2}}{{{\beta _2}}}\;} ,$$

where ${\beta _2}$ is the group velocity dispersion parameter of the fiber. For estimating the typical phase shift $\theta$ an amplifier stage of the fiber CPA laser could be a representative example of such medium, where stretched pulses are propagating. Substituting typical values for the fiber ${\beta _2} \sim 20\;{ps^2}/km,\;\gamma \sim 1\;{W^{ - 1}}/km,\;L \sim 1\;m$ and for the $10\;\mu J$ pulse with $500\;ps$ duration typical ripple parameters $\omega \sim 0.5\;{ps^{ - 1}},\;A_0^2 \sim 20\;kW$ we obtain $\theta \sim 0.3 \ll 2\pi$. It means that harmonic ripple with time period $2\pi /\omega = 12.6\;ps$ will shift by $0.6\;ps$ after propagating through the amplifier. In this case comparison of phase shift with $2 \pi$ points that the ripple phase shift due to the group velocity dispersion is small but one cannot stand, that it is negligible.

GVD effect can be eliminated from pulse propagation by increasing the effective mode area of the radiation inside the Kerr nonlinear medium, as $\gamma$ is inversely proportional to effective mode area. For the bulk medium it can be done by collimating the beam in larger diameter.

Using the fact, that the function in parentheses $r + {r^*} = 2\operatorname {Re}r$ in the Eq. (19) is the integral of motion, the solution of this differential equation is:

(21)$${{r}} = {{i\gamma }}{\left| {{{{A}}_{{{slow}}}}} \right|^2}\left( {{{{r}}_{{{in}}}} + {{r}}_{{{in}}}^{{*}}} \right){{z}} + {{{r}}_{{{in}}}} = {{i}}2{{\gamma }}{\left| {{{{A}}_{{{slow}}}}} \right|^2}{{z}}\operatorname{Re} {{r}}_{{{in}}} + {{{r}}_{{{in}}}},{{\;\;}}{{{r}}_{{{in}}}} = {{r}}\left( {{{t}},0} \right).$$

Rather obvious though very important conclusions can be made considering the solution (21). If the complex envelope function of the pulse has only a phase ripple at the medium input $(z=0)$, i.e. $\operatorname {Re}{r_{in}} \equiv 0,\operatorname {Im}{r_{in}} \ll 1$, then the pulse will propagate through the nonlinear medium without any change. In the case of a pure amplitude ripple, i.e. $\operatorname {Im}{r_{in}} \equiv 0,\operatorname {Re}{r_{in}} \ll 1$, the spectrum shape ripple will change during its propagation inside the medium, or, equivalently, with the input pulse energy scaling. In this case at $z = 0$ only an amplitude ripple exists and for $z > 0$ both amplitude and phase ripples of the complex envelope occur, as both real and imaginary parts of the function $r$ are nonzero. In accordance with the Eqs. (10) and (14) these two kinds of ripple affect the pulse spectrum shape in different ways, amplitude-like and phase-like. A conjecture can be made that at large distances $z$ the phase ripple will prevail and the ripple of the pulse spectrum will just increase, while its overall shape will be maintained phase-like. However, in such cases the initial assumptions made for the Eq. (21) may be invalid, so that this conjecture should be accurately verified. That is why for a pure amplitude ripple at the medium input the spectrum shape ripple will transfer from an amplitude-like to a phase-like, changing its shape.

6. Pulse retrieval

As it was introduced above, the amplitude ripple of an optical pulse complex envelope leads to the ripple in its spectrum amplitude according to the Eq. (10). In turn, the phase ripple of the pulse complex envelope should be treated using the Eq. (14). In general, when the ripple of the pulse is experimentally observed using a spectrum analyzer, it is impossible to clarify the ripple origin and, consequently, to determine the pulse complex envelope in a time domain. Nevertheless, the theoretical consideration introduced below reveal that the complex envelope of an arbitrary rippled chirped pulse can be retrieved using a spectrum analyzer and a Kerr nonlinear medium. According to the Eq. (21) after the transmission through the Kerr nonlinear medium of the $z$ meters length the difference between two measured spectrum amplitudes can be expressed as:

(22)$$\begin{aligned} \left| {{{S}}\left( {{{\omega }},{{z}}} \right)} \right| &- \left| {{{S}}\left( {{{\omega }},0} \right)} \right| = \nonumber\\ &\quad= \left| {{{\cal F}_{{t}}}\left\{ {{{{A}}_{{{slow}}}}\left( {1 + {{i}}2{{\gamma }}{{\left| {{{{A}}_{{{slow}}}}} \right|}^2}z\operatorname{Re}{{r}}_{{{in}}} + {{{r}}_{{{in}}}}} \right)\exp \left( {{{i\gamma }}{{\left| {{{{A}}_{{{slow}}}}} \right|}^2}{{z}}} \right)} \right\}\left( {{\omega }} \right)} \right| - \nonumber\\ &\quad\quad-\left| {{{\cal F}_{{t}}}\left\{ {{{{A}}_{{{slow}}}}\left( {1 + {{{r}}_{{{in}}}}} \right)} \right\}\left( {{\omega }} \right)} \right|. \end{aligned}$$

Further we will neglect the exponential term $\exp \left ( {i\gamma {{\left | {{A_{slow}}} \right |}^2}z} \right )$, which describes the self-phase modulation of the whole pulse and the adjacent spectral broadening of the pulse edges due to a nonlinear chirping, and it is not connected somehow with the ripple. In this paper we are concentrating on a ripple behavior during the pulse propagation, and as a rule for ultrafast pulses a spectrum amplitude ripple grows much earlier compared to any spectral broadening. That is why in order to remain the relation (22) still valid and at the same time to neglect the exponential term on the right hand side of it we should properly apodize the spectrum amplitude difference on the left hand side.

Eliminating the exponent, using the linearity of Fourier transform and the approximation $\left | {\alpha + \beta } \right | \approx \left | \alpha \right | + \left | \beta \right |\cos \left ( {\arg \alpha - \arg \beta } \right )$, we can rewrite the Eq. (22) as follows:

(23)$$\begin{aligned} \left| {{{S}}\left( {{{\omega }},{{z}}} \right)} \right| - \left| {{{S}}\left( {{{\omega }},0} \right)} \right| &\approx \left| {{{{c}}_1}} \right| - \left| {{{{c}}_3}} \right| + \left| {{{{c}}_2}} \right|\left[ {\cos \left( {\arg {{{c}}_1} - \arg {{{c}}_2}} \right) - \cos \left( {\arg {{{c}}_3} - \arg {{{c}}_2}} \right)} \right], \nonumber\\ {c_1} &= {{\cal F}_t}\left\{ {{A_{slow}}\left( {1 + i2\gamma {{\left| {{A_{slow}}} \right|}^2}z\operatorname{Re}{r_{in}}} \right)} \right\}\left( \omega \right), \nonumber\\ {c_2} &= {{\cal F}_t}\left\{ {{A_{slow}}{r_{in}}} \right\}\left( \omega \right), \nonumber\\ {c_3} &= {{\cal F}_t}\left\{ {{A_{slow}}} \right\}\left( \omega \right). \end{aligned}$$

It can be seen from the Eq. (8), that the argument of the rippled spectrum function is represented mainly by a greatly varying parabolic function (typically an argument of the spectrum function can change for thousands of radians from one part of the spectrum to another) and by some ripple, which is negligibly small. That is why we can consider that $\arg {c_1} \approx \arg {c_3}$. Taking into account that in this case $\cos \left ( {\arg {c_1} - \arg {c_2}} \right ) - \cos \left ( {\arg {c_3} - \arg {c_2}} \right ) \approx 0$ we will finally obtain

(24)$$\begin{aligned} \left| {S\left( {\omega ,z} \right)} \right| &- \left| {S\left( {\omega ,0} \right)} \right| \approx \nonumber\\ &\quad\quad\left| {{{\cal F}_t}\left\{ {{A_{slow}}\left( t \right)\left( {1 + i2\gamma {{\left| {{A_{slow}}} \right|}^2}z\operatorname{Re}{r_{in}}} \right)} \right\}\left( \omega \right)} \right| - \left| {{{\cal F}_t}\left\{ {{A_{slow}}\left( t \right)} \right\}\left( \omega \right)} \right|. \end{aligned}$$

Profound look at the Eq. (24) reveals that its right hand side is just a spectrum amplitude of the phase modulated pulse complex envelope with the slowly varying part excluded, i.e. the function ${\left | S \right |_{ripple}}\left ( \omega \right )$. According to the Eqs. (14) and (15) for a linearly chirped pulse the difference between its spectrum amplitudes measured at the nonlinear medium output and input follows the formula:

(25)$${{\cal F}_t}\left\{ {\left| {S\left( {\frac{t}{{{D_2}}},z} \right)} \right| - \left| {S\left( {\frac{t}{{{D_2}}},0} \right)} \right|} \right\}\left( \omega \right) \approx{-} \sqrt {\left| {{D_2}} \right|} {{\cal F}_t}\left\{ {2\gamma {{\left| {{A_{slow}}} \right|}^2}z\operatorname{Re}{r_{in}}} \right\}\left( \omega \right)\sin \frac{{{\omega ^2}{D_2}}}{2}.$$

The Eq. (25) has a lot in common with the Eq. (14). This is natural, as the SPM effect induces only a phase modulation of the pulse complex envelope. After dividing both parts of the last equation by the sine function and performing an inverse Fourier transform, we shall obtain the following expression for the $\operatorname {Re}{r_{in}}$:

(26)$$\operatorname{Re}{r_{in}} \approx{-} \frac{1}{{2\sqrt {\left| {{D_2}} \right|} \gamma {{\left| {{A_{slow}}} \right|}^2}z}}{\cal F}_\omega ^{ - 1}\left\{ {{{\cal F}_t}\left\{ {\left| {S\left( {\frac{t}{{{D_2}}},z} \right)} \right| - \left| {S\left( {\frac{t}{{{D_2}}},0} \right)} \right|} \right\}\left( \omega \right)/\sin \frac{{{\omega ^2}{D_2}}}{2}} \right\}\left( t \right).$$

Certain assumptions should be made for the transition from the Eq. (25) to the Eq. (26). First of all, the function on the left hand side of the Eq. (25) should be zero for the frequencies which correspond to zeros of the sine function $\sin ({\omega ^2}{D_2}/2)$. When the spectrum amplitudes are measured experimentally many different factors such as the noise, environmental disturbances and other disregarded physical effects (e.g. group delay dispersion) can violate this requirement. This is the key point of the retrieving procedure, as it requires experiments to be carried out very carefully. For example, in order to apply the developed theory to pulse retrieving problem one should first provide an estimation of the ripple shift $\theta$ due to GVD effect. If $\theta$ is comparable with $\pi$, it could be reduced by increasing the effective mode area of the radiation inside the Kerr nonlinear medium. Also special attention should be payed to sufficient resolution of measured spectrum amplitudes. Poor resolution could induce DFT aliasing defect, when calculating the right hand side of the Eq. (26) on the computer.

Zero values of the sine function in (25) formally mean that the infinitely countable discrete set of values ${{\cal F}_t}\left \{ {{{\left | {{A_{slow}}} \right |}^2}\operatorname {Re}{r_{in}}} \right \}\left ( {{\omega _z}} \right )$ is undetermined. Fortunately we can recover the ${{\cal F}_t}\left \{ {{{\left | {{A_{slow}}} \right |}^2}\operatorname {Re}{r_{in}}} \right \}\left ( \omega \right )$ function using its continuity property. For example, ${{\cal F}_t}\left \{ {{{\left | {{A_{slow}}} \right |}^2}\operatorname {Re}{r_{in}}} \right \}\left ( 0 \right )$ is undetermined, it means that we cannot find a constant part of $\operatorname {Re}{r_{in}}$ using the Eq. (26). However, this part is zero by the definition of a ripple. This issue means that in practice an inaccuracy of deriving the ${{\cal F}_t}\left \{ {{{\left | {{A_{slow}}} \right |}^2}\operatorname {Re}{r_{in}}} \right \}\left ( \omega \right )$ function is greater around ${\omega _z}$ arguments.

Naturally, the Eq.s (25) and (26) are valid for a small ripple. As the phase ripple tends to grow with the pulse energy and the length of the medium (see the Eq. (21)), in experiments one should try to minimize these gaining factors of the ripple. In addition, the Eq. (26) makes sense only for the time intervals corresponding to a nonzero pulse amplitude, as the division by ${\left | {{A_{slow}}} \right |^2}$ takes place.

A verification of the Eq. (26) was conducted using numerical simulations for the pulse with the arbitrary assigned amplitude and phase ripple ${r_{in}}$ of the complex envelope $A\left ( {t,0} \right )$. The $A\left ( {t,z} \right )$ function obtained as a numerical solution of the Eq. (16) was used for the calculation of the $\left | {S\left ( {{t}/{{{D_2}}},z} \right )} \right |$ function via Fast Fourier Transform. Modeling results are introduced in Fig. 6, where two curves representing the left hand side and the right hand side of the Eq. (26) are plotted together, $\gamma = 1.25\;W/km,\;z = 0.25\;km$. Except the constant value ambiguity and the edge effects, discussed earlier in this section, it can be seen, that a good matching is observed between two curves. This fact actually proves the validity of the Eq. (26).

Fig. 6. Retrieving of the initial amplitude ripple of the pulse. The blue curve is the initially assumed amplitude ripple function $\operatorname {Re}{r_{in}}$ and the yellow curve is the retrieved amplitude ripple calculated according to the Eq. (26) using a difference of the two spectrum shapes of the pulse. Constant value mismatch and the edge effects are discussed earlier in this section.

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It is clear, that the right-hand side of the Eq. (26) comprises parameters that can be measured experimentally. As the low-frequency components of the spectrum amplitude $\left | {S\left ( \omega \right )} \right |$ and the pulse amplitude $\left | {{A_{slow}}\left ( t \right )} \right |$ coincide, according to the Eq. (3), then the input amplitude of the chirped pulse complex envelope follows the formula

(27)$$\left| {{{A}}\left( {{t}} \right)} \right| = \left| {{{{A}}_{{{slow}}}}} \right|\left( {1 + \operatorname{Re}{{{r}}_{{{in}}}}} \right).$$

As a result, in a temporal domain the function $\operatorname {Re}{r_{in}}$ can be retrieved relying only onto the measured spectra of the pulse at the input and the output of the Kerr nonlinear medium. In the situation when the pulse amplitude $\left | {A\left ( t \right )} \right |$ is calculated using the formula (27) and the spectrum amplitude $\left | {S\left ( \omega \right )} \right |$ is measured, the phases of both the pulse complex envelope and its spectrum functions can be retrieved using, for example, well known Gershberg-Saxton algorithm or one of its modifications [17]. It means that a pulse retrieving problem can be ultimately solved.

The solvability of the pulse retrieving problem using only two spectra of the pulse measured before and after the transmission through a Kerr nonlinear medium was demonstrated earlier using a complex iterative algorithm [18–20] or even a cross-phase modulation [21]. Especially it should be noted, that in [18] it is shown, that group velocity dispersion impact is small for the pulse retrieving procedure, and it also could be even more reduced by the proper choice of Kerr nonlinear medium. However, an analytic approach for the pulse retrieving problem, which is introduced in the present paper, was made for the first time, as a result two simple Eqs. (26) and (27) were derived.

7. Conclusions

A theoretical analysis of the chirped optical pulse amplitude and phase ripples was performed. It was demonstrated that an amplitude ripple of the pulse complex envelope leads to the ripple of its spectrum amplitude. The interrelation between these ripples is described by the Eq. (10). In turn, a pure phase ripple of the pulse complex envelope also induces the spectrum amplitude ripple that is described by the Eq. (14). An evolution of the amplitude and the phase ripples of the chirped pulse in Kerr nonlinear medium was considered. As a result, for the first time to my knowledge an experimentally simple retrieving procedure of the linearly chirped pulse complex envelope was analytically described. The facilities required for the pulse retrieval are a spectrum analyzer and the Kerr nonlinear medium, which induces only a self-phase modulation of the transmitted pulse. The absolute value of the pulse complex envelope can be determined using the Eqs. (3), (26) and (27). A principle opportunity of the application of the pulse retrieving procedure is theoretically proved by modelling, still experimental verifications of the introduced approach should be carried out in future.

References

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Theoretical analysis of the stretched optical pulse ripple and novel chirped pulse retrieving algorithm

Abstract

1. Introduction

2. Time and spectral domains

3. Amplitude ripples in a time domain

4. Phase ripples in a time domain

5. Evolution of the ripple

6. Pulse retrieval

7. Conclusions

References

Cited By

Figures (6)

Equations (27)

Optics Express