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In this article, we analytically investigate the spectral broadening by self-phase modulation of strongly chirped optical pulses. The dispersion due to the nonlinear optical process is expressed as functions of a linear and a nonlinear initial chirp. As a result, it is found that the third-order dispersion strongly depends on the initial linear chirp and the nonlinearity for self-phase modulation.

Optical pulse compression using self-phase modulation (SPM) and subsequent dispersion compensation is a widely used technique for ultrashort pulse generation, first demonstrated with an optical fiber in 1987 [

In this article, we present analytical expressions derived for spectral broadening by chirped pulse SPM and investigate the chirp of the output pulse. In particular, we argue that the influence of nonlinear chirp on the third-order dispersion, which must be compensated for in order to generate ultrashort pulses in the few optical cycle regime.

We consider that an optical pulse is subject to temporal phase modulation depending on its intensity profile,

where a = 2ln2/Δt^{2 }and Δt is the pulsewidth (full width at half maximum: FWHM). Applying a group delay dispersion (GDD)

Here,

which is a dimensionless quantity roughly equal to a pulse stretching ratio with regard to the FTL pulse having the same spectral width. For example,

When a linearly chirped pulse is subject to SPM, the temporal profile of the optical intensity is given by

where _{0}

Considering only the time-dependent term,

where _{0}

The GDD of the output pulse resulting from SPM can be obtained from the imaginary part of the exponential function as

where as the real part gives the spectral width (FWHM)

It is noted that _{OUT}

Applying both GDD and third-order dispersion (TOD)

Here,

where _{0}^{2}^{3}^{2}^{3}

The same treatment given in the previous section is applied to Equation (13), including the time-dependent refractive index.

The Fourier transform of Equation (17) based on Equation (16) yields

The TOD of the output pulse resulting from SPM can be obtained from the imaginary part of the exponential function as

It should be noted again that this is a dimensionless quantity. Both the GDD and spectral width of the output pulse are independent of the initial TOD, being the same as shown in Equations (10) and (11), respectively. This is because the contribution of TOD is assumed to be small. Strictly, the GDD and spectral width would be weak functions of the initial TOD.

SPM is the lowest-order nonlinear phenomenon based on the optical Kerr effect that provides symmetric spectral broadening. The next higher-order nonlinear phenomenon is self-steepening, which gives asymmetric spectral broadening. Here, we consider self-steepening together with TOD, while GDD is neglected for simplicity.

A nonlinearly chirped pulse including TOD is expressed in the time domain as,

The optical intensity corresponding to the electric field is

From Equations (20) and (21)

and the temporal profile including the self-steepening can be represented by

Neglecting ^{4} and higher-order terms, the Fourier transform of Equation (23) results in

The TOD of the output pulse resulting from SPM and self-steeping can be obtained from the imaginary part of the exponential function as

SPM of a FTL pulse results in an up-chirped pulse. This is also obvious from Equation (8). In this case, the sign of the GDD is positive, which requires a negative GDD for dispersion compensation. Here, we consider the SPM of strongly chirped pulses.

Group Delay Dispersion (GDD) of the output pulse _{OUT}

In order to generate few-cycle pulses after spectral broadening by SPM, it is important to carefully compensate the phase of the output pulse, not only GDD but also TOD. The origin of TOD generated during the SPM process is asymmetry of the temporal intensity profile that gives a time-dependent nonlinear refractive index. One of the factors that provide an asymmetric temporal profile is a nonlinear chirp,

Third-Order Dispersion (TOD) of the output pulse _{OUT}

Summarizing the results shown in

In order to determine the effect of self-steepening, we compare the TOD of the output pulse as a function of initial TOD with and without self-steepening.

Same as

The use of chirped pulses in hollow-fiber pulse compression is proposed as a means to further increase the energy and intensity of few-cycle pulses. In this paper, we derived analytical expressions for chirped-pulse SPM, focusing on the relationship between the initial chirp and the dispersion remaining in the output pulse. The results show that a strongly chirped pulse with large initial GDD retains the initial TOD. On the other hand, the initial TOD decreases in a FTL or weakly chirped pulse. Such characteristics are convenient for dispersion compensation in the SPM-based pulse compression technique. It has generally been considered that pre-compensation of the chirp in an optical system including nonlinear processes such as SPM is difficult. However, the present result shows that dispersion can be roughly compensated for by a pair of prisms or chirped mirrors after a hollow fiber, while the fine-adjustment can be carried out with a programmable phase modulator prior to the hollow fiber, or even before a chirped-pulse amplifier.