1. Introduction
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 [
1]. However, the energy generated in that case was several nJ. Then, the advent of gas-filled hollow fibers enabled us to generate few-cycle optical pulses with energies in excess of 100 nJ [
2]. This development has led to the generation of attosecond x-ray pulses by high-order harmonic conversion of few-cycle optical pulses [
3]. One of the main features of the hollow fiber compression technique is that a broadband spectrum together with a monotonic spectral phase suitable for dispersion compensation can readily be obtained by SPM. For this reason, monocycle to few-cycle pulses can be generated without any serious technical difficulties. Other features are the excellent beam quality and the focusability, which arise because of SPM taking place inside the hollow fiber during propagation. On the other hand, the drawback is energy scaling. As the energy incident on a hollow fiber increases, the coupling to the fiber mode is degraded by the poor beam quality due to self-focusing and multi-photon ionization, both of which take place near the entrance to the fiber. In order to avoid this problem, we have developed a pressure-gradient hollow fiber technique that can generate TW-class few-cycle pulses [
4]. Further increases in energy require suppression of multi-photon ionization during propagation inside the fiber. The use of chirped pulses in spectral broadening by SPM was proposed as a solution to this problem and some experimental results indicated that this was possible as long as the pulse intensity was suppressed below the ionization threshold [
5,
6,
7]. In order to realize this, the key technology is dispersion compensation over a wide range of the spectrum corresponding to a few optical cycles in the time domain. For this opportunity, we need to understand the relationship between the initial chirp imposed on the incident pulse and the dispersion after spectral broadening by SPM.
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.
2. Analytical Derivation
2.1. Self-Phase Modulation by a Linearly Chirped Pulse
We consider that an optical pulse is subject to temporal phase modulation depending on its intensity profile, i.e., self-phase modulation (SPM). Assuming that the temporal profile is Gaussian, the Fourier-transform-limited (FTL) pulse is represented in the time and frequency domains, respectively, as,
(1)
(2)
where a = 2ln2/Δt2 and Δt is the pulsewidth (full width at half maximum: FWHM). Applying a group delay dispersion (GDD) Ф′′ to Equations (1) and (2), the corresponding chirped pulses are expressed in the frequency and time domains, respectively, as follows.
(3)
(4)
Here, Ф′′ is the normalized GDD given by
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, Ф′′ = 10 implies that the pulse is stretched by a factor of approximately 10.
When a linearly chirped pulse is subject to SPM, the temporal profile of the optical intensity is given by
(6)
where I0 is the peak intensity. It is noted that in Equation (6) the main part of the pulse can be approximated by a quadratic function. The nonlinear refractive index is expressed as follows.
(7)
Considering only the time-dependent term, i.e., the third term, Equation (4) can be rewritten as,
(8)
where
is the
B-integral,
λ0 is the center wavelength, and
L is the interaction length. The Fourier transform of Equation (8) yields
(9)
The GDD of the output pulse resulting from SPM can be obtained from the imaginary part of the exponential function as
(10)
where as the real part gives the spectral width (FWHM)
(11)
It is noted that Ф′′OUT is a dimensionless normalized quantity, being the same as in Equation (5).
2.2. Self-Phase Modulation by a Nonlinearly Chirped Pulse
Applying both GDD and third-order dispersion (TOD) Ф′′′ to a FTL pulse, the chirped pulse is expressed in the frequency and time domains, respectively, as follows.
(12)
(13)
Here, Ф′′′ is the normalized TOD given by
(14)
where ω0 is the center wavelength. It should be noted that this is also a dimensionless quantity. Although the Fourier transform of the exponential function including the cubic function is the Airy function, we use following approximations assuming that the contribution of the cubic term is relatively small, i.e., |bt2| > |ct3|and |bω2| > |cω3| in the following equations, respectively.
(15)
(16)
The same treatment given in the previous section is applied to Equation (13), including the time-dependent refractive index.
(17)
The Fourier transform of Equation (17) based on Equation (16) yields
(18)
The TOD of the output pulse resulting from SPM can be obtained from the imaginary part of the exponential function as
(19)
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.
2.3. Effects of Self-Steepening
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,
(20)
The optical intensity corresponding to the electric field is
(21)
From Equations (20) and (21)
(22)
and the temporal profile including the self-steepening can be represented by
(23)
Neglecting t4 and higher-order terms, the Fourier transform of Equation (23) results in
(24)
The TOD of the output pulse resulting from SPM and self-steeping can be obtained from the imaginary part of the exponential function as
(25)