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Following Kerr21, Buckingham22, 23, Duguay28, and Alfano et al.1, 4, the general form for the index of refraction based on the index of refraction becomes electric field E dependent:

$$n= {n}_{0 }+ {n}_{2}{E(t)}^{2},$$

(1)

where n0 is the index of refraction, n2 is the nonlinear index from various mechanisms, and E is the electric field. Our ansatz is that the index of refraction (n) is a function of angular frequency () and time (t): n(,t).

In the EM model, the Kerr index of refraction n2 of the material depends on the time response of the underlying mechanisms of the material to the electric field of the laser:({n}_{2}= sum_{i}{n}_{i}), where i=mechanisms (such as electronic (~1017s), molecular redistribution (~1014s), plasma (~1013s), rotational (~1012s), librational, and other slower mechanism)28,29,30,31,32. The work of KenneyWallace showed the various temporal components of the Kerr index in CS231, 32.

Based on the Kerr effect, the electric field of the light is distorted in the CEP after an intense light beam propagates a distance z into the material and the electric field of the light has the form:

$$begin{aligned} Eleft( {t,omega } right) = & E_{0} e^{{ - frac{{t^{2} }}{{T^{2} }}}} cosleft[ {phi left( {t,omega } right)} right] \ = & E_{0} e^{{ - frac{{t^{2} }}{{T^{2} }}}} cosleft[ {omega _{0} t - kz + varphi } right], \ end{aligned}$$

(2)

where the exponential time is T = (frac{{tau }_{p}}{sqrt{2mathrm{ln}2}}); p is the full width half maximum (FWHM) of the pulse; and the bracket is the phase (upphi left(mathrm{t},omega right).) The phase (upphi left(mathrm{t},omega right)) is modulated by the index of refraction due to Kerr effect. The high laser intensity induces changes in the refractive index from electronic and molecular distortion. The propagation constant becomes time and frequency dependent: (k=frac{nomega }{c}). Expanding about ({upomega }_{0}), the modulated instantaneous phase of Carrier Envelope Phase (CEP) under the envelope becomes:

$$phi left(t,omega right)= {omega }_{0}left{t-frac{nleft(t,omega right)z}{c}right}+varphi ,$$

(3)

where ({upomega }_{0}) is the central angular frequency of the laser, n(t,) is the refractive index, z is the propagating distance, and is the offset phase (set (varphi =0)). This phase is the key for the generation of the Supercontinuum and Higher Harmonics where the response time of the materials index of refraction is critical to the generation of HHG. The offset CEP phase is set to be zero for the cosine-like pulse which drives HHG modes. The nonlinear refractive index with quadratic field dependence and the material response time is given by:

$$nleft(tright)={n}_{0}+{int }_{-infty }^{t}{int }_{-infty }^{t}fleft({t}{prime},{t}^{{prime}{prime}}right)Eleft(t-{t}{prime}right)Eleft(t-{t}^{{prime}{prime}}right)d{t}{prime}d{t}^{{prime}{prime}},$$

(4)

where n0 is the ordinary index, E the electric field and,

$$fleft({t}{prime},{t}^{{prime}{prime}}right)=left(frac{{n}_{2}}{tau }right){e}^{- frac{{t}{prime}}{tau }} delta left(t-{t}^{{prime}{prime}}right).$$

(5)

Here, n2 is the nonlinear index and is the response time (tau). Equation(4) may be simplified to:

$$nleft(tright)= {n}_{0}+ left(frac{{n}_{2}}{tau }right){int }_{-infty }^{t}{e}^{- frac{left(t-{t}{prime}right)}{tau }}{E}^{2}left({t}{prime}right)d{t}{prime}.$$

(6)

The pure electronic mechanism of n2 is the instantaneous index of refraction for rare noble gases like Ar, Kr, and Ne and solids involving no translation of nuclei or rotation of atomic cluster. It is expected to have relaxation response time much less than the optical period (<<(frac{1}{{omega }_{0}})), faster than few femtoseconds. For this case, the index n(t) responses to E(t) at optical frequencies. Hence the weighting function ((frac{1}{tau }){e}^{- frac{left(t-{t}{prime}right)}{tau }}) may be replaced by (updelta left(mathrm{t}-{mathrm{t}}^{mathrm{^{prime}}}right)). Following the Kerr effect, the electronic response of the instantaneous response time on 50fs nonlinear index for ultrafast laser pulses causes the HHG and responsible for ESPM to become:

$${n}_{instantaneous}(t)={n}_{0}+{n}_{2}{[{E}_{0}{e}^{- frac{{t}^{2}}{{T}^{2}}}cosphi left(tright)]}^{2}.$$

(7)

Equation(7) represents the instantaneous response of the index of refraction. This is the ansatz that has been used before in the form of n by luminaries like Kerr21 and Buckingham22. The ansatz n(t) follows the modulation optical cycles of the phase of E. The instantaneous response is used to follow the optical cycle rather than the envelope of the CEP without time averaging was proposed by Buckingham22. On the other hand, the average index of refractionfollowing the envelope of the electric field will reveal the supercontinuum without HHG.

The general form for the nonlinear refractive index with quadratic field dependence is,

$$nleft(tright)={n}_{0}+delta n= {n}_{0}+ {int }_{-infty }^{t}f({t}{prime},t){E}^{2}left({t}{prime}right)d{t}{prime},$$

(8)

where n0 is the normal index, E(t) is the laser electric field assumed to have a Gaussian envelope (({E}_{0}left(tright)={E}_{0}{e}^{-frac{{t}^{2}}{{T}^{2}}})) and f(t) is the weighting function describing the response of the system; f(t) assumes the form (frac{{e}^{-frac{t}{T}}}{tau }) where is the response time of the material. The incident lasers electric field has the form given by Eq.(2). Following the ESPM model for the electric field E(t) gives:

$$Eleft(tright)={E}_{0}{e}^{- frac{{t}^{2}}{{T}^{2}}}cosleft(omega t-frac{omega nleft(tright)z}{c}right),$$

(9)

with the instantaneous n(t):

$$nleft(tright)={n}_{0}+{n}_{2}{[{E}_{0}{e}^{- frac{{t}^{2}}{{T}^{2}}}cosleft({omega }_{0}tright)]}^{2}.$$

(10)

Using an averaging procedure on n(t) to simulate the generation of the signal caused by the material response resulting in HHG and the associated characteristics that can be found in the experimental results such as the number of odd harmonics and the cutoff frequency.

We have numerically applied an averaging procedure using response filter to the n(t) for the Kerr effect in different media:

$$=frac{1}{tau }{int }_{t}^{t+tau }nleft({t}{prime}right)d{t}{prime},$$

(11)

to generate HHG and SC for the instantaneous electronic cloud response time on about 50 as; the fast response time of ionization and molecular redistribution on about 1fs; and the slower rotational and vibrational relaxation times of 110ps or greater for different pulse durations.

After substituting Eq.(11),for a response time into Eq.(9) to get E(t), the modified electronic self-phase modulated spectral E() is obtained by the Fast Fourier Transform (FFT) technique. The spectral density S() of the phase-modulated light at is:

$$Sleft(omega right)=frac{c}{4pi }{left|E(omega )right|}^{2},$$

(12)

where E() is the Fourier transform of E(t).

For a material with response time slower than pure electronic on the order of such as molecular redistribution, plasma, librational, orientational, and vibrational motion ((t>tau >frac{10}{{omega }_{L}})), the envelope of the pulse is reflected in n(t) and no HHG is produced. For slow response time of the material , the average n(t) becomes the classical SPM following the envelope of the pulse:

$${n}_{slow}left(tright)={n}_{0}+frac{{n}_{2}}{2}{left[{E}_{0}{e}^{- frac{{t}^{2}}{{T}^{2}}}right]}^{2}.$$

(13)

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