Self-phase-modulation

Valdmanis J A and Fork R L 1986 Design considerations for a femtosecond pulse laser balancing self phase modulation, group velocity dispersion, saturable absorption, and saturable gain IEEE J. Quantum. Electron. 22 112-18... 

The study confirmed that the action of the precursors of the main pulse has to be carefully considered in the ultrashort intense laser pulse interactions. Nevertheless, a regime of rather stable propagation [34] of a laser pulse of tens of femtoseconds was found in a broad window of the laser intensity/medium density diagram, in which (1) the ionization occurs in one or a few optical cycles, avoiding effects of self-phase modulation and defocusing for most of the... 

Self-phase Modulation White Light from Monochromatic Lasers... 

When an intense pulse of monochromatic laser light is focussed on a transparent liquid or solid, there is an emission of white light over a wide continuous spectral range. This process is known as self-phase modulation . We will not consider its physics. For our purpose it is important to note its photochemical implications. On the one hand, this pulse of white light can be used to provide a probe light in ps and fs flash photolysis (sections 8.1 and 8.2). On the other hand, it can be a source of stray light in some luminescence measurements. This comes as a surprise to many users of lasers for luminescence kinetics measurements, but it is an unavoidable problem. 

Equations (16) and (17) describe second-harmonic generation (SHG) and third-harmonic generation (THG) of one laser beam with a single polarization. Self-phase modulation (SPM) of a single laser beam is described in Eq.(18) as e.g. employed in z-scan experiments [6]. Equation (19) is the cross-phase modulation (XPM) process between two laser beams and Eq.(20) describes the four wave mixing with degenerate frequencies (DFWM). 

The factors 2 and 4 in the denominators are due to the definition of the field amplitudes (Eq.(14) and Eq.(15)). In order to prevent these factors some authors drop the factor 1/2 in the definitions of the amplitudes. The disadvantage of this convention is the unusual convergence behaviour of the electric field as the frequency to approaches zero. This different field definition of course additionally complicates the comparison of different hyperpolarizability values. The factors in the numerator arise from the different possibilities to permute the input frequencies. As an example, in self-phase modulation the three input electric fields each provide a factor 1/2 which, with the factor 1/2 from the polarization, results in a denominator of 4. The negative frequency for SPM allows three permutations yielding finally a prefactor of 3/4. 

For all-optical signal processing (based on pure y nonlinearities) the important quantity is the nonlinear refractive index n2. The resulting refractive index change can be induced either by the beam at the frequency co1 itself (self-phase modulation) or by a beam at another frequency to2 (cross-phase modulation). 

In a CPA, because a large secondary dispersion is added to the stretcher, the spectrum is mapped to the temporal waveform in the amplifier. For that reason, the phase change added to the pulse shaper cannot cause a large change in the pulse shape in the CPA. Self-phase modulation depends on the temporal amplitude transition. Therefore, the phase perturbation can be regarded as almost linear in the amplifier. Thus, the inverse of the spectrum phase difference between the acquired and the target phase is added to the pulse shaper. 

Self-Phase Modulation and White-Light Generation... 

The high intensity within the filaments (4 to 6 x 1013 W/cm2) [43] generates an efficient self-phase modulation (Fig. 14.4), resulting in the emission of... 

Fig. 14.4. Effect of self-phase modulation. The initial pulse (top) is deformed along the propagation. The centre of the beam, which is more intense, is retarded relative to its head and trail by the non-linear refractive index. The resulting change in the envelope generates low frequencies on the rising front and high frequencies on the trailing front...

R. R. Alfano, S. L. Shapiro, Observation of Self-Phase Modulation and Small-Scale Filaments in Crystals and Glasses, Physical Review Letters 24, 592 (1970)... 

This time dependent phase shift leads to a frequency modulation that is proportional to the time derivative of the self induced phase shift fused silica with its positive Kerr coefficient rio = 2.5 x 10 16 cm2/W [28] the leading edges of the pulses are creating extra frequencies shifted to the red ( jvi(t) < 0) while the trailing edges causes blue shifted frequencies to emerge. Self-phase modulation modifies the envelope function according to... 

Because NL(t) has the same periodicity as A(t) the comb structure of the spectrum, as derived in section 3, is not affected. In an optical fiber self-phase modulation can be quite efficient even though the nonlinear coefficient in fused silica is comparatively small. This is because the fiber core carries a high intensity over an extended length. 

This simplified picture of self-phase modulation neglects dispersion, time-delayed nonlinearities and shock formation which is all known to occur in optical fibers. While no in fused silica is at least as fast as a few fs, the GVD broadens the pulses as they travel along the fiber so that the available peak power Pa is decreased. Effective self-phase modulation however takes place when the so called dispersion length is much smaller then the nonlinear length whose ratio is given by [27]... 

The intensity that can be used is ultimately limited by self-focusing and self-phase modulation in the sample. An early study concluded that these problems would preclude Raman echo experiments (31), but careful experimental design has pushed these limits back. However, the self-focusing problem remains the primary practical limitation in expanding the application of the Raman echo. 

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