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Generalized multiple-prism dispersion theory for laser pulse compression: Higher order phase derivatives

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Abstract

An exact, and explicit, expression for the second derivative of the generalized multiple-prism angular dispersion is provided. This corresponds to the third derivative of the generalized exit angle with respect to the refractive index (∇ n 3 φ 2,m ). Higher derivatives, in abstract notation, are also given. The generalized equations are presented in a format applicable to practical prismatic configurations utilized in laser pulse compression schemes in the femtosecond domain. Exact values, as a function of the refractive index, are given for the first, second, and third angular derivatives for compensating double-prism and four-prism configurations of practical interest.

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... can be applied to restate the generalized multiple-prism dispersion as[49,50]( ) ...
... the third and higher derivatives are given by Duarte[39,50]. Observing the progression of the higher derivatives a generalized expression can be found to be ...
... Next, the attention was focused on higher order dispersions as needed, and applied, in prismatic pulse compressors for femtosecond lasers. In this regard, recent results obtained by Duarte[39,50]have been extended to obtain a succinct and elegant general equation for the generalized multiple-prism dispersion valid up to the rth derivative ...
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... Eq. (4.7) can be restated as[87] ...
... the third derivative is given by Duarte[87]) )( ...
... and the fourth derivative is given by[87]) ...
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... To numerically determine the GDD and third-order dispersion (TOD) of our BPC, we used a toolbox. 14,15 It determines GDD and TOD for some central wavelength λ 0 based on a prism compressor's geometry (Fig. 5), involving quantities such as incidence angle ϑ 0 , prism apex angle α, distance from apex to ray 1 (l 1 ), and distance from apex to ray 2 (l 2 ). (L/2) is the length projection perpendicular to the incident prism face, for the length from the prism's apex to the corner of CCM. ...
... Here, GDD and TOD are given by 14 where P = 2[n(λ)P AB + P BC + n(λ)P CD + P DE ]. An extensive phase delay analysis for two prisms has been done in Ref. 15. ...
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... It is possible to compensate the temporal chirp in vortex pulses using normal dispersion compensation techniques. Such techniques to compensate dispersion includes prisms [33], semiconductor saturable absorber mirrors SESAMs, among others but specifically multilayer interference coatings, such as chirped mirrors [34] (CMs) are a compact solution with a much larger bandwidth and custom compensation of high order dispersion [35]. In addition computer optimized designs of multilayer structures have been demonstrated [36]. ...
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... It means that all outgoing wavelengths from the back surface of last prism must be parallel. In this condition, the scanning of whole wavelength range can be possible (Duarte and Piper 1983;Duarte 2006Duarte , 2009. ...
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