Complete classification and additional saddle-point solutions for high-order above-threshold ionization induced by a strong laser field

Ionization of atoms by a strong laser field can be described using the improved strong-field approximation. The corresponding transition amplitude of high-order above-threshold ionization is presented in the form of a two-dimensional integral over the electron ionization time <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi>t</a:mi><a:mn>0</a:mn></a:msub></a:math> and the rescattering time <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>t</b:mi></b:math>. This integral can be solved using the saddle-point (SP) method and the resulting <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>T</c:mi></c:math>-matrix element is expressed as a sum (over the SP times <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:msub><d:mi>t</d:mi><d:mn>0</d:mn></d:msub></d:math> and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>t</e:mi></e:math>) of the partial transition amplitudes. We address the problem of finding the solutions of the system of SP equations for the times <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:msub><f:mi>t</f:mi><f:mn>0</f:mn></f:msub></f:math> and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mi>t</g:mi></g:math>. For a bichromatic linearly polarized laser field with the frequencies <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:mi>r</h:mi><h:mi>ω</h:mi></h:mrow></h:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:mi>s</i:mi><i:mi>ω</i:mi></i:mrow></i:math> (<j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mi>r</j:mi></j:math> and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mi>s</k:mi></k:math> are integers, <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mrow><l:mi>s</l:mi><l:mo>&gt;</l:mo><l:mi>r</l:mi></l:mrow></l:math>, and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>ω</m:mi></m:math> is the fundamental frequency) we found that there are <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:mrow><n:mn>8</n:mn><n:msup><n:mi>s</n:mi><n:mn>2</n:mn></n:msup></n:mrow></n:math> SP solutions per optical cycle. For one half of them the velocity of the electron emitted in the laser field polarization direction changes the sign at the rescattering time (we call such solutions backward-scattering solutions), while for the other half this velocity remains unchanged (these solutions we call forward-scattering SP solutions). For very short (or even negative) electron travel time we call these solutions backward-like and forward-like scattering SP solutions. For these solutions the imaginary parts of the times <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"><o:msub><o:mi>t</o:mi><o:mn>0</o:mn></o:msub></o:math> and <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"><p:mi>t</p:mi></p:math> become large so that the concept of real electron trajectories becomes questionable. Having such a classification, we found additional SP solutions even for the simplest case of a monochromatic linearly polarized laser field. For a bichromatic linearly polarized laser field with <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"><q:mrow><q:mi>s</q:mi><q:mo>=</q:mo><q:mn>2</q:mn></q:mrow></q:math> and equal component intensities we presented a detailed analysis of all 32 solutions per optical cycle, showing how the SP times <r:math xmlns:r="http://www.w3.org/1998/Math/MathML"><r:msub><r:mi>t</r:mi><r:mn>0</r:mn></r:msub></r:math> and <s:math xmlns:s="http://www.w3.org/1998/Math/MathML"><s:mi>t</s:mi></s:math> and the corresponding differential ionization rates depend on the photoelectron energy. We have also analyzed the case where the intensity of the second component decreases while the sum of the component intensities remains fixed.