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    Journals >Acta Physica Sinica >Volume 69 >Issue 2 >Page 020201-1 > Article
    • Acta Physica Sinica
    • Vol. 69, Issue 2, 020201-1 (2020)
    Analytical solution of three-dimensional Fourier transform frequency spectrum for three-level potassium atomic gas
    Chao-Ying Zhao1、2、* and Wei-Han Tan3
    Author Affiliations
  • 1School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, China
  • 2State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China
  • 3Department of Physics, Shanghai University, Shanghai 200444, China
    • show less
    DOI: 10.7498/aps.69.20190964 Cite this Article
    Chao-Ying Zhao, Wei-Han Tan. Analytical solution of three-dimensional Fourier transform frequency spectrum for three-level potassium atomic gas[J]. Acta Physica Sinica, 2020, 69(2): 020201-1 Copy Citation Text show less

    Abstract

    With the development of laser technology in the field of optics, ultra-fast optics has become an important research field. Compared with the traditional technology, ultrafast optics can be realized not only under shorter pulse function, but also on a smaller scale, which can more quickly reflect the dynamic process. We present an analytical calculation of the full three-dimensional (3D) coherent spectrum with a finite duration two-dimensional (2D) Gaussian pulse envelope. Our starting point is the solution of the optical Bloch equations for three-level potassium atomic gas in the 3D time domain by using the projection-slice theorem, error function and Fourier-shift theorem of 3D Fourier transform. These principles are used to calculate and simplify the third-order polarization equation generated by the device, and the analytical calculation of three-dimensional Fourier transform frequency spectrum at T = 0 is obtained. We simulate the analytic solution by using mathematics software. By comparing the simulations with the experimental results, with the homogeneous line-width fixed, we can obtain the relationship among the in-homogeneous broadening, the correlation diagonal coefficients and the three-dimensional spectrum characteristics, which can be identified quantitatively by fitting the slices of three-dimensional Fourier transform spectrum peaks in an appropriate direction. The results show that the three-dimensional Fourier transform spectrum will extend along the diagonal direction with the increasing of the in-homogeneous broadening, and the spectrogram progressively becomes a circle with the increasing of the diagonal correlation coefficient, and the amplitude also gradually turns smaller. According to the analytical solution, we give a complete two-dimensional spectrum of the T = 0 interface. The results can be fit to the experimental 3D coherent spectrum for arbitrary inhomogeneity.

    Keywords

    $\begin{split} & {P^{(3)}}(\tau,T,t) \\= \, &\int\nolimits_0^\infty {\int\nolimits_0^\infty {\int\nolimits_0^\infty {{\rm{d}}t_A'{\rm{d}}t_B'{\rm{d}}} } } {R^{(3)}}(t_A',t_B',t_C')\\ & \times E_A^*(t_A' - {t_1}){E_B}(t_B' - {t_2}){E_C}(t_C' - {t_3}), \end{split}$(1)

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    $\dot \rho = - \frac{{\rm{i}}}{\hbar }(H\rho - \rho H) - \frac{{\rm{1}}}{{\rm{2}}}(\varGamma \rho + \rho \varGamma ),$(2)

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    $\begin{split} g({\omega _{10}},{\omega _{20}}) =\, & \frac{{\sqrt {{a_{11}}{a_{22}} - a_{12}^2} }}{{\text{π}}}\\ & \times{{\rm{e}}^{ - ({a_{11}}{\omega _{10}^2} - 2{a_{12}}{\omega _{10}}{\omega _{20}} + {a_{22}}{\omega _{20}^2})}}, \end{split}$(3)

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    $\begin{split} & {\text{δ}} {\omega _{10}} = \sqrt {\frac{{\ln 2}}{{{a_{11}}(1 - {R^2})}}},\\& {\text{δ}} {\omega _{20}} = \sqrt {\frac{{\ln 2}}{{{a_{1{\rm{2}}}}(1 - {R^2})}}}, \\ & R = \frac{{{a_{12}}}}{{\sqrt {{a_{11}}{a_{22}}} }},\end{split}$ (4)

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    $\begin{split} \rho _A^{{\rm{(3)}}}(\tau,{\rm{0}},t) \propto\, & \rho _{A{\rm{0}}}^{{\rm{(3)}}}\varTheta (t)\varTheta (\tau ){{\rm{e}}^{ - {\varGamma _{{\rm{10}}}}(t + \tau )}}{{\rm{e}}^{ - \frac{{{\text{δ}} \omega _{{\rm{10}}}^2}}{{{\rm{4ln2}}}}{{(t - \tau )}^2}}}\\ & \times{{\rm{e}}^{ - {\rm{i}}(\omega - \omega _{{\rm{10}}}^c)(t - \tau )}}, \end{split}$(5)

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    ${\rm{Erf}}\,[x] = \frac{2}{{\sqrt {\text{π}} }}\int_{\rm{0}}^x {{{\rm{e}}^{ - {\eta ^2}}}{\rm{d}}\eta } .$(6)

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    $\begin{split} &\rho _A^{\left( 3 \right)}\left( {t',{\omega _0} + {\Delta}\omega } \right) \\ =\,& \frac{{\sqrt {{\rm{\text{π} ln}}2} }}{{{\text{δ}} {\omega _{10}}}}\varTheta \left( {t'} \right){{\rm{e}}^{ - \left( {{\varGamma _{10}}t' + \ln 2\tfrac{{{\Delta}{\omega ^2}}}{{{\text{δ}} \omega _{10}^2}}} \right)}}\\& \times {\rm{Erf}}\left[ {\frac{{{\text{δ}} {\omega _{10}}}}{{2\sqrt {\ln 2} }}t' - {\rm{i}}\frac{{\sqrt {\ln 2} }}{{{\text{δ}} {\omega _{10}}}}{\Delta}\omega } \right]\\ & +{\rm{ Erf}}\left[ {\frac{{{\text{δ}} {\omega _{10}}}}{{2\sqrt {\ln 2} }}t' + {\rm{i}}\frac{{\sqrt {\ln 2} }}{{{\text{δ}} {\omega _{10}}}}{\Delta}\omega } \right], \end{split} $ (7)

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    $\begin{split} & \rho _A^{\left( 3 \right)}\left( {{\omega _{t'}},{\omega _0} + {\Delta}\omega } \right)\\ =\, & \sqrt {\ln 2} \frac{{{{\rm{e}}^{\ln 2\tfrac{{{{\left[ {{\varGamma _{10}} + {\rm{i}}\left( {{\omega _{t'}} - {\Delta}\omega } \right)} \right]}^2}}}{{{\text{δ}} \omega _{10}^2}}}}}}{{{\text{δ}} {\omega _{10}}\left( {2{\varGamma _{10}} - {\rm{i}}\sqrt 2 {\omega _{t'}}} \right)}}\\& \times {\rm{Erfc}}\left[ {2\sqrt {\ln 2} \frac{{{\varGamma _{10}} + {\rm{i}}\left( {{\omega _{t'}} - {\Delta}\omega } \right)}}{{{\text{δ}} {\omega _{10}}}}} \right]\\ & + {{\rm{e}}^{16\ln 2\tfrac{{{{\left[ {{\varGamma _{10}} - {\rm{i}}\left( {{\omega _{t'}} + {\Delta}\omega } \right)} \right]}^2}}}{{{\text{δ}} \omega _{10}^2}}}}\\& \times {\rm{Erfc}}\left[ {2\sqrt {\ln 2} \frac{{{\varGamma _{10}} - {\rm{i}}\left( {{\omega _{t'}} + {\Delta}\omega } \right)}}{{{\text{δ}} {\omega _{10}}}}} \right], \end{split} $ (8)

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    $ \sqrt 2 {\Delta}\omega = {\omega _\tau } - {\omega _t} + 2{\omega _0},\; \sqrt 2 {\omega _{t'}} = {\omega _\tau } + {\omega _t}, $ (9)

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    $ \begin{split} & {S_A}\left( {{\omega _t},{\omega _\tau }} \right) \\ = \, &\sqrt {\ln 2} \frac{{{{\rm{e}}^{2\ln 2\tfrac{{{{\left[ {{\varGamma _{10}} - {\rm{i}}\left( {{\omega _t} - {\omega _0}} \right)} \right]}^2}}}{{{\text{δ}} \omega _{10}^2}}}}}}{{{\text{δ}} {\omega _{10}}\left[ {2{\varGamma _{10}} - {\rm{i}}\left( {{\omega _t} + {\omega _\tau }} \right)} \right]}}\\& \times {\rm{Erfc}}\left[ {\sqrt {2\ln 2} \frac{{{\varGamma _{10}} - {\rm{i}}\left( {{\omega _t} - {\omega _0}} \right)}}{{{\text{δ}} {\omega _{10}}}}} \right]\\&+ {{\rm{e}}^{2\ln 2\tfrac{{{{\left[ {{\varGamma _{10}} - {\rm{i}}\left( {{\omega _t} + {\omega _0}} \right)} \right]}^2}}}{{{\text{δ}} \omega _{10}^2}}}}\\& \times {\rm{Erfc}}\left[ {\sqrt {2\ln 2} \frac{{{\varGamma _{10}} - {\rm{i}}\left( {{\omega _t} + {\omega _0}} \right)}}{{{\text{δ}} {\omega _{10}}}}} \right], \end{split} $ (10)

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    $\begin{split} & {S_B}\left( {{\omega _t},{\omega _\tau }} \right) \\=\, & \sqrt {\ln 2} \frac{{{{\rm{e}}^{2\ln 2\tfrac{{{{\left[ {{\varGamma _{20}} - {\rm{i}}\left( {{\omega _t} - {\omega _0}} \right)} \right]}^2}}}{{{\text{δ}} \omega _{20}^2}}}}}}{{{\text{δ}} {\omega _{20}}\left[ {2{\varGamma _{20}} - {\rm{i}}\left( {{\omega _t} + {\omega _\tau }} \right)} \right]}}\\& \times {\rm{Erfc}}\left[ {\sqrt {2\ln 2} \frac{{{\varGamma _{20}} - {\rm{i}}\left( {{\omega _t} - {\omega _0}} \right)}}{{{\text{δ}} {\omega _{20}}}}} \right]\\& + {{\rm{e}}^{2\ln 2\tfrac{{{{\left[ {{\varGamma _{20}} - {\rm{i}}\left( {{\omega _t} + {\omega _0}} \right)} \right]}^2}}}{{{\text{δ}} \omega _{20}^2}}}}\\& \times {\rm{Erfc}}\left[ {\sqrt {2\ln 2} \frac{{{\varGamma _{20}} - {\rm{i}}\left( {{\omega _t} + {\omega _0}} \right)}}{{{\text{δ}} {\omega _{20}}}}} \right], \end{split} $ (11)

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    $\begin{split} & \rho _C^{\left( 3 \right)}(\tau ',t') \\ =\, & \varTheta (t' - \tau ')\varTheta (t' + \tau '){{\rm{e}}^{ - {\varGamma _{{\rm{10}}}}t'}}{{\rm{e}}^{ - \frac{{(xt{'^2} + 2z\tau 't' + y\tau {'^2})}}{{{\rm{4ln2}}}}}}, \end{split}$(12)

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    $\begin{split} & {S_{C{\rm{2}}}}\left( {{\omega _t},{\omega _\tau }} \right) \\=\, & \frac{{\sqrt {2{\rm{ln}}2} }}{{\sqrt {1 + R} }}\frac{{{{\rm{e}}^{\tfrac{{4{\rm{ln}}2}}{{{\rm{1}} + R}}\tfrac{{{{\left[ {{\varGamma _{10}} - {\rm{i}}\left( {{\omega _t} - {\omega _0}} \right)} \right]}^2}}}{{{\text{δ}} \omega _{10}^2}}}}}}{{{\text{δ}} {\omega _{10}}\left[ {2{\varGamma _{10}} - {\rm{i}}\left( {{\omega _t} + {\omega _\tau }} \right)} \right]}}\\& \times {\rm{Erfc}}\left[ {\frac{{2\sqrt {{\rm{ln}}2} }}{{{\rm{1}} + R}}\frac{{{\varGamma _{10}} - {\rm{i}}\left( {{\omega _t} - {\omega _0}} \right)}}{{{\text{δ}} {\omega _{10}}}}} \right]\\& + {{\rm{e}}^{\tfrac{{4\ln 2}}{{{\rm{1 + }}R}}\tfrac{{{{\left[ {{\varGamma _{10}} - {\rm{i}}\left( {{\omega _t} + {\omega _0}} \right)} \right]}^2}}}{{{\text{δ}} \omega _{10}^2}}}}\\& \times {\rm{Erfc}}\left[ {\frac{{2\sqrt {\ln 2} }}{{1 + R}}\frac{{{\varGamma _{10}} - {\rm{i}}\left( {{\omega _t} + {\omega _0}} \right)}}{{{\text{δ}} {\omega _{10}}}}} \right], \end{split} $ (13)

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    $\begin{split} & {S_{C3,E{\rm{3}}}}\left( {{\omega _t},{\omega _\tau }} \right) \\= \, & \frac{{2\sqrt {\ln 2} }}{{m + {\rm{1}}}}\frac{{{{\rm{e}}^{\tfrac{{8\ln 2}}{{m + 1}}\tfrac{{{{\left[ {{\varGamma _{{\rm{1}}0}} - {\rm{i}}\left( {{\omega _t} + {\omega _0}} \right)} \right]}^2}}}{{{\text{δ}} \omega _{10}^2}}}}}}{{{\text{δ}} {\omega _{10}}\left[ {2{\varGamma _{10}} - {\rm{i}}\left( {{\omega _t} + {\omega _\tau }} \right)} \right]}}\\& \times {\rm{Erfc}}\left[ {\frac{{2\sqrt {2\ln 2} }}{{m + 1}}\frac{{{\varGamma _{{\rm{1}}0}} - {\rm{i}}\left( {{\omega _t} - {\omega _0}} \right)}}{{{\text{δ}} {\omega _{{\rm{1}}0}}}}} \right]\\& + {{\rm{e}}^{\tfrac{{8\ln 2}}{{m + 1}}\tfrac{{{{\left[ {{\varGamma _{{\rm{1}}0}} - {\rm{i}}\left( {{\omega _t} + {\omega _0}} \right)} \right]}^2}}}{{{\text{δ}} \omega _{10}^2}}}}\\& \times {\rm{Erfc}}\left[ {\frac{{2\sqrt {2\ln 2} }}{{m + 1}}\frac{{m{\varGamma _{10}} - {\rm{i}}\left( {{\omega _t} + {\omega _0}} \right)}}{{{\text{δ}} {\omega _{10}}}}} \right], \end{split} $ (14)

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    $\begin{split} & {S_{D3,F3}}\left( {{\omega _t},{\omega _\tau }} \right) \\= \, & \frac{{2\sqrt {\ln 2} }}{{m + 1}}\frac{{{{\rm{e}}^{\tfrac{{8\ln 2}}{{m + 1}}\tfrac{{{{\left[ {{\varGamma _{20}} - {\rm{i}}\left( {{\omega _t} + {\omega _0}} \right)} \right]}^2}}}{{{\text{δ}} \omega _{20}^2}}}}}}{{{\text{δ}} {\omega _{20}}\left[ {2{\varGamma _{20}} - {\rm{i}}\left( {{\omega _t} + {\omega _\tau }} \right)} \right]}}\\& \times {\rm{Erfc}}\left[ {\frac{{2\sqrt {2\ln 2} }}{{m + 1}}\frac{{{\varGamma _{20}} - {\rm{i}}\left( {{\omega _t} - {\omega _0}} \right)}}{{{\text{δ}} {\omega _{20}}}}} \right]\\& + {{\rm{e}}^{\tfrac{{8\ln 2}}{{m + 1}}\tfrac{{{{\left( {{\varGamma _{20}} - {\rm{i}}\left( {{\omega _t} + {\omega _0}} \right)} \right)}^2}}}{{{\text{δ}} \omega _{20}^2}}}}\\& \times {\rm{Erfc}}\left[ {\frac{{2\sqrt {2\ln 2} }}{{m + {\rm{1}}}}\frac{{m{\varGamma _{20}} - {\rm{i}}\left( {{\omega _\tau } + {\omega _0}} \right)}}{{{\text{δ}} {\omega _{20}}}}} \right]. \end{split} $ (15)

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    $\begin{split} & {S_{C4,E4}}\left( {{\omega _t},{\omega _\tau }} \right) \\ =\, & \frac{{2\sqrt {\ln 2} }}{{\sqrt {{m^2} + 2mR + 1} }}\frac{{{{\rm{e}}^{\tfrac{{8\ln 2}}{{\sqrt {{m^2} + 2mR + 1} }}\tfrac{{{{\left[ {{\varGamma _{10}} - {\rm{i}}\left( {{\omega _t} - {\omega _0}} \right)} \right]}^2}}}{{{\text{δ}} \omega _{10}^2}}}}}}{{{\text{δ}} {\omega _{10}}\left[ {2{\varGamma _{10}} - {\rm{i}}\left( {{\omega _t} + {\omega _\tau }} \right)} \right]}}\\& \times {\rm{Erfc}}\left[ {\frac{{2\sqrt {2\ln 2} }}{{\sqrt {{m^2} + 2mR + 1} }}\frac{{{\varGamma _{10}} - {\rm{i}}\left( {{\omega _t} - {\omega _0}} \right)}}{{{\text{δ}} {\omega _{10}}}}} \right]\\& + {{\rm{e}}^{\tfrac{{8\ln 2}}{{\sqrt {{m^2} + 2mR + 1} }}\tfrac{{{{\left[ {{\varGamma _{10}} - {\rm{i}}\left( {{\omega _t} + {\omega _0}} \right)} \right]}^2}}}{{{\text{δ}} \omega _{10}^2}}}}\\& \times {\rm{Erfc}}\left[ {\frac{{2\sqrt {2\ln 2} }}{{\sqrt {{m^2} + 2mR + 1} }}\frac{{m{\varGamma _{10}} - {\rm{i}}\left( {{\omega _t} + {\omega _0}} \right)}}{{{\text{δ}} {\omega _{10}}}}} \right]. \end{split} $ (16)

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    $\begin{split} & {S_{D4,F4}}\left( {{\omega _t},{\omega _\tau }} \right) \\= \, &\frac{{2\sqrt {\ln 2} }}{{\sqrt {{m^2} + 2Rm + 1} }}\frac{{{{\rm{e}}^{\tfrac{{8\ln 2}}{{\sqrt {{m^2} + 2mR + 1} }}\tfrac{{{{\left[ {{\varGamma _{20}} - {\rm{i}}\left( {{\omega _t} - {\omega _0}} \right)} \right]}^2}}}{{{\text{δ}} \omega _{20}^2}}}}}}{{{\text{δ}} {\omega _{20}}\left[ {2{\varGamma _{20}} - {\rm{i}}\left( {{\omega _t} + {\omega _\tau }} \right)} \right]}}\\& \times {\rm{Erfc}}\left[ {\frac{{2\sqrt {2\ln 2} }}{{\sqrt {{m^2} + 2Rm + 1} }}\frac{{{\varGamma _{20}} - {\rm{i}}\left( {{\omega _t} - {\omega _0}} \right)}}{{{\text{δ}} {\omega _{20}}}}} \right]\\& + {{\rm{e}}^{\tfrac{{8\ln 2}}{{\sqrt {{m^2} + 2mR + 1} }}\tfrac{{{{\left[ {{\varGamma _{20}} - {\rm{i}}\left( {{\omega _t} + {\omega _0}} \right)} \right]}^2}}}{{{\text{δ}} \omega _{20}^2}}}}\\& \times{\rm{Erfc}}\left[ {\frac{{2\sqrt {2\ln 2} }}{{\sqrt {{m^2} + 2Rm + 1} }}\frac{{m{\varGamma _{20}} - {\rm{i}}\left( {{\omega _t} + {\omega _0}} \right)}}{{{\text{δ}} {\omega _{20}}}}} \right]. \end{split} $ (17)

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    Chao-Ying Zhao, Wei-Han Tan. Analytical solution of three-dimensional Fourier transform frequency spectrum for three-level potassium atomic gas[J]. Acta Physica Sinica, 2020, 69(2): 020201-1
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