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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.
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Keywords:
- Fourier transform spectrum /
- four-wave-mixing /
- atomic gas
[1] Ernst R R, Bodenhausen G, Wokaun A 1987 Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford: Clarendon Press)
[2] Jonas D M 2003 Annu. Rev. Phys. Chem. 54 425
Google Scholar
[3] Siemens M E, Moody G, Li H B, Bristow A D, Cundiff S T 2010 Opt. Express 18 17699
Google Scholar
[4] Fecko C J, Eaves J D, Loparo J J, Tokmakoff A, Geissler P L 2003 Science 301 1698
Google Scholar
[5] Turner D B, Wen P, Arias D H, Nelson K A, Li H B, Moody G, Siemens M E, Cundiff S T 2012 Phys. Rev. B 85 201303
Google Scholar
[6] Cundiff S T, Bristow A D, Siemen M, Li H B, Moody G, Karaiskaj D, Dai X C, Zhang T H 2012 IEEE J. Sel. Top Quant. 18 318
Google Scholar
[7] Nardin G, Moody G, Singh R, Autry T M, Li H B, Morier-Genoud F, Cundiff S T 2014 Phys. Rev. Lett. 112 046402
Google Scholar
[8] Moody G, Akimov I A, Li H B, Singh R, Yakovlev D R, Karczewski G, Wiater M, Wojtowicz T, Bayer M, Cundiff S T 2014 Phys. Rev. Lett. 112 097401
Google Scholar
[9] Li H B, Bristow A D, Siemens M E, Moody G, Cundiff S T 2013 Nat. Commun. 4 1390
Google Scholar
[10] Bell J D, Conrad R, Siemens M E 2015 Opt. Lett. 4 1157
[11] Titze M, Li H B 2017 Phys. Rev. A 96 032508
Google Scholar
[12] Dai X C, Bristow A D, Cundiff S T 2010 Phys. Rev. A 82 052503
Google Scholar
[13] Dai X C, Richter M, Li H B, Bristow A D, Falvo C, Mukamel S, Cundiff S T 2012 Phys. Rev. Lett. 108 193201
Google Scholar
[14] 赵威, 周肇宇, 杨金新, 戴星灿 2015 物理学进展 35 177
Zhao W, Zhou Z Y, Yang J X, Dai X C 2015 Prog. Phys. 35 177
[15] Zhu W D, Wang R, Zhang C F, Wang G D, Liu Y L, Zhao W, Dai X C, Wang X Y, Cerullo G, Cundiff S T, Xiao M 2017 Opt. Express 25 21115
Google Scholar
[16] Zhao W, Qin Z Y, Zhang C F, Wang G D, Li B, Dai X C, Xiao M 2019 J. Phys. Chem. Lett. 10 1251
Google Scholar
[17] Huang T Y, Li X H, Shum P P, Wang Q J, Shao X G, Wang L L, Li H Z, Wu Z F, Dong X Y 2015 Opt. Express 23 340
Google Scholar
[18] Wang L, Li X H, Wang C, Luo W F, Feng T C, Zhang Y, Zhang H 2019 Chem. Nanomater. Bio. 5 1233
[19] Liu J S, Li X H, Guo Y X, Qyyum A, Shi Z J, Feng T C, Zhang Y, Jiang C X, Liu X F 2019 Small 15 1902811
Google Scholar
[20] Zhao Y, Guo P L, Li X H, Jin Z W 2019 Carbon 149 336
Google Scholar
[21] Garrett-Roe S, Hamm P 2009 J. Chem. Phys. 130 164510
Google Scholar
[22] Mukherjee S S, Skoff D R, Middleton C T, Zanni M T 2013 J. Chem. Phys. 139 144205
Google Scholar
[23] 李淳飞 2009 非线性光学 (北京: 电子工业出版社) 第57页
Li C F 2009 Nonlinear Optics (Beijing: Electronics industry Press) p57 (in Chinese)
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图 1 四波混频原理图
Figure 1. Four wave mixing schematic.
图 2 (a) 二维时域; (b) 光子回波信号的频率坐标; (c) 二维时域投影在对应于沿
${\hat \omega _{{t'}}}$ 的切片的对角线上; (d)沿${\hat \omega _{{\tau '}}}$ 的切片对应的交叉对角线上的二维时域投影Figure 2. (a) 2D time; (b) frequency coordinates for photon echo signals; (c) 2D time projection onto the diagonal corresponding to a slice along
${\hat \omega _{{t'}}}$ ; (d) 2D time projection onto the cross diagonal corresponding to a slice along${\hat \omega _{{\tau '}}}$ .
图 3 当
${\text{δ}} {\omega _{10}} = {\text{δ}} {\omega _{20}} = 0.2\;{\rm{ THz}}$ ,${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = 0.05\;{\rm{THz}}$ ,$R = 1$ 时${S_{C{\rm{1}}}}\left( {{\omega _t}, {\omega _\tau }} \right)$ 频谱图 (a)实部; (b)虚部; (c)模Figure 3. The three-dimensional Fourier transform spectrum
${S_{C{\rm{1}}}}\left( {{\omega _t}, {\omega _\tau }} \right)$ with${\text{δ}} {\omega _{10}} = {\text{δ}} {\omega _{20}} = 0.2\;{\rm{ THz}}$ ,${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = 0.05\;{\rm{THz}}$ ,$R = 1$ : (a) Real part; (b) imaginary part; (c) module.
图 4 当
${\text{δ}} {\omega _{10}} = {\text{δ}} {\omega _{20}} = 0.2\;{\rm{THz}}$ ,${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = 0.05\;{\rm{THz}}$ ,$R = 0.5$ 时${S_{C2}}\left( {{\omega _t}, {\omega _\tau }} \right)$ 频谱图 (a)实部; (b)虚部; (c)模Figure 4. The three-dimensional Fourier transform spectrum
${S_{C2}}\left( {{\omega _t}, {\omega _\tau }} \right)$ with${\text{δ}} {\omega _{10}} = {\text{δ}} {\omega _{20}} = 0.2\;{\rm{ THz}}$ ,${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = 0.05\;{\rm{THz}}$ ,$R = 0.5$ : (a) Real part; (b) imaginary part; (c) module.
图 5 当
${\text{δ}} {\omega _{10}} = 0.3\;{\rm{THz}}$ ,${\text{δ}} {\omega _{20}} = 0.2\;{\rm{THz}}$ ,${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = 0.05 \;{\rm{THz}}$ ,$R = 1$ 时${S_{C3, E{\rm{3}}}}\left( {{\omega _t}, {\omega _\tau }} \right)$ 频谱图 (a)实部; (b)虚部; (c)模Figure 5. The three-dimensional Fourier transform spectrum
${S_{C3, E{\rm{3}}}}\left( {{\omega _t}, {\omega _\tau }} \right)$ with${\text{δ}} {\omega _{10}} = 0.3\;{\rm{THz}}$ ,${\text{δ}} {\omega _{20}} = 0.2\;{\rm{THz}}$ ,${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = $ 0.05 THz,$R = 1$ : (a) Real part; (b) imaginary part; (c) module.
图 6 当
${\text{δ}} {\omega _{10}} = 0.3\;{\rm{THz}}$ ,${\text{δ}} {\omega _{20}} = 0.2\;{\rm{THz}}$ ,${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = 0.05\;{\rm{THz}}$ ,$R = 0.5$ 时${S_{C4, E4}}\left( {{\omega _t}, {\omega _\tau }} \right)$ 频谱图 (a)实部; (b)虚部; (c)模Figure 6. The three-dimensional Fourier transform spectrum
${S_{C4, E4}}\left( {{\omega _t}, {\omega _\tau }} \right)$ with${\text{δ}} {\omega _{10}} = 0.3\;{\rm{THz}}$ ,${\text{δ}} {\omega _{20}} = 0.2\;{\rm{THz}}$ ,${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = $ 0.05 THz,$R = 0.5$ (a) Real part; (b) imaginary part; (c) module.
图 7 三维傅里叶转换频谱图 (a) 参考文献[11]中的图5(a),
${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = 0.05 \;{\rm{THz}}$ ,${\text{δ}} {\omega _{10}} = {\text{δ}} {\omega _{20}} = 0.2\;{\rm{THz}}$ , (b)$R = 1$ ; (c)$R = 0.5$ Figure 7. Three-dimensional Fourier transform spectrum: (a) Fig. 5(a) in Ref. [11],
${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = 0.05 \;{\rm{THz}}$ ,${\text{δ}} {\omega _{10}} = {\text{δ}} {\omega _{20}} = $ 0.2 THz; (b)$R = 1$ ; (c)$R = 0.5$ .
图 8 R不同时, 三维傅里叶转换频谱,
${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = 0.05\;{\rm{THz}}$ ,${\text{δ}} {\omega _{10}} = 0.3\;{\rm{THz}}$ ,${\text{δ}} {\omega _{20}} = 0.2\;{\rm{THz}}$ (a)$R = 1$ ; (b)$R = 0.5$ Figure 8. The three-dimensional Fourier transform spectrum with
${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = 0.05\;{\rm{THz}}$ ,${\text{δ}} {\omega _{10}} = 0.3\;{\rm{THz}}$ ,${\text{δ}} {\omega _{20}} = 0.2\;{\rm{THz}}$ for different R: (a)$R = 1$ ; (b)$R = 0.5$ .表 1 非均匀展宽和对角线相关系数之间的关系
Table 1. The relation between in-homogeneous line-width and the diagonal correlation coefficient.
x y z ${\text{δ}} {\omega _{10}} = {\text{δ}} {\omega _{20}}$, $R = {\rm{1}}$ 0 ${\rm{4}}{\text{δ}} \omega _{10}^2$ 0 ${\text{δ}} {\omega _{10}} = {\text{δ}} {\omega _{20}}$, $R \ne {\rm{1}}$ $2\left( {1 - R} \right){\text{δ}} \omega _{10}^2$ $2\left( {{\rm{1}} + R} \right){\text{δ}} \omega _{10}^2$ 0 ${\text{δ} } {\omega _{10} } = m{\text{δ} } {\omega _{20} },$$R = {\rm{1}}$ ${\left(1 - \dfrac{1}{m}\right)^2}{\text{δ}} \omega _{10}^2$ ${\left(1 + \dfrac{1}{m}\right)^2}{\text{δ}} \omega _{10}^2$ $\left(1- \dfrac{1}{{{m^2}}}\right){\text{δ}} \omega _{10}^2$ ${\text{δ} } {\omega _{10} } = m{\text{δ} } {\omega _{20} },$$R \ne {\rm{1}}$ $\dfrac{{({m^2} - 2 Rm + 1)}}{{{m^2}}}{\text{δ}} \omega _{10}^2$ $\dfrac{{({m^2} + 2 Rm + 1)}}{{{m^2}}}{\text{δ}} \omega _{10}^2$ $\left(1 - \dfrac{1}{{{m^2}}}\right){\text{δ}} \omega _{10}^2$ 
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[1] Ernst R R, Bodenhausen G, Wokaun A 1987 Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford: Clarendon Press)
[2] Jonas D M 2003 Annu. Rev. Phys. Chem. 54 425
Google Scholar
[3] Siemens M E, Moody G, Li H B, Bristow A D, Cundiff S T 2010 Opt. Express 18 17699
Google Scholar
[4] Fecko C J, Eaves J D, Loparo J J, Tokmakoff A, Geissler P L 2003 Science 301 1698
Google Scholar
[5] Turner D B, Wen P, Arias D H, Nelson K A, Li H B, Moody G, Siemens M E, Cundiff S T 2012 Phys. Rev. B 85 201303
Google Scholar
[6] Cundiff S T, Bristow A D, Siemen M, Li H B, Moody G, Karaiskaj D, Dai X C, Zhang T H 2012 IEEE J. Sel. Top Quant. 18 318
Google Scholar
[7] Nardin G, Moody G, Singh R, Autry T M, Li H B, Morier-Genoud F, Cundiff S T 2014 Phys. Rev. Lett. 112 046402
Google Scholar
[8] Moody G, Akimov I A, Li H B, Singh R, Yakovlev D R, Karczewski G, Wiater M, Wojtowicz T, Bayer M, Cundiff S T 2014 Phys. Rev. Lett. 112 097401
Google Scholar
[9] Li H B, Bristow A D, Siemens M E, Moody G, Cundiff S T 2013 Nat. Commun. 4 1390
Google Scholar
[10] Bell J D, Conrad R, Siemens M E 2015 Opt. Lett. 4 1157
[11] Titze M, Li H B 2017 Phys. Rev. A 96 032508
Google Scholar
[12] Dai X C, Bristow A D, Cundiff S T 2010 Phys. Rev. A 82 052503
Google Scholar
[13] Dai X C, Richter M, Li H B, Bristow A D, Falvo C, Mukamel S, Cundiff S T 2012 Phys. Rev. Lett. 108 193201
Google Scholar
[14] 赵威, 周肇宇, 杨金新, 戴星灿 2015 物理学进展 35 177
Zhao W, Zhou Z Y, Yang J X, Dai X C 2015 Prog. Phys. 35 177
[15] Zhu W D, Wang R, Zhang C F, Wang G D, Liu Y L, Zhao W, Dai X C, Wang X Y, Cerullo G, Cundiff S T, Xiao M 2017 Opt. Express 25 21115
Google Scholar
[16] Zhao W, Qin Z Y, Zhang C F, Wang G D, Li B, Dai X C, Xiao M 2019 J. Phys. Chem. Lett. 10 1251
Google Scholar
[17] Huang T Y, Li X H, Shum P P, Wang Q J, Shao X G, Wang L L, Li H Z, Wu Z F, Dong X Y 2015 Opt. Express 23 340
Google Scholar
[18] Wang L, Li X H, Wang C, Luo W F, Feng T C, Zhang Y, Zhang H 2019 Chem. Nanomater. Bio. 5 1233
[19] Liu J S, Li X H, Guo Y X, Qyyum A, Shi Z J, Feng T C, Zhang Y, Jiang C X, Liu X F 2019 Small 15 1902811
Google Scholar
[20] Zhao Y, Guo P L, Li X H, Jin Z W 2019 Carbon 149 336
Google Scholar
[21] Garrett-Roe S, Hamm P 2009 J. Chem. Phys. 130 164510
Google Scholar
[22] Mukherjee S S, Skoff D R, Middleton C T, Zanni M T 2013 J. Chem. Phys. 139 144205
Google Scholar
[23] 李淳飞 2009 非线性光学 (北京: 电子工业出版社) 第57页
Li C F 2009 Nonlinear Optics (Beijing: Electronics industry Press) p57 (in Chinese)
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