高階諧波是一種從極紫外光延伸至 X 光的高相干性光源，
它是藉由高強度雷射游離氣體原子而產生。高階諧波的轉換效
率與入射光源在介質中作用的相位匹配有關，主要包括四個因
素：中性氣體色散 (neutral gas dispersion)、電漿色散 (plasma
dispersion)、光束幾何相位變化 (geometrical phase shift) 與內
在偶極子相位變化 (intrinsic dipole phase variation)。在低游
離率的情況下，可以透過調整雷射強度游離適當比列的原子，
使電漿色散與中性氣體色散達成平衡，藉此達成相位匹配，然
而為了獲得更高階數的高階諧波，需要提高雷射的強度來游離
氣體，以達到更高的截止能量，但是此時游離率會過高而無法
達成相位匹配，導致能量轉換效率降低。
本實驗為了尋找解決能量轉換效率問題的方法，我們針對第
27 階高階諧波進行實驗，詳細紀錄高階諧波產生時的雷射狀
態、氣體狀態、兩者產生作用時的游離狀態，以此直接測量出
高階諧波的相位變化，然後使用一道側向雷射控制作用長度，
得到高階諧波與雷射作用距離的關係，最終統整高階諧波產生
時的詳細條件，以此補助在高游離態下高階諧波產生的研究。

High-order harmonic generation (HHG) is a kind of ultrashort coherent EUV/x-ray source. It is produced by ionizing
gas atoms with a high-intensity laser. The conversion efficiency of high-order harmonic generation is determined by
the relative phase between the driving field and the harmonic
field in the medium. It is determined with four factors: neutral gas dispersion, plasma dispersion, geometrical phase shift,
and intrinsic dipole phase variation. In low ionization condition, phase-matching condition can be achieved by balancing
the negative plasma dispersion and the positive neutral gas
dispersion.
In this thesis we focus on the generation of the 27th order
harmonic. We made a tomographic measurement of its growing process, and characterized its complete phase-matching
condition. We found that the phase-matching condition is
nonuniforn in both longitudinal and transverse directions, and
the major contribution of the 27th harmonic yield comes from
the short-trajectory emission from the central part of the driving pulse. These techniques could be beneficial for pursuing
efficient high-harmonic generation in shorter wavelengths.

[1] C. Winterfeldt, C. Spielmann, and G. Gerber, “Colloquium: Optimal control of high-harmonic generation,” Rev. Mod. Phys. 80,
117–140 (2008).
http://link.aps.org/doi/10.1103/RevModPhys.80.117
[2] T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers
of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[3] P. B. Corkum and F. Krausz, “Attosecond science,” Nature Physics
3, 381–387 (2007).
[4] H. Kapteyn, O. Cohen, I. Christov, and M. Murnane, “Harnessing
Attosecond Science in the Quest for Coherent X-rays,” Science
317, 775–778 (2007).
https://science.sciencemag.org/content/317/5839/775
[5] A. McPherson, G. Gibson, H. Jara, U. Johann, T. S. Luk, I. A.
McIntyre, K. Boyer, and C. K. Rhodes, “Studies of multiphoton
production of vacuum-ultraviolet radiation in the rare gases,” J.

Opt. Soc. Am. B 4, 595–601 (1987).
http://josab.osa.org/abstract.cfm?URI=josab-4-4-595
[6] A. L＇Huillier and P. Balcou, “High-order harmonic generation in
rare gases with a 1-ps 1053-nm laser,” Phys. Rev. Lett. 70, 774–
777 (1993).
[7] Z. Chang, A. Rundquist, H. Wang, M. M. Murnane, and H. C.
Kapteyn, “Generation of coherent soft x rays at 2.7 nm using high
harmonics,” Phys. Rev. Lett. 79, 2967–2970 (1997).
[8] C. Spielmann, N. H. Burnett, S. Sartania, R. Koppitsch,
M. Schnurer, C. Kan, M. Lenzner, P. Wobrauschek, and F. Krausz,
“Generation of coherent x-rays in the water window using 5-
femtosecond laser pulses,” Science 278, 661–664 (1997).
[9] P. B. Corkum, “Plasma perspective on strong field multiphoton
ionization,” Phys. Rev. Lett. 71, 1994–1997 (1993).
http://link.aps.org/doi/10.1103/PhysRevLett.71.1994
[10] M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B.
Corkum, “Theory of high-harmonic generation by low-frequency
laser fields,” Phys. Rev. A 49, 2117–2132 (1994).
http://link.aps.org/doi/10.1103/PhysRevA.49.2117

[11] A. Rundquist, C. G. D. III, Z. Chang, C. Herne, S. Backus, M. M.
Murnane, and H. C. Kapteyn, “Phase-Matched Generation of Coherent Soft X-rays,” Science 280, 1412–1415 (1998).
http://www.sciencemag.org/content/280/5368/1412.abstract
[12] C. G. Durfee, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C. Kapteyn, “Phase Matching of High-Order Harmonics in Hollow Waveguides,” Phys. Rev. Lett. 83, 2187–2190
(1999).
http://link.aps.org/doi/10.1103/PhysRevLett.83.2187
[13] Kazamias, S., Weihe, F., Douillet, D., Valentin, C., Planchon, T.,
Sebban, S., Grillon, G., Augé, F., Hulin, D., and Balcou, Ph.,
“High order harmonic generation optimization with an apertured
laser beam,” Eur. Phys. J. D 21, 353–359 (2002).
https://doi.org/10.1140/epjd/e2002-00193-0
[14] H.-W. Sun, P.-C. Huang, Y.-H. Tzeng, J.-T. Huang, C. D. Lin,
C. Jin, and M.-C. Chen, “Extended phase matching of high harmonic generation by plasma-induced defocusing,” Optica 4, 976–
981 (2017).
[15] C. M. Heyl, J. Güdde, A. L’Huillier, and U. Höfer, “High-order
harmonic generation with µJ laser pulses at high repetition rates,”

J. Phys. B: At. Mol. Opt. Phys. 45, 074 020 (2012).
https://doi.org/10.1088
[16] J. Peatross, S. Voronov, and I. Prokopovich, “Selective zoning of
high harmonic emission using counter-propagating light,” Opt. Express 1, 114–125 (1997).
[17] S. L. Voronov, I. Kohl, J. B. Madsen, J. Simmons, N. Terry,
J. Titensor, Q. Wang, and J. Peatross, “Control of Laser HighHarmonic Generation with Counterpropagating Light,” Phys. Rev.
Lett. 87, 133 902 (2001).
[18] X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn,
M. M. Murnane, and O. Cohen, “Quasi-phase-matching and
quantum-path control of high-harmonic generation using counterpropagating light,” Nature Physics 3, 270–275 (2007).
[19] J. B. Madsen, L. A. Hancock, S. L. Voronov, and J. Peatross,
“High-order harmonic generation in crossed laser beams,” J. Opt.
Soc. Am. B 20, 166–170 (2003).
http://josab.osa.org/abstract.cfm?URI=josab-20-1-166
[20] J.-C. Chanteloup, “Multiple-wave lateral shearing interferometry
for wave-front sensing,” Appl. Opt. 44, 1559–1571 (2005).
http://ao.osa.org/abstract.cfm?URI=ao-44-9-1559

[21] T.-S. Hung, C.-H. Yang, J. Wang, S. yuan Chen, J.-Y. Lin, and
H. hsin Chu, “A 110-TW multiple-beam laser system with a 5-
TW wavelength-tunable auxiliary beam for versatile control of laserplasma interaction,” Appl. Phys. B 117, 1189–1200 (2014).
http://link.springer.com/article/10.1007%2Fs00340-014-5943-6
[22] E. A. Gibson, Quasi-Phase Matching of Soft X-ray Light from
High-Order Harmonic Generation using, Ph.D. thesis, University
of Colorado (2004).
[23] C.-H. Yang, S.-C. Kao, J. Wang, and H.-H. Chu, “Synthesis of a
Beat-Wave Pulse Train with Increasing Pulse Separation for QuasiPhase-Matched High-Harmonic Generation,” J. Opt. Soc. Am. B
31, 1793–1800 (2014).
[24] E. Constant, D. Garzella, P. Breger, E. Mével, C. Dorrer, C. L.
Blanc, F. Salin, and P. Agostini, “Optimizing High Harmonic Generation in Absorbing Gases: Model and Experiment,” Phys. Rev.
Lett 82 (1999).