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Mutual compensation of higher-order dispersion in chirped pulse amplification system with regenerative amplifier
ARTICLE IN PRESS
Optics & Laser Technology 39 (2007) 29–33 www.elsevier.com/locate/optlastec
Mutual compensation of higher-order dispersion in chirped pulse amplification system with regenerative amplifier Zhenhong Suna, Lu Chaia,, Zhigang Zhanga, Chingyue Wanga, Weili Zhanga,b, Xudong Xiea,c, Xiaojun Huanga,c, Xiaodong Yuanc a
School of Precision Instruments and Optoelectronics Engineering, University of Tianjin, Key Laboratory of Optoelectronic Information Technical Science, EMC, Tianjin 300072, PR China b School of Electrical and Computer Engineering, Oklahoma State University, OK 74078, USA c Research Center for Laser Fusion, CAEP, Mianyang 621900, PR China Received 15 December 2004; received in revised form 20 May 2005; accepted 23 May 2005 Available online 14 July 2005
Abstract In this paper, based on ray tracing, the approach of mutual compensation that introduced properly the negative third-order dispersion to balance the higher-order dispersion in the chirped pulse amplification (CPA) system with regenerative amplifier is presented. A shorter pulse with near-transform-limitation can be generated by this method than by the conventional approach that zeros the second- and the third-order dispersion successively in the CPA system. r 2005 Elsevier Ltd. All rights reserved. PACS: 42.65.Re; 42.15.Dp Keywords: Mutual compensation; Chirped pulse amplification; Ray tracing
1. Introduction Chirped pulse amplification [1] (CPA) has become a standard technique to achieve amplification of ultrashort laser pulses to the terawatt power level [2]. A CPA laser system that produces a peak power of 0.85 PW for the pulse duration of 33 fs has been reported [3]. In the CPA system, the seed pulses, which come from the oscillator, are first stretched by a typical factor of 10000 with a grating stretcher, then amplified in the regenerative amplifier or the multi-pass amplifier, and eventually recompressed to near the original pulse duration with a Treacy-type compressor [4]. With the goal being to obtain the shortest pulse, the compressor must precisely compensate for the dispersion introduced by the stretcher and the amplifier. There are two Corresponding author.
E-mail address: [email protected] (L. Chai). 0030-3992/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2005.05.014
important factors in a CPA laser system that significantly affects the precise compensation of the dispersion: the precise calculation of the dispersion of the whole CPA system and the proper selection of the system’s parameters to compensate for the dispersion of the CPA system. Based on the ray-tracing method, the conventional theoretical work on the propagation of light pulse in a CPA system is to use a polynomial expansion of the phase modulation and to try to cancel the different orders one by one. In this way, a Treacy-type compressor can compensate for phase modulation up to the third order by control of the grating pair separation and the incident angle [5,6]. With this understanding, a fourth-order dispersion-limited CPA system can be obtained by selecting the proper separation of a stretcher’s grating pair. However, when the system is fourth-order dispersion-limited, the fifth-order dispersion will become the dominating factor so that the
ARTICLE IN PRESS Z. Sun et al. / Optics & Laser Technology 39 (2007) 29–33
2. Conventional way of dispersion compensation When the seed pulse propagates through the dispersive elements of the CPA system, the spectral phase is accumulated, which can be represented by a Taylor’s series expansion as qf 1 q2 f fðoÞ ¼ fðo0 Þ þ ðo o0 Þ þ ðo o0 Þ2 qo o0 2! qo2 o0 1 q3 f 1 q4 f 3 þ ðo o0 Þ þ ðo o0 Þ4 3! qo3 o0 4! qo4 o0 þ þ ,
ð1Þ
where o is the optical frequency and o0 is the central frequency of the expansion. The first term, fðo0 Þ, is a constant, and qf=qojo0 , q2 f=qo2 jo0 , q3 f=qo3 jo0 , q4 f=qo4 jo0 are called the group delay (GD), the group delay dispersion (GDD), the thirdorder dispersion (TOD), and the fourth dispersion (FOD), respectively. GD is only an overall time shift to the shape of the pulse. GDD and the higher-order dispersion produce a pulse broadening or distortion, where the effect of the latter term is significantly smaller than the former term. For the conventional way of dispersion compensation, the second- and third-order terms are canceled and the fourth-order dispersion is limited. This is enough for pulses longer than 30 fs [7], but for shorter pulses (10 fs pulses) the higher-order terms must be considered. It is difficult to eliminate the higher-order dispersion because of the limitation of the controllable parameters in the CPA system, nevertheless one can still optimize the parameters to partially counteract the effect of the higher-order terms by introducing the lower-order dispersion. The model of the CPA system consists of an allreflective Martinez stretcher [8], a regenerative amplifier, and a Treacy-type compressor. We did not here consider the effect of gain narrowing and self-phase-modulation (SPM). SPM can be negative during regenerative amplification due to the low peak intensity after the pulse is stretched to several tens or hundreds of picoseconds. The gain narrowing is an important concern in the multi-TW laser system but is not the focus of this paper with the single-stage regenerative
amplifier. The focus of this paper is the effect of higherorder phase errors induced in the regenerative cavity, stretcher and compressor, as well as the ability to compensate them. The conventional way of dispersion compensation is to adjust the grating pair separation and the angle of incidence in the compressor to cancel the dispersion up to the third order. Meanwhile, one adjusts the grating pair separation in the stretcher to obtain the flattest phase curve and group delay curve shown in Fig. 1. We choose typical parameters for the model system: the regenerative amplifier contains 10 mm long Ti:sapphire, 30 mm long KD P Pockels cell, and 14 mm long calcite polarizer, and 10 mm long TGG (terbium gallium garnet) in the isolator. A pulse takes 15 round tips in the regenerative cavity before it is released. The curvature radius of the mirror used in the stretcher is 1000 cm:s1 , the distance from the mirror to the diffraction grating [5], is set at 296 mm. The groove density of the grating is 1200 lines/mm for both the stretcher and the compressor. The incident angle of the stretcher is 10 larger than the Littrow angle at 800 nm. If a 10 fs of Fouriertransform limit Gaussian pulse was input through the system, a 17.8 fs of output pulse could be obtained analytically. The broadening to the pulse came from the residual higher-order dispersion and the limitation of bandwidth (calculating range of wavelength from 680 to 930 nm). Moreover, the dispersion from second to seventh order of the system and the broadening to the 10 fs Gaussian pulse every order were calculated, respectively, as shown in Table 1. In Table 1, we can observe two features. First, the conventional way of dispersion compensation is well behaved, i.e. the second- and third-order dispersion approach zero. They have hardly any broadening to the pulse (little broadening comes from the limitation of the bandwidth). Second, when the system parameters are optimized, the fourth-order dispersion-limited CPA system is achieved, and the fifth-order dispersion has
1000 40 800
Phase 20
GD 600
0 400 -20 200 -40 0 -60 650
700
750 800 850 Wavelength (nm)
900
950
Fig. 1. Phase curve and group delay curve of the CPA system.
Group delay(fs)
optimum result is not achieved. In this paper, we present a global optimization with mutual compensation by introducing proper third-order dispersion (not canceling it to zero) to balance the higher-order dispersion in the CPA system with regenerative amplifier. A shorter pulse with near-transform-limitation can be generated by this method than that by the conventional approach which zeros the second- and the third-order dispersion successively in the CPA system.
Phase (rad)
30
ARTICLE IN PRESS Z. Sun et al. / Optics & Laser Technology 39 (2007) 29–33
31
Table 1 Dispersion from second to seventh order of the system and broadening to the 10 fs Gaussian pulse every order, respectively Orders
2
3
4
5
6
7
Magnitude of each order FWHM
0 ðfs2 Þ 11.0 fs
11 ðfs3 Þ 11.0 fs
2274 ðfs4 Þ 11.5 fs
890 822 ðfs5 Þ 17.6 fs
7.92E6 ðfs6 Þ 15.8 fs
6.25E7 ðfs7 Þ 13.1 fs
than at the longer wavelength region, the total GD of all orders is smaller at the shorter wavelength region than at the longer wavelength region because of the mutual canceling. That is why the synthetical broadening to the pulse is smaller than the simple accumulation of each order of broadening. Based on the analysis above, if we are not able to exactly eliminate all second- and third-order dispersion but leave some of them to partially cancel the higherorder dispersion, we should get a wider range of flat dispersive bandwidth and obtain a shorter pulse output than the conventional way. The method can be called mutual dispersion compensation.
1000 800
3rd order 4th order 5th order 6th order 7th order Total
Group delay (fs)
600 400 200 0 -200 -400 -600 650
700
750 800 850 Wavelength (nm)
900
950
Fig. 2. GD curves generated, respectively, by each order of dispersion and the total GD curve of all orders.
the maximal effect to the pulse. We find in Table 1 that only the fifth-order dispersion itself has already broadened the pulse to 17.6 fs, yet the synthetical effect of all order of dispersion merely broadens the pulse to 17.8 fs as above. The synthetical effect is much better than the simple accumulation of each order of broadening. The explanation for this is as follows. The group delay is related to spectral phase by qfðoÞ . qo Substituting (1) into (2) then it is given by qf q2 f tðoÞ ¼ þ ðo o0 Þ qo o0 qo2 o0 1 q3 f 1 q4 f 2 ðo o0 Þ þ ðo o0 Þ3 þ 2! qo3 o0 3! qo4 o0
tðoÞ ¼
þ þ .
(2)
ð3Þ
Substituting the dispersion of Table 1 each order into (3) was shown in Fig. 2. In Fig. 2, the values of GD generated by each order dispersion are different at the higher frequency ðo4o0 Þ and the lower frequency ðooo0 Þ. At the higher frequency region, the values of GD are partially positive and partially negative, while the values of GD are all positive at the lower frequency region. Therefore, in Fig. 2, even if the absolute value of GD each order is larger at the shorter wavelength region
3. Mutual dispersion compensation method From Eq. (3), it is known that the effect of odd order dispersion to GD curve is symmetrical and even order dispersion to GD curve is dissymmetrical. To the Ushape of GD curve shown in Fig. 1, the number of TOD can increase or decrease both sides of the curve in the same time, so that wider flat dispersive bandwidth can be obtained by adding the proper number of TOD. However, the GDD increases one side of the curve and decreases another side simultaneously, so that the flat dispersive bandwidth of GD curve cannot be extended by adding the GDD. Based on the GD curve of Fig. 1, we introduce different numbers of GDD or TOD to the curve, respectively. Fig. 3 shows several typical GD curves among the calculations adding GDD or TOD comparing with the original. Table 2 shows the typical five cases of the FWHM of the output pulse broadening for 10 fs of transform-limited Gaussian pulse. In Table 2, introducing a little positive or negative GDD does not have evident change to the FWHM of the output pulse (introducing a large number of GDD will have significant broadening to the pulse). Introducing positive TOD results in more broadening to the pulse than the conventional way does. Only introducing negative TOD, a shorter pulse can be obtained. From another point of view, fifth-order dispersion has the maximal effect on the pulse. By introducing opposite sign of TOD, it can partially cancel the effect of fifthorder dispersion. The magnitude of negative TOD must be properly introduced, otherwise the intensity profile quality of the output pulse will become bad. Fig. 4
ARTICLE IN PRESS Z. Sun et al. / Optics & Laser Technology 39 (2007) 29–33
32
400
25 20
0fs3 0fs2
-1500fs3 Group delay (fs)
Group delay (fs)
15
-50fs2 10 +50fs2 5
-3000fs3
200
-5000fs3
0 0 -5 -10 760
780
(a)
800 820 Wavelength (nm)
700
840
750
(a)
30
800 Wavelength (nm)
-1500fs3(15.4fs)
25 Normallized intensity
0fs3 Group delay (fs)
-1500fs3 15
+1500fs3
10 5
-3000fs3(13.7fs)
0.8
-5000fs3(14.2fs)
0.6 0.4 0.2
0 -5
0.0
-10
-20 740
(b)
900
0fs3(17.6fs)
1.0
20
850
760
780 800 820 Wavelength (nm)
840
860
Fig. 3. Comparison of GD curves with different magnitudes of GDD or TOD: (a) adding GDD; (b) adding TOD.
Table 2 FWHM of the output pulse produced by different cases for an initially spectral-limited 10 fs Gaussian pulse Magnitude of 50 fs2 þ50 fs2 1500 fs3 þ1500 fs3 0 fs2 introduced dispersion 0 fs3 FWHM of output 17.6 fs 17.7 fs 18.0 fs 15.4 fs 19.7 fs pulse
shows the comparison of GD curves with different numbers of TOD and corresponding intensity profiles of output pulse for an initially Fourier-transform-limited 10 fs Gaussian pulse. The FWHM of output pulse becomes even shorter (13.7 fs) with the TOD of 3000 fs3 , but sidelobes of the pulse have increased significantly. The intensity profile of output pulse becomes worse when the TOD is 5000 fs3 . Only when the two factors of dispersive bandwidth of GD curve, flatness and width, are considered synthetically, can we get simultaneously a shorter output pulse and good
(b)
-10
0 10 Time (fs)
20
30
Fig. 4. Comparison of GD curves with different numbers of TOD and corresponding intensity profiles of output pulse for an initially spectral limited 10 fs Gaussian pulse: (a) GD curves; (b) intensity profiles.
intensity profile. Thereby, only introducing 1500 fs3 of TOD is the best value to exhibit a small wings pulse profile and a markedly broadened flat dispersive bandwidth of GD curve under the flat dispersive bandwidth of the spectrum range was defined in the scope of 5 fs.
4. Conclusion Based on the ray-tracing method, we analyzed the effect of each order of dispersion to the pulses in a CPA system with regenerative amplifier. It is shown that fifthorder dispersion has the maximal effect on the pulse with the conventional way of dispersion compensation. Instead of zeroing each order in succession, we have adopted the method of mutual dispersion compensation. The effect of higher-order dispersion can be partially balanced by introducing the proper number of negative TOD, so that wider flat dispersive bandwidth and shorter output pulse were obtained.
ARTICLE IN PRESS Z. Sun et al. / Optics & Laser Technology 39 (2007) 29–33
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant no. 60178007), the Foundation of LLF (Grant no. 51480040103JW1401), the National Key Basic Research Special Foundation (Grant no. G1999075201-2), and the Foundation of Chinese Ministry of Education for Outstanding Young Teachers in Universities. References [1] Strickland D, Mourou G. Compression of amplified chirped optical pulses. Opt Commun 1985;56:219–21. [2] Yamakawa K, Barty CPJ. Ultrafast, ultrahigh-peak, and highaverage power Ti:sapphire laser system and its applications. IEEE J Sel Top Quantum Electron 2000;6:658–75.
33
[3] Aoyama M, Yamakawa K, Akahane Y, Ma J, Inoue N, Ueda H, Kiriyama H. 0.85-PW,33fs Ti:sapphire laser. Opt Lett 2003; 28:1594–6. [4] Treacy EB. Optical pulse compression with diffraction gratings. IEEE J Quantum Electron 1969;5:454–8. [5] Zhang Z, Yagi T, Arisawa T. Ray-tracing model for stretcher dispersion calculation. Appl Opt 1997;6:3393–9. [6] Jiang J, Zhang Z, Hasama T. Evaluation of chirped-pulseamplification systems with Offner triplet telescope stretchers. J Opt Soc Am B 2002;19:678–83. [7] Kane S, Squier J. Fourth-order-dispersion limitations of aberration-free chirped-pulse amplification systems. J Opt Soc Am B 1997;14:1237–44. [8] Martinez OE. 3000 times grating compressor with positive group velocity dispersion: application to fiber compensation in 1.3–1:6 mm region. IEEE J Quantum Electron 1987;23:59–64.
Optics & Laser Technology 39 (2007) 29–33 www.elsevier.com/locate/optlastec
Mutual compensation of higher-order dispersion in chirped pulse amplification system with regenerative amplifier Zhenhong Suna, Lu Chaia,, Zhigang Zhanga, Chingyue Wanga, Weili Zhanga,b, Xudong Xiea,c, Xiaojun Huanga,c, Xiaodong Yuanc a
School of Precision Instruments and Optoelectronics Engineering, University of Tianjin, Key Laboratory of Optoelectronic Information Technical Science, EMC, Tianjin 300072, PR China b School of Electrical and Computer Engineering, Oklahoma State University, OK 74078, USA c Research Center for Laser Fusion, CAEP, Mianyang 621900, PR China Received 15 December 2004; received in revised form 20 May 2005; accepted 23 May 2005 Available online 14 July 2005
Abstract In this paper, based on ray tracing, the approach of mutual compensation that introduced properly the negative third-order dispersion to balance the higher-order dispersion in the chirped pulse amplification (CPA) system with regenerative amplifier is presented. A shorter pulse with near-transform-limitation can be generated by this method than by the conventional approach that zeros the second- and the third-order dispersion successively in the CPA system. r 2005 Elsevier Ltd. All rights reserved. PACS: 42.65.Re; 42.15.Dp Keywords: Mutual compensation; Chirped pulse amplification; Ray tracing
1. Introduction Chirped pulse amplification [1] (CPA) has become a standard technique to achieve amplification of ultrashort laser pulses to the terawatt power level [2]. A CPA laser system that produces a peak power of 0.85 PW for the pulse duration of 33 fs has been reported [3]. In the CPA system, the seed pulses, which come from the oscillator, are first stretched by a typical factor of 10000 with a grating stretcher, then amplified in the regenerative amplifier or the multi-pass amplifier, and eventually recompressed to near the original pulse duration with a Treacy-type compressor [4]. With the goal being to obtain the shortest pulse, the compressor must precisely compensate for the dispersion introduced by the stretcher and the amplifier. There are two Corresponding author.
E-mail address: [email protected] (L. Chai). 0030-3992/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2005.05.014
important factors in a CPA laser system that significantly affects the precise compensation of the dispersion: the precise calculation of the dispersion of the whole CPA system and the proper selection of the system’s parameters to compensate for the dispersion of the CPA system. Based on the ray-tracing method, the conventional theoretical work on the propagation of light pulse in a CPA system is to use a polynomial expansion of the phase modulation and to try to cancel the different orders one by one. In this way, a Treacy-type compressor can compensate for phase modulation up to the third order by control of the grating pair separation and the incident angle [5,6]. With this understanding, a fourth-order dispersion-limited CPA system can be obtained by selecting the proper separation of a stretcher’s grating pair. However, when the system is fourth-order dispersion-limited, the fifth-order dispersion will become the dominating factor so that the
ARTICLE IN PRESS Z. Sun et al. / Optics & Laser Technology 39 (2007) 29–33
2. Conventional way of dispersion compensation When the seed pulse propagates through the dispersive elements of the CPA system, the spectral phase is accumulated, which can be represented by a Taylor’s series expansion as qf 1 q2 f fðoÞ ¼ fðo0 Þ þ ðo o0 Þ þ ðo o0 Þ2 qo o0 2! qo2 o0 1 q3 f 1 q4 f 3 þ ðo o0 Þ þ ðo o0 Þ4 3! qo3 o0 4! qo4 o0 þ þ ,
ð1Þ
where o is the optical frequency and o0 is the central frequency of the expansion. The first term, fðo0 Þ, is a constant, and qf=qojo0 , q2 f=qo2 jo0 , q3 f=qo3 jo0 , q4 f=qo4 jo0 are called the group delay (GD), the group delay dispersion (GDD), the thirdorder dispersion (TOD), and the fourth dispersion (FOD), respectively. GD is only an overall time shift to the shape of the pulse. GDD and the higher-order dispersion produce a pulse broadening or distortion, where the effect of the latter term is significantly smaller than the former term. For the conventional way of dispersion compensation, the second- and third-order terms are canceled and the fourth-order dispersion is limited. This is enough for pulses longer than 30 fs [7], but for shorter pulses (10 fs pulses) the higher-order terms must be considered. It is difficult to eliminate the higher-order dispersion because of the limitation of the controllable parameters in the CPA system, nevertheless one can still optimize the parameters to partially counteract the effect of the higher-order terms by introducing the lower-order dispersion. The model of the CPA system consists of an allreflective Martinez stretcher [8], a regenerative amplifier, and a Treacy-type compressor. We did not here consider the effect of gain narrowing and self-phase-modulation (SPM). SPM can be negative during regenerative amplification due to the low peak intensity after the pulse is stretched to several tens or hundreds of picoseconds. The gain narrowing is an important concern in the multi-TW laser system but is not the focus of this paper with the single-stage regenerative
amplifier. The focus of this paper is the effect of higherorder phase errors induced in the regenerative cavity, stretcher and compressor, as well as the ability to compensate them. The conventional way of dispersion compensation is to adjust the grating pair separation and the angle of incidence in the compressor to cancel the dispersion up to the third order. Meanwhile, one adjusts the grating pair separation in the stretcher to obtain the flattest phase curve and group delay curve shown in Fig. 1. We choose typical parameters for the model system: the regenerative amplifier contains 10 mm long Ti:sapphire, 30 mm long KD P Pockels cell, and 14 mm long calcite polarizer, and 10 mm long TGG (terbium gallium garnet) in the isolator. A pulse takes 15 round tips in the regenerative cavity before it is released. The curvature radius of the mirror used in the stretcher is 1000 cm:s1 , the distance from the mirror to the diffraction grating [5], is set at 296 mm. The groove density of the grating is 1200 lines/mm for both the stretcher and the compressor. The incident angle of the stretcher is 10 larger than the Littrow angle at 800 nm. If a 10 fs of Fouriertransform limit Gaussian pulse was input through the system, a 17.8 fs of output pulse could be obtained analytically. The broadening to the pulse came from the residual higher-order dispersion and the limitation of bandwidth (calculating range of wavelength from 680 to 930 nm). Moreover, the dispersion from second to seventh order of the system and the broadening to the 10 fs Gaussian pulse every order were calculated, respectively, as shown in Table 1. In Table 1, we can observe two features. First, the conventional way of dispersion compensation is well behaved, i.e. the second- and third-order dispersion approach zero. They have hardly any broadening to the pulse (little broadening comes from the limitation of the bandwidth). Second, when the system parameters are optimized, the fourth-order dispersion-limited CPA system is achieved, and the fifth-order dispersion has
1000 40 800
Phase 20
GD 600
0 400 -20 200 -40 0 -60 650
700
750 800 850 Wavelength (nm)
900
950
Fig. 1. Phase curve and group delay curve of the CPA system.
Group delay(fs)
optimum result is not achieved. In this paper, we present a global optimization with mutual compensation by introducing proper third-order dispersion (not canceling it to zero) to balance the higher-order dispersion in the CPA system with regenerative amplifier. A shorter pulse with near-transform-limitation can be generated by this method than that by the conventional approach which zeros the second- and the third-order dispersion successively in the CPA system.
Phase (rad)
30
ARTICLE IN PRESS Z. Sun et al. / Optics & Laser Technology 39 (2007) 29–33
31
Table 1 Dispersion from second to seventh order of the system and broadening to the 10 fs Gaussian pulse every order, respectively Orders
2
3
4
5
6
7
Magnitude of each order FWHM
0 ðfs2 Þ 11.0 fs
11 ðfs3 Þ 11.0 fs
2274 ðfs4 Þ 11.5 fs
890 822 ðfs5 Þ 17.6 fs
7.92E6 ðfs6 Þ 15.8 fs
6.25E7 ðfs7 Þ 13.1 fs
than at the longer wavelength region, the total GD of all orders is smaller at the shorter wavelength region than at the longer wavelength region because of the mutual canceling. That is why the synthetical broadening to the pulse is smaller than the simple accumulation of each order of broadening. Based on the analysis above, if we are not able to exactly eliminate all second- and third-order dispersion but leave some of them to partially cancel the higherorder dispersion, we should get a wider range of flat dispersive bandwidth and obtain a shorter pulse output than the conventional way. The method can be called mutual dispersion compensation.
1000 800
3rd order 4th order 5th order 6th order 7th order Total
Group delay (fs)
600 400 200 0 -200 -400 -600 650
700
750 800 850 Wavelength (nm)
900
950
Fig. 2. GD curves generated, respectively, by each order of dispersion and the total GD curve of all orders.
the maximal effect to the pulse. We find in Table 1 that only the fifth-order dispersion itself has already broadened the pulse to 17.6 fs, yet the synthetical effect of all order of dispersion merely broadens the pulse to 17.8 fs as above. The synthetical effect is much better than the simple accumulation of each order of broadening. The explanation for this is as follows. The group delay is related to spectral phase by qfðoÞ . qo Substituting (1) into (2) then it is given by qf q2 f tðoÞ ¼ þ ðo o0 Þ qo o0 qo2 o0 1 q3 f 1 q4 f 2 ðo o0 Þ þ ðo o0 Þ3 þ 2! qo3 o0 3! qo4 o0
tðoÞ ¼
þ þ .
(2)
ð3Þ
Substituting the dispersion of Table 1 each order into (3) was shown in Fig. 2. In Fig. 2, the values of GD generated by each order dispersion are different at the higher frequency ðo4o0 Þ and the lower frequency ðooo0 Þ. At the higher frequency region, the values of GD are partially positive and partially negative, while the values of GD are all positive at the lower frequency region. Therefore, in Fig. 2, even if the absolute value of GD each order is larger at the shorter wavelength region
3. Mutual dispersion compensation method From Eq. (3), it is known that the effect of odd order dispersion to GD curve is symmetrical and even order dispersion to GD curve is dissymmetrical. To the Ushape of GD curve shown in Fig. 1, the number of TOD can increase or decrease both sides of the curve in the same time, so that wider flat dispersive bandwidth can be obtained by adding the proper number of TOD. However, the GDD increases one side of the curve and decreases another side simultaneously, so that the flat dispersive bandwidth of GD curve cannot be extended by adding the GDD. Based on the GD curve of Fig. 1, we introduce different numbers of GDD or TOD to the curve, respectively. Fig. 3 shows several typical GD curves among the calculations adding GDD or TOD comparing with the original. Table 2 shows the typical five cases of the FWHM of the output pulse broadening for 10 fs of transform-limited Gaussian pulse. In Table 2, introducing a little positive or negative GDD does not have evident change to the FWHM of the output pulse (introducing a large number of GDD will have significant broadening to the pulse). Introducing positive TOD results in more broadening to the pulse than the conventional way does. Only introducing negative TOD, a shorter pulse can be obtained. From another point of view, fifth-order dispersion has the maximal effect on the pulse. By introducing opposite sign of TOD, it can partially cancel the effect of fifthorder dispersion. The magnitude of negative TOD must be properly introduced, otherwise the intensity profile quality of the output pulse will become bad. Fig. 4
ARTICLE IN PRESS Z. Sun et al. / Optics & Laser Technology 39 (2007) 29–33
32
400
25 20
0fs3 0fs2
-1500fs3 Group delay (fs)
Group delay (fs)
15
-50fs2 10 +50fs2 5
-3000fs3
200
-5000fs3
0 0 -5 -10 760
780
(a)
800 820 Wavelength (nm)
700
840
750
(a)
30
800 Wavelength (nm)
-1500fs3(15.4fs)
25 Normallized intensity
0fs3 Group delay (fs)
-1500fs3 15
+1500fs3
10 5
-3000fs3(13.7fs)
0.8
-5000fs3(14.2fs)
0.6 0.4 0.2
0 -5
0.0
-10
-20 740
(b)
900
0fs3(17.6fs)
1.0
20
850
760
780 800 820 Wavelength (nm)
840
860
Fig. 3. Comparison of GD curves with different magnitudes of GDD or TOD: (a) adding GDD; (b) adding TOD.
Table 2 FWHM of the output pulse produced by different cases for an initially spectral-limited 10 fs Gaussian pulse Magnitude of 50 fs2 þ50 fs2 1500 fs3 þ1500 fs3 0 fs2 introduced dispersion 0 fs3 FWHM of output 17.6 fs 17.7 fs 18.0 fs 15.4 fs 19.7 fs pulse
shows the comparison of GD curves with different numbers of TOD and corresponding intensity profiles of output pulse for an initially Fourier-transform-limited 10 fs Gaussian pulse. The FWHM of output pulse becomes even shorter (13.7 fs) with the TOD of 3000 fs3 , but sidelobes of the pulse have increased significantly. The intensity profile of output pulse becomes worse when the TOD is 5000 fs3 . Only when the two factors of dispersive bandwidth of GD curve, flatness and width, are considered synthetically, can we get simultaneously a shorter output pulse and good
(b)
-10
0 10 Time (fs)
20
30
Fig. 4. Comparison of GD curves with different numbers of TOD and corresponding intensity profiles of output pulse for an initially spectral limited 10 fs Gaussian pulse: (a) GD curves; (b) intensity profiles.
intensity profile. Thereby, only introducing 1500 fs3 of TOD is the best value to exhibit a small wings pulse profile and a markedly broadened flat dispersive bandwidth of GD curve under the flat dispersive bandwidth of the spectrum range was defined in the scope of 5 fs.
4. Conclusion Based on the ray-tracing method, we analyzed the effect of each order of dispersion to the pulses in a CPA system with regenerative amplifier. It is shown that fifthorder dispersion has the maximal effect on the pulse with the conventional way of dispersion compensation. Instead of zeroing each order in succession, we have adopted the method of mutual dispersion compensation. The effect of higher-order dispersion can be partially balanced by introducing the proper number of negative TOD, so that wider flat dispersive bandwidth and shorter output pulse were obtained.
ARTICLE IN PRESS Z. Sun et al. / Optics & Laser Technology 39 (2007) 29–33
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant no. 60178007), the Foundation of LLF (Grant no. 51480040103JW1401), the National Key Basic Research Special Foundation (Grant no. G1999075201-2), and the Foundation of Chinese Ministry of Education for Outstanding Young Teachers in Universities. References [1] Strickland D, Mourou G. Compression of amplified chirped optical pulses. Opt Commun 1985;56:219–21. [2] Yamakawa K, Barty CPJ. Ultrafast, ultrahigh-peak, and highaverage power Ti:sapphire laser system and its applications. IEEE J Sel Top Quantum Electron 2000;6:658–75.
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