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The inverse problem of chirp pulse propagating through Ti:sapphire amplifier
Optik 115, No. 5 (2004) 201–204 http://www.elsevier.de/ijleo
International Journal for Light and Electron Optics
The inverse problem of chirp pulse propagating through Ti:sapphire amplifier Xiaoliang Chu1, Bin Zhang1, Xiaofeng Wei2, Xiaodong Yuan2, Xiaojun Huang2 1 2
College of Electronics and Information, Sichuan University, Chengdu 610064, China Research Center of Laser Fusion, CAEP, P. O. Box 919-988, Mianyang 612900, China
Abstract: By using the nonlinear Schro¨dinger equation model, the inverse problem of the chirp pulse through the Ti:sapphire amplifier was studied, in which the effects of gain narrowing, gain saturation and amplified spontaneous emission on the pulse were considered. The results were useful for the design of the pulse shaping of the ultrashort pulse. Key words: The inverse problem –– nonlinear Schro¨dinger equation –– gain narrowing –– gain saturation –– B integral
1. Introduction In recent years, as the Ti:sapphire is used as amplified media, the ultrashort pulse has been developing quickly. Ti:sapphire has some desirable characteristics including wide gain bandwidth, relatively large peak gain cross section, high thermal conductivity, high quantum efficiency, etc. Consequently, it is used in ultrashort pulse system popularly. By using the chirp pulse amplification (CPA) technology [1], the peak power of the ultrashort pulse using Ti:sapphire as amplified media has been reaching 100 TW [2–3]. Recently, it is reported that the peak power produced in Ti:sapphire chirp pulse amplification system is approaching 1 PW [4]. It is well known that the CPA system is composed of three parts: stretcher, amplifier and compressor. However, the pulse will be affected by some effects such as gain narrowing, gain saturation, dispersion and B integral etc. in the amplified procedure. It not only results in the pulse distortion and affects the beam quality but also makes the pulse longer after the compressor. Consequently, it is necessary to study the inverse problem of the pulse propagating in the amplifier and to take some ways to shape the input pulse in order to obtain the required pulse in practice, namely, from the required output pulse and
Received 9 December 2003; accepted 17 March 2004. Correspondence to: X. Chu E-mail: [email protected]
the parameters of Ti:sapphire amplifier to find the temporal, spatial and spectrum profiles of the input pulse by inverse numerical calculation. Several models have been built to study for the propagation of the chirp pulse [5–10] to solve the problem in hand. We choose the model of the nonlinear Schro¨dinger equation [5]. The inverse problem of the ultrashort pulse propagating in the Ti:sapphire medium has been solved, in which the effects of gain narrowing, gain saturation and amplified spontaneous emission, group velocity and B integral were taken into account. The results of inverse calculation would be useful for the design of the pulse shaping of the ultrashort pulse.
2. Theoretical model From the Maxwell’s equation, the wave propagation equation can be written as [11] r2 E~ð~ r; wÞ þ eðwÞ
w2 ~ Eð~ r; wÞ ¼ 0 ; c2
ð1Þ
where E~ð~ r; wÞ is the electric field in the frequency field, r~ ¼ ðx; y; zÞ is the position vector, z is the propagation distance, w is the temporal frequency, c is the velocity of light in the vacuum, and eðwÞ is the permittivity. Since the Ti:sapphire host is birefringent, the polarization of amplified pulse must be made p polarization in order to obtain the highest gain. Then, suppose the amplified pulse is linear polarization, and the electric field can be written as Eðx; y; z; TÞ ¼ Aðx; y; z; TÞ exp ðib0 z iw0 TÞ ; ð2Þ where b0 ¼ n0 w0 =c is the linear transfer constant, n0 is the linear refractive index, w0 is the center frequency of the input pulse. Taking the Fourier transform of eq. (2), we get Eðx; y; z; wÞ ¼ Aðx; y; z; w w0 Þ exp ðib0 zÞ:
ð3Þ
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Xiaoliang Chu et al., The inverse problem of chirp pulse propagating through Ti:sapphire amplifier
Substituting eq. (3) into eq. (1) and using the similar method to reference [5, 11], the equation of the ultrashort pulse propagating through the amplified medium can be expressed as @A i @2 i @2 ib2 @ 2 ¼ þ 2 @t 2 @z 2b0 @x2 2b0 @y2 3 b3 @ iw0 n2 ga 2 jAj þ þ þ A; ð4Þ 2 6 @t 3 c where b2 and b3 are group velocity dispersion and third-order dispersion parameters, respectively, n2 is the nonlinear parameter, g is the gain coefficient, and a is the loss coefficient. For the simplicity, only the third-order dispersion is given in the eq. (4). Considering the effects of the gain narrowing and gain saturation on the gain coefficient, the gain coefficient g can be expressed as gðx; y; tÞ ¼ g0 gw ðx; y; wðtÞÞ gs ðx; y; tÞ 2 3 ðt 1 ¼ g0 gw ðx; y; wðtÞÞ exp 4 Iðx; y; tÞ dt5 ; Js 0
For the simplification, we suppose the output field of the amplifier is the super-Gaussian beams with linear chirp, i.e., " # x 10 1 t 10 Aðx; tÞ ¼ E0 exp exp w0 2 t b t 2 exp i ; ð7Þ 2 t where E0 is the amplitude constant, w0 is the waist width of the input beams, t is the half of pulse width at e1 intensity, b ¼ DwL t=2 is the linear chirp parameter of the input pulse, DwL is the bandwidth of the input pulse. The calculation parameters are as follows: the intensity of output pulse is 2.0 GW/cm2, w0 ¼ 1.0 cm, t ¼ 150 ps, n0 ¼ 1.76, l0 ¼ 800 nm, b2 ¼ 581:179 fs2 =cm, n02 ¼ 2:5 1016 cm2 =W [12]. The net gain that the pulse obtains is approximately 107. In the calculation, the distribution of gain was satisfied with the possion distribution showing as in fig. 1a [13]. The effect of the amplified spontaneous emission on the input pulse was considered and the distribution of the gain in the x direction is shown as in fig. 1b.
(5) where g0 is the small signal coefficient, gw and gs are the gain factor relative to the gain narrowing and gain saturation, respectively, Js is the saturation fluence. The B integral used to express the retardation of the phase caused by the nonlinear refractive index can be written as ðL 2pn02 Iðx; t; zÞ dz ; ð6Þ Bðx; tÞ ¼ l 0
n02
where ¼ 2n2 =e0 cn, e0 is the permittivity in vacuum, and n is the linear part of the refractive index.
3. Simulation and result analysis a)
In general, for the pulse propagating through media, the split-step Fourier method can be used to calculate the eq. (4) [11]. In the inverse calculation, the split-step Fourier method and iteration were used together, i.e., from the given optical field AðzÞ in the output plane, we can calculate the Aðz hÞ ¼ AðzÞ exp ½Mh, where M expresses the part of the bracket on the right-hand side of the eq. (4), h denotes the step length of the medium. However, we can not calculate the Aðz hÞ directly because the M has the term including the unknown Aðz hÞ. So we must calculate M firstly. We can use the iteration procedure, namely, use the AðzÞ in place of the Aðz hÞ and get the M, and then calculate the A1 ðz hÞ according to the equation Aðz hÞ ¼ AðzÞ exp ½Mh. And then use the new A1 ðz hÞ taking the place of the Aðz hÞ and get the new M, and calculate A2 ðz hÞ. Such iteration procedure carries on until the difference between Ai ðz hÞ and Aiþ1 ðz hÞ is small enough.
b) Fig. 1. a) The profile of the gain factor gw as a function of wavelength; b) The gain distribution in the x direction.
Xiaoliang Chu et al., The inverse problem of chirp pulse propagating through Ti:sapphire amplifier
Fig. 2. The spatiotemporal distribution of the given output pulse.
203
Fig. 4. The temporal profile of the input pulse by inverse calculation with Dl ¼ 10 nm, 25 nm, 50 nm and their corresponding output pulse at x ¼ 0.
The calculation results are compiled in figs. 2–6. Fig. 2 shows the spatiotemporal distribution of the given output pulse. Fig. 3 shows the input pulse by inverse calculation corresponding to the given output pulse. In fig. 3, the shape of the pulse is mainly resulted in by gain narrowing, gain saturation and amplified spontaneous emission. Because when the chirp pulse propagates in the gain medium, both gain narrowing and gain saturation distort the chirp pulse. Gain narrowing pulls the pulse towards center and make the spectrum narrow. Gain saturation leads to the leading edge of the pulse undergoes the high gain than the trailing edge due to the depletion of the population inversion and pulls the pulse towards the long wavelength. Amplified spontaneous emission results in gain nonuniformity whose effect on the pulse is similar with gain narrowing. In the inverse calculation, how-
ever, these effects will produce the opposite effects. Therefore, if output pulse like fig. 2 is required, the input pulse needs to be shaped like fig. 3 to compensate the effects of the gain narrow, gain saturation and amplified spontaneous emission. Several ways were put forward and obtained the good results [3, 14–16]. To illustrate the effects of the gain narrowing, gain saturation and amplified spontaneous emission on the input pulse more clearly, fig. 4 gives the incident pulse profile with Dl ¼ 10 nm, 25 nm, 50 nm at x ¼ 0, as well as their corresponding output pulse. Fig. 5 shows the power spectrum of the input pulse with Dl ¼ 50 nm at x ¼ 0. Fig. 6 gives the spatial distribution of the input pulse with Dl ¼ 50 nm at t ¼ 0. It can be shown from fig. 4 that for the three input chirp pulse, the relative intensity of leading edge is lower than the trailing edge. The shape of the input pulse is inversed to the
Fig. 3. The spatiotemporal distribution of the input pulse.
Fig. 5. The power spectrum of the input pulse and output pulse with Dl ¼ 50 nm at x ¼ 0.
204
Xiaoliang Chu et al., The inverse problem of chirp pulse propagating through Ti:sapphire amplifier
4. Conclusion By using the nonlinear Schro¨dinger equation model, the inverse problem of the chirp pulse propagating through the Ti:sapphire amplifier was studied, i.e. from the given output pulse to get the input pulse. The effects of gain narrowing, gain saturation, as well as amplified spontaneous emission on the pulse were considered and analyzed. The effects of the group velocity and the B integral on the pulse were also discussed. Our calculation method and results obtained in this paper will be useful for the design of pulse shaping of the ultrashort pulse.
References Fig. 6. The spatial distribution of the input and output pulse with Dl ¼ 50 nm in the x direction at t ¼ 0.
effect of gain saturation on the pulse and used to compensate the effect of the gain saturation. At the same time, the input chirp pulse appears depression near the center of the pulse as the spectrum width increases, which is inversed to the effect of gain narrowing on the pulse. Moreover, the depression will become larger as the ratio of the input bandwidth and the gain bandwidth increase. In addition, from the fig. 4 we found the group velocity and B integral hardly affect the input pulse profiles. The value of B integral are calculated to be approximately 0.0339, 0.0336, 0.0335, for Dl ¼ 50 nm, 25 nm, 10 nm, respectively. This would not lead to the distinct effects of the self-phase and self-focusing. To verify the validity of the inverse calculation, we use the three input pulse in fig. 4 as input field to calculate the output pulses shown in fig. 4. We can see that the three profiles of the output pulses superpose together. The profiles coincide with the one from eq. (7) and it shows that the input pulses we got through inverse calculation are correct. From fig. 5 we can see that the spectrum profile shifts towards the short wavelength (blue shift) and a valley exists near l ¼ 800 nm in the profile, which is oppositional to the effect of gain saturation and gain narrowing, respectively. Whereas the above tow effects make the spectrum shift to long wavelength (red shift) and lead to a peak near l ¼ 800 nm during the pulse propagation. Like the fig. 4, the output spectrum corresponding the input spectrum was given in fig. 5. Fig. 6 implies that the spatial profile of input pulse is required to be with a valley in the center because of the nonuniformity of the gain distribution with higher gain in the center than in the edge. The phenomenon of amplified spontaneous emission resulting in the gain nonuniformity is similar with the effect of gain narrowing. In practice, amplified spontaneous emission must be controlled through some ways. To get the uniform output intensity, the filter can be put in front of the amplifier and make the distribution of the output intensity uniform [17].
[1] Strickland D, Mourou G: Compression of amplified chirped optical pulses. Opt. Commun. 56 (1985) 219–221 [2] Yamakawa K, Aoyama M, Matsuoka S et al.: 100-TW sub-20-fs Ti:sapphire laser system operating at a 10-Hzrepetition rate. Opt. Lett. 23 (1998) 1468–1470 [3] Pittman M, Ferre´ S, Rousseau JP: Design and characterization of a near-diffraction-limited femtosecond 100-TW 10-Hz-high-intensity laser system. Appl. Phys. B 74 (2002) 529–535 [4] Aoyama M, Yamakawa K, Akahane Y et al.: 0.85-PW, 33fs Ti:sapphire laser. Opt. Lett. 28 (2003) 1594–1596 [5] Bridges RE, Boyd RW, Agrawal GP: Multidimensional coupling owing to optical nonlinearities. I. General formulation. J. Opt. Soc. Am B 13 (1996) 553–559 [6] Chuang Y-H, Zheng L, Meyerhofer DD: Propagation of light pulses in a chirped-pulse amplification laser. IEEE J. Quantum Electron. 29 (1993) 270–280 [7] Matsuoka S, Yamakawa K: Development of a model for chirped-pulse amplification of sub-20 fs laser pulses. Jpn. J. App. Phys. 37 (1998) 5997–6000 [8] Blane CL, Curley P, Salin F: Gain-narrowing and gainshifting of ultra-short pulses in Ti:sapphire amplifiers. Opt. Commun. 131 (1996) 391–398 [9] Gogoleva NG, Gorbunov VA: Modelling of chirped pulse amplification laser. Proc. SPIE 2770 (1996) 23–30 [10] Lu XQ, Fan DY, Qian LJ: Theory high power Ti:sapphire laser amplifier. Acta Optica Sinica 22 (2002) 1059–1062 (in Chinese) [11] Agrawal GP: Nonlinear fiber optics. 3rd edition. Academic Press, Boston 2001 [12] Backus S, Durfee CG, Murnane MM et al.: High power ultrafast lasers. Rev. Sci. Instrum. 69 (1998) 1207–1223 [13] Eggleston JM, Deshazer LG, Kangas KW: Characteristics and kinetics of laser-pumped Ti:sapphire oscillators. J. Quantum Electron. 24 (1988) 1009–1015 [14] Backus S, Durfee CG, Mourou G et al.: 0.2-TW laser system at 1 KHz. Opt. Lett. 22 (1997) 1256–1258 [15] Yamakawa K, Aoyama M, Matsuoka S et al.: Generation of 16-fs, 10-TW pulses at a 10-Hz repetition rate with efficient Ti:sapphire amplifiers. Opt. Lett. 23 (1998) 525–527 [16] Cha YH, Kang Yil, Nam CH: Generation of a broad amplified spectrum in a femtosecond terawatt Ti:sapphire laser by a long-wavelength injection method. J. Opt. Soc. Am. B 16 (1999) 1220–1223 [17] Erlandson AC, Jancaitis KS, McCracken RW et al.: Gain uniformity and amplified spontaneous emission in multisegment amplifiers. UCRL-LR-105821-92-3, pp. 105–114
International Journal for Light and Electron Optics
The inverse problem of chirp pulse propagating through Ti:sapphire amplifier Xiaoliang Chu1, Bin Zhang1, Xiaofeng Wei2, Xiaodong Yuan2, Xiaojun Huang2 1 2
College of Electronics and Information, Sichuan University, Chengdu 610064, China Research Center of Laser Fusion, CAEP, P. O. Box 919-988, Mianyang 612900, China
Abstract: By using the nonlinear Schro¨dinger equation model, the inverse problem of the chirp pulse through the Ti:sapphire amplifier was studied, in which the effects of gain narrowing, gain saturation and amplified spontaneous emission on the pulse were considered. The results were useful for the design of the pulse shaping of the ultrashort pulse. Key words: The inverse problem –– nonlinear Schro¨dinger equation –– gain narrowing –– gain saturation –– B integral
1. Introduction In recent years, as the Ti:sapphire is used as amplified media, the ultrashort pulse has been developing quickly. Ti:sapphire has some desirable characteristics including wide gain bandwidth, relatively large peak gain cross section, high thermal conductivity, high quantum efficiency, etc. Consequently, it is used in ultrashort pulse system popularly. By using the chirp pulse amplification (CPA) technology [1], the peak power of the ultrashort pulse using Ti:sapphire as amplified media has been reaching 100 TW [2–3]. Recently, it is reported that the peak power produced in Ti:sapphire chirp pulse amplification system is approaching 1 PW [4]. It is well known that the CPA system is composed of three parts: stretcher, amplifier and compressor. However, the pulse will be affected by some effects such as gain narrowing, gain saturation, dispersion and B integral etc. in the amplified procedure. It not only results in the pulse distortion and affects the beam quality but also makes the pulse longer after the compressor. Consequently, it is necessary to study the inverse problem of the pulse propagating in the amplifier and to take some ways to shape the input pulse in order to obtain the required pulse in practice, namely, from the required output pulse and
Received 9 December 2003; accepted 17 March 2004. Correspondence to: X. Chu E-mail: [email protected]
the parameters of Ti:sapphire amplifier to find the temporal, spatial and spectrum profiles of the input pulse by inverse numerical calculation. Several models have been built to study for the propagation of the chirp pulse [5–10] to solve the problem in hand. We choose the model of the nonlinear Schro¨dinger equation [5]. The inverse problem of the ultrashort pulse propagating in the Ti:sapphire medium has been solved, in which the effects of gain narrowing, gain saturation and amplified spontaneous emission, group velocity and B integral were taken into account. The results of inverse calculation would be useful for the design of the pulse shaping of the ultrashort pulse.
2. Theoretical model From the Maxwell’s equation, the wave propagation equation can be written as [11] r2 E~ð~ r; wÞ þ eðwÞ
w2 ~ Eð~ r; wÞ ¼ 0 ; c2
ð1Þ
where E~ð~ r; wÞ is the electric field in the frequency field, r~ ¼ ðx; y; zÞ is the position vector, z is the propagation distance, w is the temporal frequency, c is the velocity of light in the vacuum, and eðwÞ is the permittivity. Since the Ti:sapphire host is birefringent, the polarization of amplified pulse must be made p polarization in order to obtain the highest gain. Then, suppose the amplified pulse is linear polarization, and the electric field can be written as Eðx; y; z; TÞ ¼ Aðx; y; z; TÞ exp ðib0 z iw0 TÞ ; ð2Þ where b0 ¼ n0 w0 =c is the linear transfer constant, n0 is the linear refractive index, w0 is the center frequency of the input pulse. Taking the Fourier transform of eq. (2), we get Eðx; y; z; wÞ ¼ Aðx; y; z; w w0 Þ exp ðib0 zÞ:
ð3Þ
0030-4026/04/115/05-201 $ 30.00/0
202
Xiaoliang Chu et al., The inverse problem of chirp pulse propagating through Ti:sapphire amplifier
Substituting eq. (3) into eq. (1) and using the similar method to reference [5, 11], the equation of the ultrashort pulse propagating through the amplified medium can be expressed as @A i @2 i @2 ib2 @ 2 ¼ þ 2 @t 2 @z 2b0 @x2 2b0 @y2 3 b3 @ iw0 n2 ga 2 jAj þ þ þ A; ð4Þ 2 6 @t 3 c where b2 and b3 are group velocity dispersion and third-order dispersion parameters, respectively, n2 is the nonlinear parameter, g is the gain coefficient, and a is the loss coefficient. For the simplicity, only the third-order dispersion is given in the eq. (4). Considering the effects of the gain narrowing and gain saturation on the gain coefficient, the gain coefficient g can be expressed as gðx; y; tÞ ¼ g0 gw ðx; y; wðtÞÞ gs ðx; y; tÞ 2 3 ðt 1 ¼ g0 gw ðx; y; wðtÞÞ exp 4 Iðx; y; tÞ dt5 ; Js 0
For the simplification, we suppose the output field of the amplifier is the super-Gaussian beams with linear chirp, i.e., " # x 10 1 t 10 Aðx; tÞ ¼ E0 exp exp w0 2 t b t 2 exp i ; ð7Þ 2 t where E0 is the amplitude constant, w0 is the waist width of the input beams, t is the half of pulse width at e1 intensity, b ¼ DwL t=2 is the linear chirp parameter of the input pulse, DwL is the bandwidth of the input pulse. The calculation parameters are as follows: the intensity of output pulse is 2.0 GW/cm2, w0 ¼ 1.0 cm, t ¼ 150 ps, n0 ¼ 1.76, l0 ¼ 800 nm, b2 ¼ 581:179 fs2 =cm, n02 ¼ 2:5 1016 cm2 =W [12]. The net gain that the pulse obtains is approximately 107. In the calculation, the distribution of gain was satisfied with the possion distribution showing as in fig. 1a [13]. The effect of the amplified spontaneous emission on the input pulse was considered and the distribution of the gain in the x direction is shown as in fig. 1b.
(5) where g0 is the small signal coefficient, gw and gs are the gain factor relative to the gain narrowing and gain saturation, respectively, Js is the saturation fluence. The B integral used to express the retardation of the phase caused by the nonlinear refractive index can be written as ðL 2pn02 Iðx; t; zÞ dz ; ð6Þ Bðx; tÞ ¼ l 0
n02
where ¼ 2n2 =e0 cn, e0 is the permittivity in vacuum, and n is the linear part of the refractive index.
3. Simulation and result analysis a)
In general, for the pulse propagating through media, the split-step Fourier method can be used to calculate the eq. (4) [11]. In the inverse calculation, the split-step Fourier method and iteration were used together, i.e., from the given optical field AðzÞ in the output plane, we can calculate the Aðz hÞ ¼ AðzÞ exp ½Mh, where M expresses the part of the bracket on the right-hand side of the eq. (4), h denotes the step length of the medium. However, we can not calculate the Aðz hÞ directly because the M has the term including the unknown Aðz hÞ. So we must calculate M firstly. We can use the iteration procedure, namely, use the AðzÞ in place of the Aðz hÞ and get the M, and then calculate the A1 ðz hÞ according to the equation Aðz hÞ ¼ AðzÞ exp ½Mh. And then use the new A1 ðz hÞ taking the place of the Aðz hÞ and get the new M, and calculate A2 ðz hÞ. Such iteration procedure carries on until the difference between Ai ðz hÞ and Aiþ1 ðz hÞ is small enough.
b) Fig. 1. a) The profile of the gain factor gw as a function of wavelength; b) The gain distribution in the x direction.
Xiaoliang Chu et al., The inverse problem of chirp pulse propagating through Ti:sapphire amplifier
Fig. 2. The spatiotemporal distribution of the given output pulse.
203
Fig. 4. The temporal profile of the input pulse by inverse calculation with Dl ¼ 10 nm, 25 nm, 50 nm and their corresponding output pulse at x ¼ 0.
The calculation results are compiled in figs. 2–6. Fig. 2 shows the spatiotemporal distribution of the given output pulse. Fig. 3 shows the input pulse by inverse calculation corresponding to the given output pulse. In fig. 3, the shape of the pulse is mainly resulted in by gain narrowing, gain saturation and amplified spontaneous emission. Because when the chirp pulse propagates in the gain medium, both gain narrowing and gain saturation distort the chirp pulse. Gain narrowing pulls the pulse towards center and make the spectrum narrow. Gain saturation leads to the leading edge of the pulse undergoes the high gain than the trailing edge due to the depletion of the population inversion and pulls the pulse towards the long wavelength. Amplified spontaneous emission results in gain nonuniformity whose effect on the pulse is similar with gain narrowing. In the inverse calculation, how-
ever, these effects will produce the opposite effects. Therefore, if output pulse like fig. 2 is required, the input pulse needs to be shaped like fig. 3 to compensate the effects of the gain narrow, gain saturation and amplified spontaneous emission. Several ways were put forward and obtained the good results [3, 14–16]. To illustrate the effects of the gain narrowing, gain saturation and amplified spontaneous emission on the input pulse more clearly, fig. 4 gives the incident pulse profile with Dl ¼ 10 nm, 25 nm, 50 nm at x ¼ 0, as well as their corresponding output pulse. Fig. 5 shows the power spectrum of the input pulse with Dl ¼ 50 nm at x ¼ 0. Fig. 6 gives the spatial distribution of the input pulse with Dl ¼ 50 nm at t ¼ 0. It can be shown from fig. 4 that for the three input chirp pulse, the relative intensity of leading edge is lower than the trailing edge. The shape of the input pulse is inversed to the
Fig. 3. The spatiotemporal distribution of the input pulse.
Fig. 5. The power spectrum of the input pulse and output pulse with Dl ¼ 50 nm at x ¼ 0.
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Xiaoliang Chu et al., The inverse problem of chirp pulse propagating through Ti:sapphire amplifier
4. Conclusion By using the nonlinear Schro¨dinger equation model, the inverse problem of the chirp pulse propagating through the Ti:sapphire amplifier was studied, i.e. from the given output pulse to get the input pulse. The effects of gain narrowing, gain saturation, as well as amplified spontaneous emission on the pulse were considered and analyzed. The effects of the group velocity and the B integral on the pulse were also discussed. Our calculation method and results obtained in this paper will be useful for the design of pulse shaping of the ultrashort pulse.
References Fig. 6. The spatial distribution of the input and output pulse with Dl ¼ 50 nm in the x direction at t ¼ 0.
effect of gain saturation on the pulse and used to compensate the effect of the gain saturation. At the same time, the input chirp pulse appears depression near the center of the pulse as the spectrum width increases, which is inversed to the effect of gain narrowing on the pulse. Moreover, the depression will become larger as the ratio of the input bandwidth and the gain bandwidth increase. In addition, from the fig. 4 we found the group velocity and B integral hardly affect the input pulse profiles. The value of B integral are calculated to be approximately 0.0339, 0.0336, 0.0335, for Dl ¼ 50 nm, 25 nm, 10 nm, respectively. This would not lead to the distinct effects of the self-phase and self-focusing. To verify the validity of the inverse calculation, we use the three input pulse in fig. 4 as input field to calculate the output pulses shown in fig. 4. We can see that the three profiles of the output pulses superpose together. The profiles coincide with the one from eq. (7) and it shows that the input pulses we got through inverse calculation are correct. From fig. 5 we can see that the spectrum profile shifts towards the short wavelength (blue shift) and a valley exists near l ¼ 800 nm in the profile, which is oppositional to the effect of gain saturation and gain narrowing, respectively. Whereas the above tow effects make the spectrum shift to long wavelength (red shift) and lead to a peak near l ¼ 800 nm during the pulse propagation. Like the fig. 4, the output spectrum corresponding the input spectrum was given in fig. 5. Fig. 6 implies that the spatial profile of input pulse is required to be with a valley in the center because of the nonuniformity of the gain distribution with higher gain in the center than in the edge. The phenomenon of amplified spontaneous emission resulting in the gain nonuniformity is similar with the effect of gain narrowing. In practice, amplified spontaneous emission must be controlled through some ways. To get the uniform output intensity, the filter can be put in front of the amplifier and make the distribution of the output intensity uniform [17].
[1] Strickland D, Mourou G: Compression of amplified chirped optical pulses. Opt. Commun. 56 (1985) 219–221 [2] Yamakawa K, Aoyama M, Matsuoka S et al.: 100-TW sub-20-fs Ti:sapphire laser system operating at a 10-Hzrepetition rate. Opt. Lett. 23 (1998) 1468–1470 [3] Pittman M, Ferre´ S, Rousseau JP: Design and characterization of a near-diffraction-limited femtosecond 100-TW 10-Hz-high-intensity laser system. Appl. Phys. B 74 (2002) 529–535 [4] Aoyama M, Yamakawa K, Akahane Y et al.: 0.85-PW, 33fs Ti:sapphire laser. Opt. Lett. 28 (2003) 1594–1596 [5] Bridges RE, Boyd RW, Agrawal GP: Multidimensional coupling owing to optical nonlinearities. I. General formulation. J. Opt. Soc. Am B 13 (1996) 553–559 [6] Chuang Y-H, Zheng L, Meyerhofer DD: Propagation of light pulses in a chirped-pulse amplification laser. IEEE J. Quantum Electron. 29 (1993) 270–280 [7] Matsuoka S, Yamakawa K: Development of a model for chirped-pulse amplification of sub-20 fs laser pulses. Jpn. J. App. Phys. 37 (1998) 5997–6000 [8] Blane CL, Curley P, Salin F: Gain-narrowing and gainshifting of ultra-short pulses in Ti:sapphire amplifiers. Opt. Commun. 131 (1996) 391–398 [9] Gogoleva NG, Gorbunov VA: Modelling of chirped pulse amplification laser. Proc. SPIE 2770 (1996) 23–30 [10] Lu XQ, Fan DY, Qian LJ: Theory high power Ti:sapphire laser amplifier. Acta Optica Sinica 22 (2002) 1059–1062 (in Chinese) [11] Agrawal GP: Nonlinear fiber optics. 3rd edition. Academic Press, Boston 2001 [12] Backus S, Durfee CG, Murnane MM et al.: High power ultrafast lasers. Rev. Sci. Instrum. 69 (1998) 1207–1223 [13] Eggleston JM, Deshazer LG, Kangas KW: Characteristics and kinetics of laser-pumped Ti:sapphire oscillators. J. Quantum Electron. 24 (1988) 1009–1015 [14] Backus S, Durfee CG, Mourou G et al.: 0.2-TW laser system at 1 KHz. Opt. Lett. 22 (1997) 1256–1258 [15] Yamakawa K, Aoyama M, Matsuoka S et al.: Generation of 16-fs, 10-TW pulses at a 10-Hz repetition rate with efficient Ti:sapphire amplifiers. Opt. Lett. 23 (1998) 525–527 [16] Cha YH, Kang Yil, Nam CH: Generation of a broad amplified spectrum in a femtosecond terawatt Ti:sapphire laser by a long-wavelength injection method. J. Opt. Soc. Am. B 16 (1999) 1220–1223 [17] Erlandson AC, Jancaitis KS, McCracken RW et al.: Gain uniformity and amplified spontaneous emission in multisegment amplifiers. UCRL-LR-105821-92-3, pp. 105–114