- Email: [email protected]
Experimental measurement of ultrashort pulse evolution at different spatial positions in nonlinear media
Optics & Laser Technology 59 (2014) 47–51
Contents lists available at ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Experimental measurement of ultrashort pulse evolution at different spatial positions in nonlinear media Chao Tan a, Xiquan Fu a,n, Yangbao Deng b a Key Laboratory for Micro-/Nano-Optoelectronic Devices of Ministry of Education, College of Information Science and Engineering, Hunan University, Changsha 410082, China b College of Communication and Electronic Engineering, Hunan City University, Yiyang 413002, China
art ic l e i nf o
a b s t r a c t
Article history: Received 22 August 2013 Received in revised form 13 December 2013 Accepted 14 December 2013 Available online 1 January 2014
The temporal evolution of ultrashort pulses before the beam splitting is very useful for filamentation theory. In this article, we experimentally demonstrated the temporal evolution of ultrashort pulses in the different spatial positions during small-scale self-focusing. The simulation results are found to be in good agreement with experiment measurements. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Temporal evolution Ultrashort pulse Small-scale self-focusing
1. Introduction The nonlinear propagation of ultrashort laser pulses through a Kerr medium is usually accompanied by several interesting phenomena, such as self-focusing (SF) [1], supercontinuum generation [2], multiphoton ionization [3] and filamentation [4]. SF of beam is a common phenomenon in nonlinear optics [5]. In the high power laser system, SF, especially small-scale self-focusing (SSSF), is a major factor for degradation of the laser beam quality and damage of optical materials [6,7]. When ultrashort pulses propagate in the nonlinear media, SF moves off-axis energy towards the peak of the pulse and compresses it both in spatial and temporal [8,9]. As the peak intensity increases, the process of self-phase modulation (SPM) also increases, as a result of the interplay between SPM-induced upchirp and group velocity dispersion, initiating the pulse splitting [10,11]. It is well known that Bespalov and Talanov [12] first explain the SSSF and the filamentation of a beam in terms of the modulation instability (MI) model, but they do not consider the perturbations in the temporal-domain. Campillo and Bliss et al. verify experimentally MI and the relationship among fastest growing frequency, SF length and light intensity [13–15]. The direct time-resolved observation of the propagation of intensity femtosecond laser pulses experiencing SF, beam filamentation is investigated extensively [16–19]. Previously, we have observed the evolution of spatial modulation in the process of SSSF of a beam, the
n
Corresponding author. Tel.: þ 867 318 882 4527. E-mail address: [email protected] (X. Fu).
0030-3992/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlastec.2013.12.010
impact of relaxation effect in the media on spatial–temporal instability [20,21]. However, these studies mainly focus on the temporal evolution of the pulse in the forming and after the formation of filaments, there is little attention to the temporal evolution of the beam in the period before filamentation. Especially, the partial spatial intensity of the beam changes impact on the temporal evolution during the period of SSSF has been rarely studied. Recently, we have demonstrated the time evolution of spatial intensity increasing zones and non-increasing zones in the process of SSSF [22]. In this paper, we first demonstrate the spatial evolution of the pump beam during SSSF in the experiment and simulation. We find that there will be different growths in the different spatial positions, especially the modulation peak and modulation bottom in space, with the increase of input power. Then, we show that the temporal evolution of some spatial positions of the pump pulse monitored by a probe pulse. Due to the effect of spatiotemporal coupling, the increase of spatial intensity at modulation peak will lead to pulse width compression. However, the pulse width of modulation bottom is broadened with the increase of power, because its zone contrast is actually decreased. 2. Experiment setup A schematic overview of our experiment setup is shown in Fig. 1. We use an amplified Ti:sapphire laser system (LibraS, Coherent, λ0 ¼ 800 nm) delivering pulses with bandwidth of 12 nm, single pulse energy of 1 mJ, duration τ0 ¼ 100 fs at 1 kHz repetition rate. The beam is split into two beams by a beam splitter, BS1, one is used
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C. Tan et al. / Optics & Laser Technology 59 (2014) 47–51
low, the influence of nonlinear effect on the beam is far less than the diffraction effect, the diffractive modulation play a major role in the spatial evolution of the pump beam. So the SSSF at D1 is not obvious when peak power is 30.5 MW [Fig. 2(a)]. The nonlinear effect enhanced by the increment of the power and a new growth at D1 appears, the SSSF at D1 is obvious to see [Fig. 2(b), peak power 90.2 MW]. When the power increased to 138.1 MW, the intensity of D1 reach maximum and a filament is formed at D1 [Fig. 2(c)]. If the power is increased continuously, the pump beam will split into multifilaments. However, the intensity of D2 not only has no significant growth with the increase of power, but its zone contrast is fall-off. Zone contrast is defined as the ratio between the partial spatial intensity and the average intensity of the beam. Using a split-step Fourier algorithm, spatial–temporal nonlinear Schrödinger equation is used for calculation. 2 ∂A i ∂ ∂2 iβ ∂2 A iω0 n2 ¼ j Aj 2 A þ 2 A 2 2 þ 2 ∂z 2β0 ∂x 2 ∂t c ∂y Fig. 1. Experimental setup. BS1 and BS2, beam splitters; M1 and M2, optical delay line; M3 and M4, silver-coated plane mirrors; M5, rotating mirror; A1 and A2, attenuators. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
as a pump beam and the other as a probe beam. The probe beam passes through a variable optical delay line before it is directed to a slit. A rotating mirror in front of the slit is used to change the direction of the probe beam, according that the pump beam can be scanned by the probe beam when M5 is rotated. The horizontal spatial profile of the probe beam after passing through the slit is a Sinc function distribution. The spatial and temporal scanning resolution can be controlled by adjusting the slit width and the pulse width of the probe pulse, respectively. We set the aperture of vertical slit to 0.57 mm in the experiment. A glass cuvette with 5 cm path length is filled with carbon disulfide (CS2). CS2, a typical relaxing Kerr media, is chosen as the nonlinear media because of its strong Kerr nonlinearity. The pump beam is directed to the CS2 reflected by M3. A variable attenuator, A1, is used to adjust the input power of the pump beam. A filament with diameter of 0.1 mm in front of CS2 is used to generate initial diffraction modulation. The evolution of spatial–temporal of the pump pulse can be monitored by two synchronized high-resolution charge-coupled device (CCD) cameras (Coherent Laser Cam-HR™ Beamview, 1280 1024 pixel, pixel size of 6:7 μm). In order to measure the spatial evolution of the pump beam in the front surface of BBO crystal, a part of the pump beam is imaged onto CCD1 reflected by a beam splitter, BS2. It follows that the optical path from the beam splitter, BS2, to BBO crystal is equal to from BS2 to CCD1. A sum-frequency beam generated in the BBO nonlinear crystal under noncollinear interaction of two beams is collected with a CCD2. So we can indirectly measure the temporal evolution of pump pulses by the sumfrequency beam based on the cross-correlation principle. The adjustable attenuators, A2 and A3, are custom-designed to protect CCD cameras from damage by high power laser beams.
3. Results and discussions 3.1. The spatial evolution of the beam during SSSF Adjusting the attenuator, A1, and keeping the position of CCD1 unchanged, the spatial evolution of the pump beam which modulated by a filament in different power is shown in Fig. 2. From Fig. 2, it is easy to see distinct SSSF of the beam appears. The arrows serve to indicate the positions of the spatial modulation peak D1 and spatial modulation bottom D2. When the power is
where A is the amplitude of the input pulse, β0 is the transmission constant, β2 is the group velocity dispersion, ω0 is the central frequency, n2 is the nonlinear index of refraction. The initial field is taken to be a Gaussian in both time and space, having an intensity full width at half maximum (FWHM) of 105 fs and 2 mm, respectively. The beam waist is located at the entrance face of the BBO. The center wavelength is λ ¼ 800 nm, the linear index of refraction is n0 ¼ 1:63, the nonlinear index of refraction is n2 ¼ 3:5 10 15 cm2 =W, and the group velocity dispersion (GVD) coefficient is β2 ¼ 1:54 10 25 s2 =m. To simplify the problem, we have chosen to neglect loss and high order dispersion. The diameter of the filament is 0.1 mm and the length of nonlinear propagation is 5 cm. The numerical simulation maps shown in Fig. 2(d)–(f) show the intensity distribution of the pump beam modulated by a filament at three different input powers. When the power is 30.5 MW, input peak power is lower than the critical power for SF, and diffractive modulation play a major role in the spatial evolution of pump beam [Fig. 2(d)]. With the increase of the power, a new growth position appears in the first modulation peak, the SSSF of beam is obvious to see [Fig. 2(e), peak power 90 MW]. Because SSSF moves energy towards the modulation peak, so we can see from Fig. 2 that the intensity of modulation peak D1 increases rapidly. When the power increased to 138.1 MW, the intensity of the growth area reaches maximum [Fig. 2(f)]. However, the spatial intensity of the modulation peak cannot increase infinitely because the total energy of the beam is constant. As the power is increased continuously, new growth points will complete with the original growth point, resulting in the filamentation of the beam. No obvious growth can be observed at the modulation bottom because the SSSF effect at D2 is very weak. The simulation results are found to be in good agreement with experimental measurements. Fig. 3(a) represents the spatial distribution of the pump beam in the horizontal plane position pointed by arrows A and B of Fig. 2(a)–(c). The numerical simulation map shown in Fig. 3(b) shows the pump beam varies with different input power. Similar to Figs. 2 and 3 displays visually the spatial evolution of laser beam modulated by a filament in the process of SSSF. With the increase of input power, nonlinear effect gradually strengthened and a new growth point in D1 appears. The intensity of D1 reach maximum when the input power increased to 138.1 MW. The intensity at D1 increased by 12% when the power is increased from 30.5 MW to 138.1 MW, the growth is a little difference between experiment and simulation at some other modulation peaks because the quality of the beam profile in the experiment is not very good. However, the zone contrast of D2 decreases with the increase of power [Fig. 3].
C. Tan et al. / Optics & Laser Technology 59 (2014) 47–51
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Fig. 2. The spatial evolution of the pump beam which modulated by a filament at peak power of (a) 30.5 MW, (b) 90.2 MW, (c) 138.1 MW in the experiment; (d) 30.5 MW, (e) 90.2 MW, (f) 138.1 MW in the simulation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3.2. The temporal evolution of ultrashort pulses with SSSF in different spatial positions In the process of nonlinear propagation of the femtosecond laser pulses, nonlinear effect gradually strengthened with the increase of input power. It finds that the SF effect of the intense femtosecond pulses will lead to the rapid growth of partial spatial intensity, the changes of spatial intensity will affect the pulse width of these areas due to spatiotemporal coupling effect. Fig. 4 shows numerical simulation curves of the pulse width in the horizontal plane position pointed by arrows A and B of Fig. 2(a)– (c) at different peak power. The experimental measured pulse width of pump pulses at positions of D1 and D2 are also displayed in Fig. 4. We only considered the left brain part of the beam because the Gaussian beam is symmetrical. Root mean square (RMS) pulse width is used to characterize the pulse width in numerical simulation [23].
RMS pulse width is defined as Z 1 Z s ¼ ½〈T 2 〉 〈T〉2 1=2 ; 〈T n 〉 ¼ T n j Aðz; TÞj 2 dT= 1
1 1
j Aðz; TÞj 2 dT
When the power is 30.5 MW and the nonlinear effect is very weak in this case. The initial pulse widths at D1 and D2 are 106 fs and 115 fs from Fig. 4, this is because the pulse widths of ultrashort pulse at different positions of wave front are different. The pulse width of D1 is compressed with the increase of power, it decreased by 9.4% when the power is increased from 30.5 MW to 138.1 MW, the experimental results confirmed the numerical simulation in Fig. 4. However, the pulse width of D2 is broadened by 4.5% when the power increased from 30.5 MW to 138.1 MW. The residual dispersion in the experiment is neglected in the numerical simulation, so the quantity of pulse width broadened of experimental result is a little bit different from numerical simulation.
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C. Tan et al. / Optics & Laser Technology 59 (2014) 47–51
Fig. 3. The spatial distribution of the pump beam in the horizontal plane position pointed by arrows A and B in Fig. 2 varies with input power (a) in the experiment; (b) in the simulation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
4. Conclusion In summary, we have investigated the spatial–temporal evolution of ultrashort pulses modulated by a filament in the process of SSSF. Our results indicate that the SSSF of the beam enhanced with the increase of power, as a result of spatiotemporal coupling effect, the increase of spatial intensity at modulation peak will lead to pulse width compression with the increase of power. However, the pulse width of modulation bottom is always broadened, because its zone contrast falls off with the increase of power. Our experimental observations are in good agreement with simulation results. For high power laser transmission and the measurement of spatial–temporal evolution of ultrashort pulses propagating in the nonlinear media, these research results have better practical reference value.
Acknowledgements
Fig. 4. The pulse width in the horizontal plane pointed by arrows A and B in Fig. 2 varies with the input power in the simulation, and the pulse widths at D1 and D2 vary with the input power in the experiment. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
As mentioned previously, the spatial intensity at different spatial positions will appear different changing trends when the ultrashort laser pulses propagate in the nonlinear media. According to (3 þ 1)-dimensional nonlinear Schrödinger equation, the temporal-domain and spatial-domain of the laser are affected by the same common nonlinear term. So we can see clearly that, differences in the changing trends of the pulse width appear at D1 and D2. The spatial intensity of modulation peak increases quickly with the increase of input power. The rapid growth of the spatial intensity leads to strong nonlinear effect. The spectrum of the pulses is broadened by SPM and then the pulse width of modulation peaks become shorter because of the strong nonlinear effect and dispersion. So the pulse width at the modulation peak in space is compressed with the increase of input power. The pulse width of D2 is broadened, because the zone contrast of D2 decreases with the increase of power.
This research is supported in part by the Specialized Research Fund for the Doctoral Program of Higher Education of China (20110161110012), the Department of Science and Technology of Hunan Province (2013TP4026), and Hunan Provincial Natural Science Foundation of China (12JJ7005). References [1] Zozulya AA, Diddams SA, Clement TS. Investigations of nonlinear femtosecond pulse propagation with the inclusion of Raman, shock, and third-order phase effects. Phys Rev A: At Mol Opt Phys 1998;58(4):3303. [2] Kaya N, Strohaber J, Kolomenskii AA, et al. White-light generation using spatially-structured beams of femtosecond radiation. Opt Express 2012;20 (12):13337–46. [3] Shi L, Li W, Wang Y, et al. Generation of high-density electrons based on plasma grating induced Bragg diffraction in air. Phys Rev Lett 2011;107 (9):95004. [4] Shen YR. Principle of nolinear optics. New York: Wiley&Sons Inc; 1984. [5] Hao Z, Stelmaszczyk K, Rohwetter P, et al. Femtosecond laser filament-fringes in fused silica. Opt Express 2011;19(8):7799–806. [6] Williams W, Trenholme J, Orth C. NIF design optimization. LLNL laser program quarterly report, 1996.UCRL-LR-105821 (96-4): 181–191. [7] Shen YR, Loy MMT. Theoretical interpretation of small-scale filaments of light originating from moving focal spots. Phys Rev A: At Mol Opt Phys 1971;3 (6):2099. [8] Marburrger J, Wagner W. Self-focusing as a pulse sharpening mechanism. IEEE J Quantum Electron 1967;3(10):415–6.
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[9] McAllister GL, Marburger JH, DeShazer LG. Observation of optical pulse shaping by the self-focusing effects. Phys Rev Lett 1968;21:1648–9. [10] Zozulya AA, Diddams SA, Clement TS. Investigations of nonlinear femtosecond pulse propagation with the inclusion of Raman, shock, and third-order phase effects. Phys Rev A: At Mol Opt Phys 1998;58(4):3303. [11] Velchev I, Pattnaik R, Toulouse J. Two-beam modulation instability in noninstantaneous nonlinear media. Phys Rev Lett 2003;91(9):093905. [12] Bespalov VI, Talanov VI. Filamentary structure of light beams in nonlinear liquids. Sov Phys JEPT Lett 1966;3:307. [13] Campillo AJ, Shapiro SL, Suydam BR. Periodic breakup of optical beams due to self‐focusing. Appl Phys Lett 1973;23(11):628–30. [14] Campillo AJ, Shapiro SL, Suydam BR. Relationship of self‐focusing to spatial instability modes. Appl Phys Lett 1974;24(4):178–80. [15] Bliss ES, Speck DR, Holzrichter JF, et al. Propagation of a high‐intensity laser pulse with small‐scale intensity modulation. Appl Phys Lett 1974;25(8): 448–50. [16] Zozulya AA, Diddams SA, Van Engen AG, et al. Propagation dynamics of intense femtosecond pulses: multiple splittings, coalescence, and continuum generation. Phys Rev Lett 1999;82:1430–3.
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[17] La Fontaine B, Vidal F, Jiang Z, et al. Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air. Phys Plasmas 1999;6:1615. [18] Minardi S, Gopal A, Couairon A, et al. Accurate retrieval of pulse-splitting dynamics of a femtosecond filament in water by time-resolved shadowgraphy. Phys Opt Lett 2009;34(19):3020–2. [19] Dachraoui H, Oberer C, Michelswirth M, et al. Direct time-domain observation of laser pulse filaments in transparent media. Phys Rev A: At Mol Opt Phys 2010;82(4):043820. [20] Zhang L, Wen S, Fu X, et al. Spatiotemporal instability in dispersive nonlinear Kerr medium with a finite response time. Opt Commun 2010;283(10):2251–7. [21] Zhang L, Fu X, Deng J, et al. Evolution of spatial modulation of broadband laser pulses with different pulse durations in medium with non-instantaneous Kerr nonlinearity. Opt Commun 2011;284(8):2210–4. [22] Deng YB, Fu XQ, Tan C, et al. Experimental investigation of spatiotemporal evolution of femtosecond laser pulses during small-scale self-focusing. Appl Phys B 2013:1–6. [23] Agrawal GP. Principle and applications of nonlinear fiber optics. Beijing: Publishing House of Electronics Industry; 2003.
Contents lists available at ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Experimental measurement of ultrashort pulse evolution at different spatial positions in nonlinear media Chao Tan a, Xiquan Fu a,n, Yangbao Deng b a Key Laboratory for Micro-/Nano-Optoelectronic Devices of Ministry of Education, College of Information Science and Engineering, Hunan University, Changsha 410082, China b College of Communication and Electronic Engineering, Hunan City University, Yiyang 413002, China
art ic l e i nf o
a b s t r a c t
Article history: Received 22 August 2013 Received in revised form 13 December 2013 Accepted 14 December 2013 Available online 1 January 2014
The temporal evolution of ultrashort pulses before the beam splitting is very useful for filamentation theory. In this article, we experimentally demonstrated the temporal evolution of ultrashort pulses in the different spatial positions during small-scale self-focusing. The simulation results are found to be in good agreement with experiment measurements. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Temporal evolution Ultrashort pulse Small-scale self-focusing
1. Introduction The nonlinear propagation of ultrashort laser pulses through a Kerr medium is usually accompanied by several interesting phenomena, such as self-focusing (SF) [1], supercontinuum generation [2], multiphoton ionization [3] and filamentation [4]. SF of beam is a common phenomenon in nonlinear optics [5]. In the high power laser system, SF, especially small-scale self-focusing (SSSF), is a major factor for degradation of the laser beam quality and damage of optical materials [6,7]. When ultrashort pulses propagate in the nonlinear media, SF moves off-axis energy towards the peak of the pulse and compresses it both in spatial and temporal [8,9]. As the peak intensity increases, the process of self-phase modulation (SPM) also increases, as a result of the interplay between SPM-induced upchirp and group velocity dispersion, initiating the pulse splitting [10,11]. It is well known that Bespalov and Talanov [12] first explain the SSSF and the filamentation of a beam in terms of the modulation instability (MI) model, but they do not consider the perturbations in the temporal-domain. Campillo and Bliss et al. verify experimentally MI and the relationship among fastest growing frequency, SF length and light intensity [13–15]. The direct time-resolved observation of the propagation of intensity femtosecond laser pulses experiencing SF, beam filamentation is investigated extensively [16–19]. Previously, we have observed the evolution of spatial modulation in the process of SSSF of a beam, the
n
Corresponding author. Tel.: þ 867 318 882 4527. E-mail address: [email protected] (X. Fu).
0030-3992/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlastec.2013.12.010
impact of relaxation effect in the media on spatial–temporal instability [20,21]. However, these studies mainly focus on the temporal evolution of the pulse in the forming and after the formation of filaments, there is little attention to the temporal evolution of the beam in the period before filamentation. Especially, the partial spatial intensity of the beam changes impact on the temporal evolution during the period of SSSF has been rarely studied. Recently, we have demonstrated the time evolution of spatial intensity increasing zones and non-increasing zones in the process of SSSF [22]. In this paper, we first demonstrate the spatial evolution of the pump beam during SSSF in the experiment and simulation. We find that there will be different growths in the different spatial positions, especially the modulation peak and modulation bottom in space, with the increase of input power. Then, we show that the temporal evolution of some spatial positions of the pump pulse monitored by a probe pulse. Due to the effect of spatiotemporal coupling, the increase of spatial intensity at modulation peak will lead to pulse width compression. However, the pulse width of modulation bottom is broadened with the increase of power, because its zone contrast is actually decreased. 2. Experiment setup A schematic overview of our experiment setup is shown in Fig. 1. We use an amplified Ti:sapphire laser system (LibraS, Coherent, λ0 ¼ 800 nm) delivering pulses with bandwidth of 12 nm, single pulse energy of 1 mJ, duration τ0 ¼ 100 fs at 1 kHz repetition rate. The beam is split into two beams by a beam splitter, BS1, one is used
48
C. Tan et al. / Optics & Laser Technology 59 (2014) 47–51
low, the influence of nonlinear effect on the beam is far less than the diffraction effect, the diffractive modulation play a major role in the spatial evolution of the pump beam. So the SSSF at D1 is not obvious when peak power is 30.5 MW [Fig. 2(a)]. The nonlinear effect enhanced by the increment of the power and a new growth at D1 appears, the SSSF at D1 is obvious to see [Fig. 2(b), peak power 90.2 MW]. When the power increased to 138.1 MW, the intensity of D1 reach maximum and a filament is formed at D1 [Fig. 2(c)]. If the power is increased continuously, the pump beam will split into multifilaments. However, the intensity of D2 not only has no significant growth with the increase of power, but its zone contrast is fall-off. Zone contrast is defined as the ratio between the partial spatial intensity and the average intensity of the beam. Using a split-step Fourier algorithm, spatial–temporal nonlinear Schrödinger equation is used for calculation. 2 ∂A i ∂ ∂2 iβ ∂2 A iω0 n2 ¼ j Aj 2 A þ 2 A 2 2 þ 2 ∂z 2β0 ∂x 2 ∂t c ∂y Fig. 1. Experimental setup. BS1 and BS2, beam splitters; M1 and M2, optical delay line; M3 and M4, silver-coated plane mirrors; M5, rotating mirror; A1 and A2, attenuators. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
as a pump beam and the other as a probe beam. The probe beam passes through a variable optical delay line before it is directed to a slit. A rotating mirror in front of the slit is used to change the direction of the probe beam, according that the pump beam can be scanned by the probe beam when M5 is rotated. The horizontal spatial profile of the probe beam after passing through the slit is a Sinc function distribution. The spatial and temporal scanning resolution can be controlled by adjusting the slit width and the pulse width of the probe pulse, respectively. We set the aperture of vertical slit to 0.57 mm in the experiment. A glass cuvette with 5 cm path length is filled with carbon disulfide (CS2). CS2, a typical relaxing Kerr media, is chosen as the nonlinear media because of its strong Kerr nonlinearity. The pump beam is directed to the CS2 reflected by M3. A variable attenuator, A1, is used to adjust the input power of the pump beam. A filament with diameter of 0.1 mm in front of CS2 is used to generate initial diffraction modulation. The evolution of spatial–temporal of the pump pulse can be monitored by two synchronized high-resolution charge-coupled device (CCD) cameras (Coherent Laser Cam-HR™ Beamview, 1280 1024 pixel, pixel size of 6:7 μm). In order to measure the spatial evolution of the pump beam in the front surface of BBO crystal, a part of the pump beam is imaged onto CCD1 reflected by a beam splitter, BS2. It follows that the optical path from the beam splitter, BS2, to BBO crystal is equal to from BS2 to CCD1. A sum-frequency beam generated in the BBO nonlinear crystal under noncollinear interaction of two beams is collected with a CCD2. So we can indirectly measure the temporal evolution of pump pulses by the sumfrequency beam based on the cross-correlation principle. The adjustable attenuators, A2 and A3, are custom-designed to protect CCD cameras from damage by high power laser beams.
3. Results and discussions 3.1. The spatial evolution of the beam during SSSF Adjusting the attenuator, A1, and keeping the position of CCD1 unchanged, the spatial evolution of the pump beam which modulated by a filament in different power is shown in Fig. 2. From Fig. 2, it is easy to see distinct SSSF of the beam appears. The arrows serve to indicate the positions of the spatial modulation peak D1 and spatial modulation bottom D2. When the power is
where A is the amplitude of the input pulse, β0 is the transmission constant, β2 is the group velocity dispersion, ω0 is the central frequency, n2 is the nonlinear index of refraction. The initial field is taken to be a Gaussian in both time and space, having an intensity full width at half maximum (FWHM) of 105 fs and 2 mm, respectively. The beam waist is located at the entrance face of the BBO. The center wavelength is λ ¼ 800 nm, the linear index of refraction is n0 ¼ 1:63, the nonlinear index of refraction is n2 ¼ 3:5 10 15 cm2 =W, and the group velocity dispersion (GVD) coefficient is β2 ¼ 1:54 10 25 s2 =m. To simplify the problem, we have chosen to neglect loss and high order dispersion. The diameter of the filament is 0.1 mm and the length of nonlinear propagation is 5 cm. The numerical simulation maps shown in Fig. 2(d)–(f) show the intensity distribution of the pump beam modulated by a filament at three different input powers. When the power is 30.5 MW, input peak power is lower than the critical power for SF, and diffractive modulation play a major role in the spatial evolution of pump beam [Fig. 2(d)]. With the increase of the power, a new growth position appears in the first modulation peak, the SSSF of beam is obvious to see [Fig. 2(e), peak power 90 MW]. Because SSSF moves energy towards the modulation peak, so we can see from Fig. 2 that the intensity of modulation peak D1 increases rapidly. When the power increased to 138.1 MW, the intensity of the growth area reaches maximum [Fig. 2(f)]. However, the spatial intensity of the modulation peak cannot increase infinitely because the total energy of the beam is constant. As the power is increased continuously, new growth points will complete with the original growth point, resulting in the filamentation of the beam. No obvious growth can be observed at the modulation bottom because the SSSF effect at D2 is very weak. The simulation results are found to be in good agreement with experimental measurements. Fig. 3(a) represents the spatial distribution of the pump beam in the horizontal plane position pointed by arrows A and B of Fig. 2(a)–(c). The numerical simulation map shown in Fig. 3(b) shows the pump beam varies with different input power. Similar to Figs. 2 and 3 displays visually the spatial evolution of laser beam modulated by a filament in the process of SSSF. With the increase of input power, nonlinear effect gradually strengthened and a new growth point in D1 appears. The intensity of D1 reach maximum when the input power increased to 138.1 MW. The intensity at D1 increased by 12% when the power is increased from 30.5 MW to 138.1 MW, the growth is a little difference between experiment and simulation at some other modulation peaks because the quality of the beam profile in the experiment is not very good. However, the zone contrast of D2 decreases with the increase of power [Fig. 3].
C. Tan et al. / Optics & Laser Technology 59 (2014) 47–51
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Fig. 2. The spatial evolution of the pump beam which modulated by a filament at peak power of (a) 30.5 MW, (b) 90.2 MW, (c) 138.1 MW in the experiment; (d) 30.5 MW, (e) 90.2 MW, (f) 138.1 MW in the simulation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3.2. The temporal evolution of ultrashort pulses with SSSF in different spatial positions In the process of nonlinear propagation of the femtosecond laser pulses, nonlinear effect gradually strengthened with the increase of input power. It finds that the SF effect of the intense femtosecond pulses will lead to the rapid growth of partial spatial intensity, the changes of spatial intensity will affect the pulse width of these areas due to spatiotemporal coupling effect. Fig. 4 shows numerical simulation curves of the pulse width in the horizontal plane position pointed by arrows A and B of Fig. 2(a)– (c) at different peak power. The experimental measured pulse width of pump pulses at positions of D1 and D2 are also displayed in Fig. 4. We only considered the left brain part of the beam because the Gaussian beam is symmetrical. Root mean square (RMS) pulse width is used to characterize the pulse width in numerical simulation [23].
RMS pulse width is defined as Z 1 Z s ¼ ½〈T 2 〉 〈T〉2 1=2 ; 〈T n 〉 ¼ T n j Aðz; TÞj 2 dT= 1
1 1
j Aðz; TÞj 2 dT
When the power is 30.5 MW and the nonlinear effect is very weak in this case. The initial pulse widths at D1 and D2 are 106 fs and 115 fs from Fig. 4, this is because the pulse widths of ultrashort pulse at different positions of wave front are different. The pulse width of D1 is compressed with the increase of power, it decreased by 9.4% when the power is increased from 30.5 MW to 138.1 MW, the experimental results confirmed the numerical simulation in Fig. 4. However, the pulse width of D2 is broadened by 4.5% when the power increased from 30.5 MW to 138.1 MW. The residual dispersion in the experiment is neglected in the numerical simulation, so the quantity of pulse width broadened of experimental result is a little bit different from numerical simulation.
50
C. Tan et al. / Optics & Laser Technology 59 (2014) 47–51
Fig. 3. The spatial distribution of the pump beam in the horizontal plane position pointed by arrows A and B in Fig. 2 varies with input power (a) in the experiment; (b) in the simulation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
4. Conclusion In summary, we have investigated the spatial–temporal evolution of ultrashort pulses modulated by a filament in the process of SSSF. Our results indicate that the SSSF of the beam enhanced with the increase of power, as a result of spatiotemporal coupling effect, the increase of spatial intensity at modulation peak will lead to pulse width compression with the increase of power. However, the pulse width of modulation bottom is always broadened, because its zone contrast falls off with the increase of power. Our experimental observations are in good agreement with simulation results. For high power laser transmission and the measurement of spatial–temporal evolution of ultrashort pulses propagating in the nonlinear media, these research results have better practical reference value.
Acknowledgements
Fig. 4. The pulse width in the horizontal plane pointed by arrows A and B in Fig. 2 varies with the input power in the simulation, and the pulse widths at D1 and D2 vary with the input power in the experiment. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
As mentioned previously, the spatial intensity at different spatial positions will appear different changing trends when the ultrashort laser pulses propagate in the nonlinear media. According to (3 þ 1)-dimensional nonlinear Schrödinger equation, the temporal-domain and spatial-domain of the laser are affected by the same common nonlinear term. So we can see clearly that, differences in the changing trends of the pulse width appear at D1 and D2. The spatial intensity of modulation peak increases quickly with the increase of input power. The rapid growth of the spatial intensity leads to strong nonlinear effect. The spectrum of the pulses is broadened by SPM and then the pulse width of modulation peaks become shorter because of the strong nonlinear effect and dispersion. So the pulse width at the modulation peak in space is compressed with the increase of input power. The pulse width of D2 is broadened, because the zone contrast of D2 decreases with the increase of power.
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