- Email: [email protected]
Fresnel diffraction and small-scale self-focusing of a phase modulated and spectrally dispersed laser beam
Optics & Laser Technology 45 (2013) 56–61
Contents lists available at SciVerse ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Fresnel diffraction and small-scale self-focusing of a phase modulated and spectrally dispersed laser beam Jianqin Deng, Xiquan Fu, Lifu Zhang, Jin Zhang, Shuangchun Wen n Key Laboratory for Micro-/Nano-Optoelectronic Devices of Ministry of Education, College of Information Science and Engineering, Hunan University, Changsha 410082, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 May 2012 Received in revised form 24 July 2012 Accepted 28 July 2012 Available online 24 August 2012
To unravel the propagation properties of the phase modulated and spectrally dispersed (PM-SD) beam in the inertial confinement fusion lasers, we investigated theoretically and numerically the Fresnel diffraction of the PM-SD beam after passing a hard-edged aperture and small-scale self-focusing (SSSF). It is found that the Fresnel diffraction pattern of the PM-SD beam can be significantly changed and the laser beam uniformity can be improved by choosing appropriate PM-SD beam’s parameters. The diffraction field of the PM-SD beam has a slower growth than the monochromatic laser beam when the beams suffer from SSSF. The PM-SD beam can be applied to improve the beam uniformity and suppress or delay the onset of SSSF in the high power laser system. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Fresnel diffraction Small-scale self-focusing Smoothing by spectral dispersion
1. Introduction In the inertial confinement fusion high-power laser system, the beam with non-uniform intensity distribution gives rise to non-uniform irradiation on the target, which may seed the Rayleigh–Taylor instability and thus seriously affect the burn efficiency of the target [1,2]. Over the years, many techniques have been proposed to improve the near-field beam quality and achieve the target irradiation uniformity in the far-field, which include soft-edged apertures [3], multi-stage spatial filters [4,5], divergent beams [6], partially coherent light beams [7], disk amplifiers [8], and broadband laser [9,10]. The smoothing techniques for target irradiation uniformity have spatial smoothing techniques which include continuous phase plates (CPP) [11,12], distributed phase plates (DPP) [13] and lens arrays (LA) [14], and temporal smoothing techniques, such as induced spatial incoherence (ISI) [15], smoothing by spectral dispersion (SSD) [16]. The strategy of SSD with phase plates (e.g., CPP or DPP) has been proved to be suitable for the solid-state glass laser driver [17–20]. Here, the SSD leads to the speckle patterns sweeping spatially on a time scale shorter than the imprinting time of the target, which can be controlled by the electro-optic (E-O) modulator and diffraction grating. The E-O modulator is used to transform the seed beam into a phase modulated laser with a small bandwidth, and then the bandwidth is spectrally dispersed by utilizing the diffraction grating. The output beam of the SSD device, termed PM-SD beam, has a special phase varying as a sinusoidal wave
n
Corresponding author. E-mail address: [email protected] (S. Wen).
0030-3992/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlastec.2012.07.034
with time and transverse space. The speckle patterns are shifted as a result of the variation of the instantaneous frequency across the beam. And here, the phase plate is used to precisely control the far-field intensity envelope on the target plane. The SSD devices are usually placed before the main amplifiers [16,19], and serves as the role for target irradiation uniformity in the far-field. The PM-SD beam was inevitably affected by Fresnel diffraction, such as the hard-edged diffraction diaphragm due to the optical elements with limited aperture. In addition, dust and other obstructions are also the diffraction sources imposing modulations. Especially, modulation ripples in the beam lead to the non-uniform intensity distribution. When modulation ripples propagate in the nonlinear medium, they will motivate the SSSF [21], even lead to the beam breakup into multi-filaments with very high light intensity. As a result, the filaments will cause damage to the laser medium and limit the output power of the laser system [22–24]. There have been many researches on SSD. Most researches focused on the function of SSD in the far-field target irradiation uniformity. Our previous work [25] has shown some interesting nonlinear propagation properties of the PM-SD beam. The properties of the PM-SD beam in the near-field, especially Fresnel diffraction, and the SSSF from the diffraction modulation are still far from full understanding. This paper addresses the issues of Fresnel diffraction and the SSSF effect with detailed theoretical and numerical investigations. The theoretical analysis shows that the near-field diffraction pattern of the PM-SD beam can be significantly changed by different SSD parameters. Comparing with the monochromatic laser beam, the PM-SD beam can be used to eliminate or smooth out the intensity peaks of the diffracted field, and improve the beam spatial distribution
J. Deng et al. / Optics & Laser Technology 45 (2013) 56–61
uniformity. Then we simulated the SSSF of the PM-SD beam by taking the diffraction field as a modulation source. And the simulation results indicate that, the PM-SD beam has much lower modulation depth, compared to the monochromatic laser beam. The paper is organized as follows. In Section 2, the expression of the PM-SD beam propagating through a hard-edged aperture is derived. In Section 3, numerical simulations are performed to verify the theoretical analysis on the Fresnel diffraction of the PM-SD beam, and the nonlinear propagation of the PM-SD beam. Finally, a concise conclusion is given out in Section 4.
2. Theoretical analysis
ð1Þ where Jn is the Bessel function of the first kind of order n and represents the amplitude of n-order sideband with frequency on ¼ o0 þnom, k¼n0o0/c is the wave number of the fundamental frequency o0, c is the speed of light in vacuum, z axis is defined as the propagation direction of the fundamental frequency, and dispersion direction x is perpendicular to the propagation direction z. For simplicity, only the one dimensional SSD is considered in this paper, d and om are the modulation amplitude and modulation frequency, respectively. The bandwidth of the PM-SD beam is approximated as Do ¼2dom, and the parameter a is
Dy om Dl o0
a
2
1 x20 Jn ðdÞexp4 2 w20 a
!5 3 5exp½inax0 exp ik ðx0 xÞ2 dx0 2z ð7Þ
The range of values allowed for integer n is d to d, which is determined by the effective bandwidth Do ¼ 2dom because the number of effective sidebands is equal to 2d. Assuming that the pulse has a Gaussian intensity profile, and then the time-averaged intensity distribution of the beam in the diffraction plane can be written as " # 2 Z Dt
1 t 2
IPM-SD_Dif ðx,zÞ ¼ E ðx,z,tÞexp ð8Þ
dt Dt 0 PM-SD_Dif 2T 2
0
The electric field of the PM-SD beam is obtained as [14] z h h ii X EPM-SD ðx,z,tÞ ¼ E0 J n ðdÞexp iðo0 tkzÞ exp in om t þ ax c n
a ¼ 2p
Z
57
ð2Þ
where Dy/Dl is the grating dispersion coefficient. Assuming that a hard-edged aperture with radius a is placed behind the diffraction grating, the incident field can be written as: X EPM-SD ðx,0,tÞ ¼ E0 ðxÞ J n ðdÞexp½io0 texp½inðom t þ axÞ ð3Þ n
For simplicity, here we have ignored the influence of pulse temporal envelope, and only consider the spatial intensity distributed with: 2 !5 3 1 x2 4 5 ð4Þ E0 ðxÞ ¼ exp 2 w20
For the monochromatic SG beam, the diffraction field distribution and the intensity distribution in the diffraction plane are given by the equations 2 !5 3 1=2 Z 2 a x i 1 0 5 exp ikz exp4 ESG_Dif ðx,z,tÞ ¼ 2 w20 lz a ik exp½io0 t exp ðx0 xÞ2 dx0 2z " # 2 Z Dt
1 t 2
ISG_Dif ðx,zÞ ¼ ðx,z,tÞexp 2 dt ð9Þ
E Dt 0 SG_Dif 2T
0
In order to compare the differences of diffraction characteristics between the PM-SD beam and the monochromatic SG beam, the transverse time-averaged intensity distributions in the diffraction plane are shown in Fig. 1. The PM-SD beam’s parameters are d ¼ 10, om ¼2pvm with vm ¼5 GHz, Dy/Dl ¼5000 mrad/nm, and the Fresnel number N ¼30 is obtained from the formula N ¼a2/lz. Compared with the monochromatic SG beam (dashed curve), the diffraction field of the PM-SD beam (solid curve) was smoothed. As can be seen, the transverse intensity distribution of the monochromatic SG beam is greatly non-uniform and many intensity spikes emerge. But for the PM-SD beam, the diffraction field exhibits a relatively uniform intensity distribution with fewer spikes and lower peak strength, especially in a certain domain near the beam center. In fact, the uniform diffraction intensity distribution results from the variation of frequency across the beam. The different diffraction patterns of the component frequency sidebands differ from one another in transverse spatial position. As a result, the peaks of some diffraction patterns
where w0 is the beam waist of initial super-Gaussian (SG) beam. In the frequency domain, according to Huygens–Fresnel principle, the diffraction field of the frequency component o ¼ o0 þnom is En ðx,zÞ ¼
Z a i 1=2 ik exp ikz En ðx0 ,0Þexp ðx0 xÞ2 dx0 2z lz a
ð5Þ
where (x0,0) are the coordinates of a point on the aperture plane and (x,z) are the coordinates of a point on the diffraction observation plane. The incident field En(x0,0) can be written as 2 !5 3 2 x 1 0 5exp½inax0 ð6Þ En ðx0 ,0Þ ¼ J n ðdÞexp4 2 w20 As the PM-SD beam has typical discrete spectrum, the whole field distribution is X i 1=2 EPM-SD_Dif ðx,z,tÞ ¼ exp ikz þ iðo0 þ nom Þt l z n
Fig. 1. Transverse diffraction intensity distributions of the SG beam (dashed curve) and the PM-SD beam (solid curve). The PM-SD beam’s parameters are d ¼ 10, om ¼ 2pvm with vm ¼ 5 GHz, Dy/Dl ¼5000 mrad/nm, and the Fresnel number N ¼ 30.
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J. Deng et al. / Optics & Laser Technology 45 (2013) 56–61
Fig. 2. Transverse diffraction intensity distributions calculated from a pair of sidebands with o0 2om (dashed curve) and o0 þ 2om (dotted curve), (a) the full diffraction patterns and (b) the detailed version of the circular region in the full diffraction patterns.
fill in the valleys of others. When the diffraction patterns of all component frequency sidebands are overlapped, the time-averaged intensity distribution of the PM-SD beam is smoothed. Fig. 2 gives the diffraction patterns of a pair of sidebands with the frequencies o0 2om and o0 þ2om. Fig. 2 shows the peaks and the valleys of the two sidebands are mutually staggered, resulting in smoothing of the overlapped diffraction field. To clarify, in Fig. 2 the two frequencies are the frequency components of the PM-SD beam shown in Fig. 1. Moreover, a better uniformity can be achieved by overlapping more diffraction patterns or improving smoothing efficiency of the frequency pairs. Our results are quite different from the previous studied in [9], where it required bigger difference of frequencies in the beam bandwidth. The bandwidth required in Ref. [9] is much bigger than the critical bandwidth. Owing to the limitations of gain bandwidth and gain saturation, the bandwidth allowed for the amplifiers cannot exceed a critical value, for example, 10 nm for the common neodymium-glass amplifiers. While in Fig. 2, the bandwidth is less than 0.1 nm. It is much more realistic than that of Ref. [9].
3. Numerical simulation When the PM-SD beam propagates in free space and nonlinear medium, the evolution process of the beam satisfies the dimen¨ sionless nonlinear Schrodinger equation @u i i LDF @2 u LDF 2 ¼ r2? u þi 9u9 u @x 4 2 LDS @t2 LNL
ð10Þ
where u ¼A/A0 is the normalized pulse envelope of the electric field, A0 is constant and equal to the amplitude in the pulse 2 center, r? ¼ @2 =@X 2 is the transverse Laplacian, t ¼t/T0, X ¼x/w0, x ¼z/LDF and T0 is the half-width (at 1/e-intensity point) of the pulse, w0 is the beam waist, LDF ¼ kw20 =2, LDS ¼ t 20 =9b2 9 and LNL ¼1/(kn2/n09A092) are diffraction length, the dispersion length and nonlinear length respectively, k is the wave number, n0 the index of refraction of the fundamental frequency o0, and n2 the Kerr coefficient, b2 ¼ @2 k=@o2 9o0 is the group-velocity dispersion (GVD) parameter. To facilitate the sampling in spatial and time domain, we can recombine Eq. (3) as z h h ii EPM-SD ¼ E0 exp io0 tikz exp idsin om t þ ax ð11Þ c Assuming that the input pulses have a Gaussian temporal profile with T0 ¼1 ns, and a transverse 5-order SG profile with
Fig. 3. Schematic diagram of the simulations.
w0 ¼2.75 mm, the incident envelope can be written as 2 " # !5 3 2 t2 1 x 5 Aðx,tÞ ¼ A0 exp 2 exp4 2 w20 2T 0 z h h ii exp io0 tikz exp idsin om t þ ax c
ð12Þ
Schematic diagram of the simulations is illustrated in Fig. 3. The SSD device generates a PM-SD beam, and then the PM-SD beam passes through a thin aperture with radius a¼2 mm. After the beam propagates through free space, it is launched into a 50 mm long nonlinear medium. The medium’s parameters are n0 ¼1.55, n2 ¼1.15 10 13 esu, and the GVD parameter is b2 ¼154 fs2/cm. A comparison of the diffraction (dashed curve) and the nonlinear propagation properties (solid curve) for the PM-SD beam and the monochromatic SG beam is shown in Fig. 4, where the initial intensity is I0 ¼5.0 109 W/cm2. The diffraction fields are used as the incident fields of the nonlinear propagation. Fig. 4(a) and (b) gives the diffraction field distributions and the output field distributions for the PM-SD beam and the monochromatic SG beam, respectively. The simulation results demonstrate that the PM-SD beam can wipe out the diffraction intensity spikes, since the diffraction field of the PM-SD beam shows a better uniformity than the monochromatic SG beam. Additionally, and most importantly, in Fig. 4(b) some modulation peaks grow from 1.2 to 4.0 approximately for the monochromatic SG beam due to serious SSSF effect. Whereas in Fig. 4(a), the output intensity peaks not even exceed 1.5 for the PM-SD beam. This indicates that the PM-SD beam can be used to suppress or delay the SSSF effect. Obviously, from Eqs. (1) and (11) we know that electric field properties of the PM-SD beam are determined by three parameters of the SSD device, namely modulation amplitude, modulation frequency, and the grating dispersion coefficient. In order to reveal the influence of the SSD device parameters on the nonlinear propagation process, we have done a series of simulations. Fig. 5 shows the time-averaged beams intensity profiles in the emergent surface of the nonlinear medium for four different modulation amplitudes [(a) d ¼5, (b) d ¼10, (c) d ¼15,
J. Deng et al. / Optics & Laser Technology 45 (2013) 56–61
59
Fig. 4. Normalized transverse intensity profiles (a) PM-SD beam and (b) monochromatic SG beam. The dashed curves are the diffraction field distributions and the solid curves are the output field distributions of the nonlinear medium. The PM-SD beam’s parameters are d ¼10, vm ¼ 5 GHz, Dy/Dl ¼5000 mrad/nm, and the Fresnel number N¼ 30.
Fig. 5. The influence of E-O modulation amplitude. (a) d ¼ 5, (b) d ¼10, (c) d ¼15 and (d) d ¼ 20, the other parameters are nm ¼3 GHz and Dy/Dl ¼5000 mrad/nm.
and (d) d ¼20] with vm ¼3 GHz and Dy/Dl ¼5000 mrad/nm. It shows that the modulation peaks decrease with increasing the modulation amplitude d. For different modulation frequencies [(a) vm ¼1 GHz, (b) vm ¼3 GHz, (c) vm ¼5 GHz, and (d) vm ¼ 7 GHz] with d ¼10 and Dy/Dl ¼5000 mrad/nm, the time-averaged beams intensity profiles are shown in Fig. 6. It is found that the effect of the modulation frequency vm is similar to the modulation amplitude d, due to the bandwidth formula Dv ¼2dvm. Although the bandwidth of the beams reflects the ability to wipe out the diffraction spikes and suppress the SSSF, there are some different mechanisms of action between d and vm. The effective frequency sidebands increase with increasing the modulation amplitude d, it means that more diffraction fields are formed in the frequency
domain. According to Eq. (7), the whole diffraction field in time domain is overlapped by the fields of each sideband, thus the more sidebands, the more uniform field distributions in the time domain and the better quality of the beam. With the same number of sidebands, the frequency interval broaden with increasing modulation frequency vm, showing a more uniform field distribution by improving the smoothing efficiency of the frequency pairs (e.g., o0 om and o0 þ om). In addition, Fig. 7 shows the time-averaged beams intensity profiles in the emergent surface of the medium for four different dispersion coefficients [(a) Dy/Dl ¼0, (b) Dy/Dl ¼ 1000 mrad/nm, (c) Dy/Dl ¼3000 mrad/nm, and (d) Dy/Dl ¼5000 mrad/nm] with d ¼10 and vm ¼5 GHz. Fig. 7(a) shows that the intensity profile for
60
J. Deng et al. / Optics & Laser Technology 45 (2013) 56–61
Fig. 6. The influence of E-O modulation frequency. (a) nm ¼1 GHz, (b) nm ¼ 3 GHz, (c) nm ¼ 5 GHz and (d) nm ¼ 7 GHz, the other parameters are d ¼ 10 and Dy/Dl ¼ 5000 mrad/nm.
Fig. 7. The influence of the grating dispersion coefficient. (a) Dy/Dl ¼ 0, (b) Dy/Dl ¼ 1000 mrad/nm, (c) Dy/Dl ¼ 3000 mrad/nm, and (d) Dy/Dl ¼ 5000 mrad/nm, with the d ¼ 10 and vm ¼5 GHz.
J. Deng et al. / Optics & Laser Technology 45 (2013) 56–61
Dy/Dl ¼0 almost have the same aspect for the monochromatic SG beam [solid line in Fig. 4(b)]. The beam is just imposed a discrete spectrum by the E-O modulator and the instantaneous frequency does not vary across the beam with the absence of dispersion. As a result, the diffraction patterns of each frequency component almost locate in the same transverse spatial position. Obviously, this kind of phase modulation beams is not able to wipe out the diffraction intensity spikes, and also the SSSF cannot be delayed or suppressed. Fig. 7(b) shows that the peaks of the diffraction intensity profiles decrease dramatically at Dy/Dl ¼1000 mrad/nm. While Fig. 7(c) and (d) shows that the intensity peaks continue to decline with the increasing of the dispersion coefficient Dy/Dl by enhancing the variation rate of the instantaneous frequency across the beam. Furthermore, it can clearly found that only a few modulation ripples have grown and the intensity profiles around the beam center almost always remain stable. According to the B-T theory [21], the ripples will grow fast only when the ripples frequency is equal to or close to the fastest growing frequency. As ripples frequency of the diffraction present obvious non-uniformity, some ripples located in the gain region grow quickly, while some ripples located out the gain region, such as the ripples around the beam center, remain stable. 4. Conclusions SSD is usually used for far-field target illumination uniformity in the high-power laser system. So the PM-SD beams are inevitably affected by Fresnel diffraction and the diffraction ripples will motivate the SSSF effect when the beams pass through nonlinear media. In this paper, Fresnel diffraction and SSSF of the PM-SD beam are investigated theoretically and numerically in detail. The theoretical analysis shows that, in comparison with the monochromatic SG beam, the diffraction field of the PM-SD beam is more uniform in the Fresnel domain. The results suggested that the PM-SD beam can smooth out the diffraction intensity spikes. The numerical simulation results demonstrated that the PM-SD beam has a slower nonlinear growth than that of the monochromatic SG beam. In addition, it is known from the results that the smoothing performance of the PM-SD beam is determined by the modulation amplitude, modulation frequency, and grating dispersion coefficient. The diffraction modulation growth rates decrease with increasing the modulation amplitude and modulation frequency. And the dispersion coefficient can also be used to suppress SSSF by enhancing the variation rate of the instantaneous frequency across the beam. Thus, the PM-SD beam can be applied as an alternative scheme to improve the beam uniformity and suppress or delay the onset of SSSF in the high power solid-state laser system.
Acknowledgments This work was supported by the National Nature Science Foundation of China (Grant No. 60890202), Hunan Provincial Natural Science Foundation of China (Grant No. 12JJ7005), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20110161110012), and the
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Fundamental Research Funds for the Central Universities, Hunan University and SRF for ROCS, SEM. References [1] Bruecker KA, Jorna S. Laser-driven fusion. Reviews of Modern Physics 1974;46:325–67. [2] Skupsky S, Lee K. Uniformity of energy deposition for laser driven fusion. Journal of Applied Physics 1983;54:3662–71. [3] Hee EWS. Fabrication of apodized apertures for laser beam attenuation. Optics & Laser Technology 1975;7:75–9. [4] Simmons WW, Murray JE, Rainer F, Guch S. Design, theory, and performance of a high-intensity spatial filter. LLNL Laser Program Annual Report UCRL50021-74; 1975. p. 169–74. [5] Mak AA, Soms LN, Fromzel VA, Iashin VE. Nd:glass lasers. Moscow: Nauka; 1990 [p. 288]. [6] Siegman AE. Small-scale self-focusing effects in tapered optical beams. Memo for File, /www.stanford.edu/ siegman/self_focusing_memo.pdfS; 2002. p. 1–13. [7] Alexandrova IV, Basov NG, Danilov AE, Mikhailov Y, Sklizkov GV, Fedotov SI. The effects of small-scale perturbations on the brightness of laser radiation in laser fusion experiments. Laser and Particle Beams 1983;1:241–50. [8] Baranova NB, Bykovsky NE, Senatsky YV, Chekalin SV. Nonlinear processes in the optical medium of a high-power neodymium laser. Journal of Russian Laser Research 1980;1:53–88. [9] Peng R, Ye Y, Tang Z, Fan D. Transverse intensity distributions of a broadband laser modulated by a hard-edged aperture. Journal of the Optical Society of America A 2005;22:1903–8. [10] Peng R, Tang Z, Ye Y, Wen S, Fan D. Effects of the frequency chirp on the fields of a chirped Gaussian pulse passing through a hard-edged aperture. Optics Communications 2006;259:474–8. [11] Arieli Y. A continuous phase plate for non-uniform illumination beam shaping using the inverse phase contrast method. Optics Communications 2000;180:239–45. [12] Lin Y, Kessler TJ, Lawrence GN. Design of continuous surface-relief phase plates by surface-based simulated annealing to achieve control of focal-plane irradiance. Optics Letters 1996;21:1703–5. [13] Lin Y, Kessler TJ, Lawrence GN. Distributed phase plates for super-Gaussian focal-plane irradiance profiles. Optics Letters 1995;20:764–6. [14] Deng X, Liang X, Chen Z, Yu W, Ma R. Uniform illumination of large targets using a lens array. Applied Optics 1986;25:377–81. [15] Lehmberg RH, Schmitt AJ, Bodner SE. Theory of induced spatial incoherence. Journal of Applied Physics 1987;62:2680–701. [16] Skupsky S, Short RW, Kessler T, Craxton RS, Letzring S, Soures JM. Improved laser beam uniformity using the angular dispersion of frequency-modulated light. Journal of Applied Physics 1989;66:3456–62. [17] Rothenberg JE. Comparison of beam-smoothing methods for direct-drive inertial confinement fusion. Journal of the Optical Society of America B 1997;14:1664–71. [18] Bodner SE, Colombant DG, Gardner JH, Lehmberg RH, Obenschain SP, Philips L, et al. Direct-drive laser fusion status and prospects. Physics of Plasmas 1998;5:1901–18. [19] Regan SP, Marozas JA, Kelly JH, Boehly TR, Donaldson WR, Jaanimagi AP, et al. Experimental investigation of smoothing by spectral dispersion. Journal of the Optical Society of America B 2000;17:1483–9. [20] Lehmberg RH, Rothenberg JE. Comparison of optical beam smoothing techniques for inertial confinement fusion and improvement of smoothing by the use of zero-correlation masks. Journal of Applied Physics 2000;87: 1012–22. [21] Bespalov VI, Talanov VI. Filamentary structure of light beams in nonlinear liquids. ZhETF Pis ma Redaktsiiu 1966;3:307–10. [22] Wimmons WW, Hunt JT, Warren WE. Light propagation through large laser systems. IEEE Journal of Quantum Electronics 1981;17:1727–44. [23] Williams W, Trenholme J, Orth C, Haney S, Sacks R, Auerbach J, et al. NIF design optimization. LLNL Laser Program Quarterly Report UCRL-LR-10582196-4; 1996. p. 181–91. [24] Fleck JA, Morris JR, Bliss ES. Small-scale self-focusing effects in a high-power glass laser amplifier. IEEE Journal of Quantum Electronics 1978;14:353–63. [25] Zhang Y, Fu X, Deng Y, Deng J. The spatiotemporal evolution of sinusoidal phase modulated laser and its competitions with noise perturbed laser during small-scale self-focusing. Optics & Laser Technology 2012;44: 2377–83.
Contents lists available at SciVerse ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Fresnel diffraction and small-scale self-focusing of a phase modulated and spectrally dispersed laser beam Jianqin Deng, Xiquan Fu, Lifu Zhang, Jin Zhang, Shuangchun Wen n Key Laboratory for Micro-/Nano-Optoelectronic Devices of Ministry of Education, College of Information Science and Engineering, Hunan University, Changsha 410082, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 May 2012 Received in revised form 24 July 2012 Accepted 28 July 2012 Available online 24 August 2012
To unravel the propagation properties of the phase modulated and spectrally dispersed (PM-SD) beam in the inertial confinement fusion lasers, we investigated theoretically and numerically the Fresnel diffraction of the PM-SD beam after passing a hard-edged aperture and small-scale self-focusing (SSSF). It is found that the Fresnel diffraction pattern of the PM-SD beam can be significantly changed and the laser beam uniformity can be improved by choosing appropriate PM-SD beam’s parameters. The diffraction field of the PM-SD beam has a slower growth than the monochromatic laser beam when the beams suffer from SSSF. The PM-SD beam can be applied to improve the beam uniformity and suppress or delay the onset of SSSF in the high power laser system. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Fresnel diffraction Small-scale self-focusing Smoothing by spectral dispersion
1. Introduction In the inertial confinement fusion high-power laser system, the beam with non-uniform intensity distribution gives rise to non-uniform irradiation on the target, which may seed the Rayleigh–Taylor instability and thus seriously affect the burn efficiency of the target [1,2]. Over the years, many techniques have been proposed to improve the near-field beam quality and achieve the target irradiation uniformity in the far-field, which include soft-edged apertures [3], multi-stage spatial filters [4,5], divergent beams [6], partially coherent light beams [7], disk amplifiers [8], and broadband laser [9,10]. The smoothing techniques for target irradiation uniformity have spatial smoothing techniques which include continuous phase plates (CPP) [11,12], distributed phase plates (DPP) [13] and lens arrays (LA) [14], and temporal smoothing techniques, such as induced spatial incoherence (ISI) [15], smoothing by spectral dispersion (SSD) [16]. The strategy of SSD with phase plates (e.g., CPP or DPP) has been proved to be suitable for the solid-state glass laser driver [17–20]. Here, the SSD leads to the speckle patterns sweeping spatially on a time scale shorter than the imprinting time of the target, which can be controlled by the electro-optic (E-O) modulator and diffraction grating. The E-O modulator is used to transform the seed beam into a phase modulated laser with a small bandwidth, and then the bandwidth is spectrally dispersed by utilizing the diffraction grating. The output beam of the SSD device, termed PM-SD beam, has a special phase varying as a sinusoidal wave
n
Corresponding author. E-mail address: [email protected] (S. Wen).
0030-3992/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlastec.2012.07.034
with time and transverse space. The speckle patterns are shifted as a result of the variation of the instantaneous frequency across the beam. And here, the phase plate is used to precisely control the far-field intensity envelope on the target plane. The SSD devices are usually placed before the main amplifiers [16,19], and serves as the role for target irradiation uniformity in the far-field. The PM-SD beam was inevitably affected by Fresnel diffraction, such as the hard-edged diffraction diaphragm due to the optical elements with limited aperture. In addition, dust and other obstructions are also the diffraction sources imposing modulations. Especially, modulation ripples in the beam lead to the non-uniform intensity distribution. When modulation ripples propagate in the nonlinear medium, they will motivate the SSSF [21], even lead to the beam breakup into multi-filaments with very high light intensity. As a result, the filaments will cause damage to the laser medium and limit the output power of the laser system [22–24]. There have been many researches on SSD. Most researches focused on the function of SSD in the far-field target irradiation uniformity. Our previous work [25] has shown some interesting nonlinear propagation properties of the PM-SD beam. The properties of the PM-SD beam in the near-field, especially Fresnel diffraction, and the SSSF from the diffraction modulation are still far from full understanding. This paper addresses the issues of Fresnel diffraction and the SSSF effect with detailed theoretical and numerical investigations. The theoretical analysis shows that the near-field diffraction pattern of the PM-SD beam can be significantly changed by different SSD parameters. Comparing with the monochromatic laser beam, the PM-SD beam can be used to eliminate or smooth out the intensity peaks of the diffracted field, and improve the beam spatial distribution
J. Deng et al. / Optics & Laser Technology 45 (2013) 56–61
uniformity. Then we simulated the SSSF of the PM-SD beam by taking the diffraction field as a modulation source. And the simulation results indicate that, the PM-SD beam has much lower modulation depth, compared to the monochromatic laser beam. The paper is organized as follows. In Section 2, the expression of the PM-SD beam propagating through a hard-edged aperture is derived. In Section 3, numerical simulations are performed to verify the theoretical analysis on the Fresnel diffraction of the PM-SD beam, and the nonlinear propagation of the PM-SD beam. Finally, a concise conclusion is given out in Section 4.
2. Theoretical analysis
ð1Þ where Jn is the Bessel function of the first kind of order n and represents the amplitude of n-order sideband with frequency on ¼ o0 þnom, k¼n0o0/c is the wave number of the fundamental frequency o0, c is the speed of light in vacuum, z axis is defined as the propagation direction of the fundamental frequency, and dispersion direction x is perpendicular to the propagation direction z. For simplicity, only the one dimensional SSD is considered in this paper, d and om are the modulation amplitude and modulation frequency, respectively. The bandwidth of the PM-SD beam is approximated as Do ¼2dom, and the parameter a is
Dy om Dl o0
a
2
1 x20 Jn ðdÞexp4 2 w20 a
!5 3 5exp½inax0 exp ik ðx0 xÞ2 dx0 2z ð7Þ
The range of values allowed for integer n is d to d, which is determined by the effective bandwidth Do ¼ 2dom because the number of effective sidebands is equal to 2d. Assuming that the pulse has a Gaussian intensity profile, and then the time-averaged intensity distribution of the beam in the diffraction plane can be written as " # 2 Z Dt
1 t 2
IPM-SD_Dif ðx,zÞ ¼ E ðx,z,tÞexp ð8Þ
dt Dt 0 PM-SD_Dif 2T 2
0
The electric field of the PM-SD beam is obtained as [14] z h h ii X EPM-SD ðx,z,tÞ ¼ E0 J n ðdÞexp iðo0 tkzÞ exp in om t þ ax c n
a ¼ 2p
Z
57
ð2Þ
where Dy/Dl is the grating dispersion coefficient. Assuming that a hard-edged aperture with radius a is placed behind the diffraction grating, the incident field can be written as: X EPM-SD ðx,0,tÞ ¼ E0 ðxÞ J n ðdÞexp½io0 texp½inðom t þ axÞ ð3Þ n
For simplicity, here we have ignored the influence of pulse temporal envelope, and only consider the spatial intensity distributed with: 2 !5 3 1 x2 4 5 ð4Þ E0 ðxÞ ¼ exp 2 w20
For the monochromatic SG beam, the diffraction field distribution and the intensity distribution in the diffraction plane are given by the equations 2 !5 3 1=2 Z 2 a x i 1 0 5 exp ikz exp4 ESG_Dif ðx,z,tÞ ¼ 2 w20 lz a ik exp½io0 t exp ðx0 xÞ2 dx0 2z " # 2 Z Dt
1 t 2
ISG_Dif ðx,zÞ ¼ ðx,z,tÞexp 2 dt ð9Þ
E Dt 0 SG_Dif 2T
0
In order to compare the differences of diffraction characteristics between the PM-SD beam and the monochromatic SG beam, the transverse time-averaged intensity distributions in the diffraction plane are shown in Fig. 1. The PM-SD beam’s parameters are d ¼ 10, om ¼2pvm with vm ¼5 GHz, Dy/Dl ¼5000 mrad/nm, and the Fresnel number N ¼30 is obtained from the formula N ¼a2/lz. Compared with the monochromatic SG beam (dashed curve), the diffraction field of the PM-SD beam (solid curve) was smoothed. As can be seen, the transverse intensity distribution of the monochromatic SG beam is greatly non-uniform and many intensity spikes emerge. But for the PM-SD beam, the diffraction field exhibits a relatively uniform intensity distribution with fewer spikes and lower peak strength, especially in a certain domain near the beam center. In fact, the uniform diffraction intensity distribution results from the variation of frequency across the beam. The different diffraction patterns of the component frequency sidebands differ from one another in transverse spatial position. As a result, the peaks of some diffraction patterns
where w0 is the beam waist of initial super-Gaussian (SG) beam. In the frequency domain, according to Huygens–Fresnel principle, the diffraction field of the frequency component o ¼ o0 þnom is En ðx,zÞ ¼
Z a i 1=2 ik exp ikz En ðx0 ,0Þexp ðx0 xÞ2 dx0 2z lz a
ð5Þ
where (x0,0) are the coordinates of a point on the aperture plane and (x,z) are the coordinates of a point on the diffraction observation plane. The incident field En(x0,0) can be written as 2 !5 3 2 x 1 0 5exp½inax0 ð6Þ En ðx0 ,0Þ ¼ J n ðdÞexp4 2 w20 As the PM-SD beam has typical discrete spectrum, the whole field distribution is X i 1=2 EPM-SD_Dif ðx,z,tÞ ¼ exp ikz þ iðo0 þ nom Þt l z n
Fig. 1. Transverse diffraction intensity distributions of the SG beam (dashed curve) and the PM-SD beam (solid curve). The PM-SD beam’s parameters are d ¼ 10, om ¼ 2pvm with vm ¼ 5 GHz, Dy/Dl ¼5000 mrad/nm, and the Fresnel number N ¼ 30.
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Fig. 2. Transverse diffraction intensity distributions calculated from a pair of sidebands with o0 2om (dashed curve) and o0 þ 2om (dotted curve), (a) the full diffraction patterns and (b) the detailed version of the circular region in the full diffraction patterns.
fill in the valleys of others. When the diffraction patterns of all component frequency sidebands are overlapped, the time-averaged intensity distribution of the PM-SD beam is smoothed. Fig. 2 gives the diffraction patterns of a pair of sidebands with the frequencies o0 2om and o0 þ2om. Fig. 2 shows the peaks and the valleys of the two sidebands are mutually staggered, resulting in smoothing of the overlapped diffraction field. To clarify, in Fig. 2 the two frequencies are the frequency components of the PM-SD beam shown in Fig. 1. Moreover, a better uniformity can be achieved by overlapping more diffraction patterns or improving smoothing efficiency of the frequency pairs. Our results are quite different from the previous studied in [9], where it required bigger difference of frequencies in the beam bandwidth. The bandwidth required in Ref. [9] is much bigger than the critical bandwidth. Owing to the limitations of gain bandwidth and gain saturation, the bandwidth allowed for the amplifiers cannot exceed a critical value, for example, 10 nm for the common neodymium-glass amplifiers. While in Fig. 2, the bandwidth is less than 0.1 nm. It is much more realistic than that of Ref. [9].
3. Numerical simulation When the PM-SD beam propagates in free space and nonlinear medium, the evolution process of the beam satisfies the dimen¨ sionless nonlinear Schrodinger equation @u i i LDF @2 u LDF 2 ¼ r2? u þi 9u9 u @x 4 2 LDS @t2 LNL
ð10Þ
where u ¼A/A0 is the normalized pulse envelope of the electric field, A0 is constant and equal to the amplitude in the pulse 2 center, r? ¼ @2 =@X 2 is the transverse Laplacian, t ¼t/T0, X ¼x/w0, x ¼z/LDF and T0 is the half-width (at 1/e-intensity point) of the pulse, w0 is the beam waist, LDF ¼ kw20 =2, LDS ¼ t 20 =9b2 9 and LNL ¼1/(kn2/n09A092) are diffraction length, the dispersion length and nonlinear length respectively, k is the wave number, n0 the index of refraction of the fundamental frequency o0, and n2 the Kerr coefficient, b2 ¼ @2 k=@o2 9o0 is the group-velocity dispersion (GVD) parameter. To facilitate the sampling in spatial and time domain, we can recombine Eq. (3) as z h h ii EPM-SD ¼ E0 exp io0 tikz exp idsin om t þ ax ð11Þ c Assuming that the input pulses have a Gaussian temporal profile with T0 ¼1 ns, and a transverse 5-order SG profile with
Fig. 3. Schematic diagram of the simulations.
w0 ¼2.75 mm, the incident envelope can be written as 2 " # !5 3 2 t2 1 x 5 Aðx,tÞ ¼ A0 exp 2 exp4 2 w20 2T 0 z h h ii exp io0 tikz exp idsin om t þ ax c
ð12Þ
Schematic diagram of the simulations is illustrated in Fig. 3. The SSD device generates a PM-SD beam, and then the PM-SD beam passes through a thin aperture with radius a¼2 mm. After the beam propagates through free space, it is launched into a 50 mm long nonlinear medium. The medium’s parameters are n0 ¼1.55, n2 ¼1.15 10 13 esu, and the GVD parameter is b2 ¼154 fs2/cm. A comparison of the diffraction (dashed curve) and the nonlinear propagation properties (solid curve) for the PM-SD beam and the monochromatic SG beam is shown in Fig. 4, where the initial intensity is I0 ¼5.0 109 W/cm2. The diffraction fields are used as the incident fields of the nonlinear propagation. Fig. 4(a) and (b) gives the diffraction field distributions and the output field distributions for the PM-SD beam and the monochromatic SG beam, respectively. The simulation results demonstrate that the PM-SD beam can wipe out the diffraction intensity spikes, since the diffraction field of the PM-SD beam shows a better uniformity than the monochromatic SG beam. Additionally, and most importantly, in Fig. 4(b) some modulation peaks grow from 1.2 to 4.0 approximately for the monochromatic SG beam due to serious SSSF effect. Whereas in Fig. 4(a), the output intensity peaks not even exceed 1.5 for the PM-SD beam. This indicates that the PM-SD beam can be used to suppress or delay the SSSF effect. Obviously, from Eqs. (1) and (11) we know that electric field properties of the PM-SD beam are determined by three parameters of the SSD device, namely modulation amplitude, modulation frequency, and the grating dispersion coefficient. In order to reveal the influence of the SSD device parameters on the nonlinear propagation process, we have done a series of simulations. Fig. 5 shows the time-averaged beams intensity profiles in the emergent surface of the nonlinear medium for four different modulation amplitudes [(a) d ¼5, (b) d ¼10, (c) d ¼15,
J. Deng et al. / Optics & Laser Technology 45 (2013) 56–61
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Fig. 4. Normalized transverse intensity profiles (a) PM-SD beam and (b) monochromatic SG beam. The dashed curves are the diffraction field distributions and the solid curves are the output field distributions of the nonlinear medium. The PM-SD beam’s parameters are d ¼10, vm ¼ 5 GHz, Dy/Dl ¼5000 mrad/nm, and the Fresnel number N¼ 30.
Fig. 5. The influence of E-O modulation amplitude. (a) d ¼ 5, (b) d ¼10, (c) d ¼15 and (d) d ¼ 20, the other parameters are nm ¼3 GHz and Dy/Dl ¼5000 mrad/nm.
and (d) d ¼20] with vm ¼3 GHz and Dy/Dl ¼5000 mrad/nm. It shows that the modulation peaks decrease with increasing the modulation amplitude d. For different modulation frequencies [(a) vm ¼1 GHz, (b) vm ¼3 GHz, (c) vm ¼5 GHz, and (d) vm ¼ 7 GHz] with d ¼10 and Dy/Dl ¼5000 mrad/nm, the time-averaged beams intensity profiles are shown in Fig. 6. It is found that the effect of the modulation frequency vm is similar to the modulation amplitude d, due to the bandwidth formula Dv ¼2dvm. Although the bandwidth of the beams reflects the ability to wipe out the diffraction spikes and suppress the SSSF, there are some different mechanisms of action between d and vm. The effective frequency sidebands increase with increasing the modulation amplitude d, it means that more diffraction fields are formed in the frequency
domain. According to Eq. (7), the whole diffraction field in time domain is overlapped by the fields of each sideband, thus the more sidebands, the more uniform field distributions in the time domain and the better quality of the beam. With the same number of sidebands, the frequency interval broaden with increasing modulation frequency vm, showing a more uniform field distribution by improving the smoothing efficiency of the frequency pairs (e.g., o0 om and o0 þ om). In addition, Fig. 7 shows the time-averaged beams intensity profiles in the emergent surface of the medium for four different dispersion coefficients [(a) Dy/Dl ¼0, (b) Dy/Dl ¼ 1000 mrad/nm, (c) Dy/Dl ¼3000 mrad/nm, and (d) Dy/Dl ¼5000 mrad/nm] with d ¼10 and vm ¼5 GHz. Fig. 7(a) shows that the intensity profile for
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Fig. 6. The influence of E-O modulation frequency. (a) nm ¼1 GHz, (b) nm ¼ 3 GHz, (c) nm ¼ 5 GHz and (d) nm ¼ 7 GHz, the other parameters are d ¼ 10 and Dy/Dl ¼ 5000 mrad/nm.
Fig. 7. The influence of the grating dispersion coefficient. (a) Dy/Dl ¼ 0, (b) Dy/Dl ¼ 1000 mrad/nm, (c) Dy/Dl ¼ 3000 mrad/nm, and (d) Dy/Dl ¼ 5000 mrad/nm, with the d ¼ 10 and vm ¼5 GHz.
J. Deng et al. / Optics & Laser Technology 45 (2013) 56–61
Dy/Dl ¼0 almost have the same aspect for the monochromatic SG beam [solid line in Fig. 4(b)]. The beam is just imposed a discrete spectrum by the E-O modulator and the instantaneous frequency does not vary across the beam with the absence of dispersion. As a result, the diffraction patterns of each frequency component almost locate in the same transverse spatial position. Obviously, this kind of phase modulation beams is not able to wipe out the diffraction intensity spikes, and also the SSSF cannot be delayed or suppressed. Fig. 7(b) shows that the peaks of the diffraction intensity profiles decrease dramatically at Dy/Dl ¼1000 mrad/nm. While Fig. 7(c) and (d) shows that the intensity peaks continue to decline with the increasing of the dispersion coefficient Dy/Dl by enhancing the variation rate of the instantaneous frequency across the beam. Furthermore, it can clearly found that only a few modulation ripples have grown and the intensity profiles around the beam center almost always remain stable. According to the B-T theory [21], the ripples will grow fast only when the ripples frequency is equal to or close to the fastest growing frequency. As ripples frequency of the diffraction present obvious non-uniformity, some ripples located in the gain region grow quickly, while some ripples located out the gain region, such as the ripples around the beam center, remain stable. 4. Conclusions SSD is usually used for far-field target illumination uniformity in the high-power laser system. So the PM-SD beams are inevitably affected by Fresnel diffraction and the diffraction ripples will motivate the SSSF effect when the beams pass through nonlinear media. In this paper, Fresnel diffraction and SSSF of the PM-SD beam are investigated theoretically and numerically in detail. The theoretical analysis shows that, in comparison with the monochromatic SG beam, the diffraction field of the PM-SD beam is more uniform in the Fresnel domain. The results suggested that the PM-SD beam can smooth out the diffraction intensity spikes. The numerical simulation results demonstrated that the PM-SD beam has a slower nonlinear growth than that of the monochromatic SG beam. In addition, it is known from the results that the smoothing performance of the PM-SD beam is determined by the modulation amplitude, modulation frequency, and grating dispersion coefficient. The diffraction modulation growth rates decrease with increasing the modulation amplitude and modulation frequency. And the dispersion coefficient can also be used to suppress SSSF by enhancing the variation rate of the instantaneous frequency across the beam. Thus, the PM-SD beam can be applied as an alternative scheme to improve the beam uniformity and suppress or delay the onset of SSSF in the high power solid-state laser system.
Acknowledgments This work was supported by the National Nature Science Foundation of China (Grant No. 60890202), Hunan Provincial Natural Science Foundation of China (Grant No. 12JJ7005), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20110161110012), and the
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