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Smoothing the side lobes of a focused pattern by spectral dispersion in the broadband laser
ARTICLE IN PRESS
Optik
Optics
Optik 118 (2007) 594–598 www.elsevier.de/ijleo
Smoothing the side lobes of a focused pattern by spectral dispersion in the broadband laser Runwu Penga,b,, Zhixiang Tanga,b, Yunxia Yeb, Chujun Zhaob, Shuangchun Wena, Dianyuan Fanb a
School of Computer and Communication, Hunan University, Changsha 410082, China Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, P.O. Box 800-211, Shanghai 201800, China
b
Received 22 March 2006; accepted 1 June 2006
Abstract Starting from the Huygens–Fresnel diffraction integral, the propagation equations of a broadband laser passing through a dispersive lens and a dispersive wedge are derived. Smoothing effect on the side lobes of the focused pattern is achieved as the broadband laser passes through the lens because of the spectral dispersion of the lens. By inserting a dispersive wedge behind the lens, better smoothing effect is realized because a relative position shift between focused patterns of different frequency components is generated due to the spectral dispersion of the wedge. Smoothing effect on the side lobe is obtained even with small bandwidth of the broadband laser as the wedge is used. r 2006 Elsevier GmbH. All rights reserved. Keywords: Spectral dispersion; Smoothing effect; Dispersive wedge; Broadband laser
1. Introduction Following the rapid advance of laser techniques, the bandwidth of laser beam increases due to the increase of gain bandwidth of the gain media and the decrease of the pulse duration. For example, 3.8-fs pulse with a bandwidth of 270 THz was produced from adaptive compression of a cashed hollow fiber supercontinuum [1]. The properties of such beams became an interesting subject of lots of papers in recent decades [2,3]. Taking advantage of the laser beams with some bandwidths and using optical elements and techniques, beam smoothing could be achieved in high power laser systems [4]. It is well known that the uniform intensity is required in laser Corresponding author. Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, P.O. Box 800-211, Shanghai 201800, China. E-mail address: [email protected] (R. Peng).
0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.06.001
driven fusion [5,6] and thus the smoothing by spectral dispersion (SSD) technique plays an important role in reducing the intensity nonuniformity in high power laser systems [7]. Recently, the SSD technique has been paid further attention to and has been developed in many works [8–10]. Also, some investigations found that the beam smoothing could be realized only by increasing the bandwidth of the broadband laser [11,12]. However, it is difficult to obtain laser beams with broad bandwidth in high power laser systems due to some limitations of techniques. Fortunately, the bandwidth of laser beams in high power laser systems tends to broaden continuously following the development of the laser techniques [13]. Therefore, investigations about some new methods such as the combination of the broadband laser with some other techniques seem to be necessary. It is known that, when a monochromatic wave is focused by a lens, the focused pattern appears as a series of concentric bright and dark rings around a central disk
ARTICLE IN PRESS R. Peng et al. / Optik 118 (2007) 594–598
called the Airy disk. The series of concentric bright and dark rings can be regarded as nonuniform intensity and it is a disadvantage for the applications of lasers in high power laser systems. Therefore, some ways should be adopted to eliminate the diffraction rings to achieve smoothing effect in the focused patterns. In this paper, smoothing effects brought by spectral dispersion of a dispersive lens and a dispersive wedge on side lobes of focused pattern are investigated. Firstly, the propagation equation of a broadband laser passing through a dispersive lens is derived. Intensity distributions of the side lobes of the broadband laser with different bandwidths are studied in detail on the basis of the propagation equation and some smoothing effect on the side lobes is found. Then, the case that the broadband laser passes the dispersive lens followed by a dispersive wedge is analyzed and better smoothing effect is discovered. Finally, a brief summary concludes the paper.
2. Propagation of a broadband laser passing through an apertured dispersive lens
where E 0 ðx0 ; 0; oÞ is the incident field, k ¼ 2p=l is the wave number, and a is half width of the rectangular aperture. Assume that the space and spectrum fields of the initial beam can be separated, the transverse mode of the beam is Gaussian shape and the beam waist is located at the same plane of the lens, thus E 0 ðx0 ; 0; oÞ can be written as x20 E 0 ðx0 ; 0; oÞ ¼ A0 exp 2 SðoÞ, (2) w0 where w0 is the width of the beam waist and A0 is a complex constant. The focal length in Eq. (1) is different for each frequency component of the broadband laser and is given by f ¼
n0 1 f , nðlÞ 1 0
silica, whose refractive index is given by [15] n2 ðlÞ ¼ 1 þ
(3)
where n0 and f0 are the refractive index and the length of the central wave, respectively, and n(l) is the refractive index of the frequency component with wavelength l. Assume that the dispersive lens is made of the fused
3 X i¼1
Bi , 1 l2i =l2
(4)
where B1 ¼ 0:6961663, B2 ¼ 0:4079426, B3 ¼ 0:8974794, l1 ¼ 0:0684043 mm, l2 ¼ 0:1162414 mm, l2 ¼ 0:116241 mm and l3 ¼ 9:896161 mm. Substituting Eq. (2) into Eq. (1) and performing integral calculation, we obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A0 SðoÞ exp ðikzÞ izR Eðx; z; oÞ ¼ 2 z þ izR ðz f Þ=f 2 ikx ðf þ izR Þz=f exp ½erfðwþ Þ 2z z þ izR ðz f Þ=f þ erfðw Þ, where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ik z f z izR x þ a w ¼ , 2z f izR a þ izR ðz f Þ=f erfðyÞ ¼
Consider a broadband laser passes through an apertured dispersive lens. From the Huygens–Fresnel diffraction integral, field of each frequency component is obtained as [14] Z expðikzÞ a Eðx; z; oÞ ¼ E 0 ðx; 0; oÞ ðilzÞ1=2 a ik zf 2 x0 2x0 x þ x2 dx0 , exp 2z f ð1Þ
595
2 p1=2
Z
ð5Þ
(6)
y
exp ðx2 Þ dx.
(7)
0
erf() is error function and zR is the Rayleigh length. The fields is given by Z l0 þDl 1 Eðx; zÞ ¼ Eðx; z; oÞ dl; (8) Dl l0 Dl where l0 is central wavelength and Dl is bandwidth (FWHM) of the laser. Then, the intensity is obtained as 2 Iðx; zÞ ¼ Eðx; zÞ . (9)
3. Influence of the spectral dispersion of the lens on the side lobes Assume that the spectrum profile of the laser is Gaussian distribution as 2 2 ðo o0 Þ SðoÞ ¼ exp ag , (10) Do2 pffiffiffiffiffiffiffiffiffiffiffiffi where ag ¼ 2 ln 2 and Do is the spectrum width (FWHM) of the laser beams. On the basis of the above equations, the influences of the dispersive lens on the intensities of the side lobes of the focused pattern are analyzed by following numerical calculations, where the calculation parameters are l0 ¼ 800 nm, f0 ¼ 1 m, w0 ¼ 10 mm, a ¼ 10 mm. Fig. 1 gives the intensity distributions of the side lobes in the focused patterns of the monochromatic wave and the broadband laser with bandwidths 20 and 100 nm at the focal plane of the central wave as the beams pass through the dispersive
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Intensity (a.u)
0.015 Aperture
0 nm 20 nm 100 nm
0.010
1 0 2
Laser 0.005 F1 0 40
F2
F0
Lens 60
80 100 120 140 Transverse position (µm)
160
Fig. 1. Side lobes of the monochromatic wave and the broadband laser with bandwidths 20 and 100 nm at the focal plane of the central wave.
Fig. 2. Each frequency component is focused at its own focal plane and different diffraction patterns are generated for the each frequency component at the focal plane of the central wave.
0.04
0.03 Intensity (a.u)
lens. It is seen from the figure that the intensity distribution of the side lobes in the broadband laser with bandwidth 20 nm differs only a little from that of the monochromatic wave and almost no smoothing effect is obtained. As the bandwidth increases to 100 nm, the amplitude of the side lobes reduces and some smoothing effect is achieved. The numerical results show that the difference between the third intensity peak and valley is 0.22 of that of the first as the bandwidth is 20 nm, whereas that of the third is only 0.08 of that of the first as the bandwidth is 100 nm. However, the intensity peaks and valleys are still exist obviously. Physically, the smoothing effect of the dispersive lens on side lobes results from different refractive index of dispersive media to each frequency component and thus it is known from Eq. (3) that all the components differ from each other in the focal length. As shown in Fig. 2, each frequency component is focused at its own focal plane, namely, the frequency components with l0, l1 (l1ol0) and l2 (l24l0) are focused at the corresponding focal plane F0, F1 and F2, respectively. Thus, the intensity distributions of the diffraction pattern of each frequency component differ from each other at the focal plane F0. The diffraction pattern of the frequency component with l0 is a typical focused pattern at focal plane, but those of the frequency components with l1 and l2 are not. The frequency component with l1 has been focused at its focal plane and is dispersing at F0; that with l2 has not arrived at its focal plane yet and is in course of focusing at F0. The side lobes of the frequency components with wavelength 750, 800 and 850 nm at F0 are depicted in Fig. 3, from which it is seen that, besides the patterns differ in the shape, the intensity peaks and valleys of the patterns are also not located at the same place. Therefore, the peaks of some patterns fill the valleys of others as the patterns are overlapped and thus smoothing effect is achieved.
800 nm 750 nm 850 nm
0.02
0.01
0 40
60
80
100
120
Transverse position (µm)
Fig. 3. Side lobes of the frequency components with wavelengths 750 nm, 800 nm and 850 nm at the focal plane of the central wave.
4. Smoothing effect brought by the dispersive wedge From the above discussions, we know that only some smoothing effect on side lobes is obtained as the broadband laser passes through the dispersive lens. To achieve better smoothing effect on the side lobes, a dispersive wedge is adopted and inserted following the lens. Assume that the wedge is also made of the fused silica. This, the field in Eq. (1) is written as Z exp ðikzÞ a Eðx; z; oÞ ¼ E 0 ðx; 0; oÞ ðilzÞ1=2 a ik zf 2 x0 2x0 x þ x2 exp 2z f þjðl; yÞ dx0 , ð11Þ
ARTICLE IN PRESS R. Peng et al. / Optik 118 (2007) 594–598
where jðl; yÞ is a phase perturbation brought by the dispersive wedge [16] and is given by d1 þ d2 x0 y . (12) jðl; yÞ ¼ ik½nðlÞ 1 2 In addition, d1 and d2 are widths of upside and underside of the dispersive wedge, respectively, y is the wedge, angle of the dispersive wedge. The integral calculation of Eq. (11) yields sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A0 SðoÞ exp ðikzÞ izR Eðx; z; oÞ ¼ 2 z þ izR ðz f Þ=f ik fx þ 2z½nðlÞ 1yg2 exp 2z ðz f Þ=f þ z=izR þx2 þ z½nðlÞ 1ðd 1 þ d 2 Þ ½erfðwþ Þ þ erfðw Þ,
ð13Þ
The relative position shift is more beneficial to the stagger of intensity peaks and valleys in different patterns, as shown in Fig. 5, where gives the side lobes of the frequency components with wavelength 798, 800 and 802 nm at F0 and the calculation parameters are the same as those in the Fig. 1 and the other parameters are d1 ¼ 3 mm, d2 ¼ 8 mm, y ¼ 0.1, additionally. As a result, the fill of the intensity peaks to valleys is also benefited from the relative position shift and better smoothing effect is obtained. Fig. 6 shows intensity distributions of the side lobes of the monochromatic wave and the broadband laser with bandwidths 3 nm and 6 nm passing through the dispersive lens and the dispersive wedge. From the figure it can be seen that changes of the amplitude of the side lobes in broadband laser with bandwidth 3 nm are smaller than those of monochromatic wave, and the peaks and valleys in broadband laser with bandwidth 6 nm are not obvious anymore, where better smoothing effect is achieved.
where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ik z f z izR fx þ 2z½nðlÞ 1yg w ¼ þ a . 2z f izR z þ izR ðz f Þ=f
0.020 800 nm 798 nm
0.015 Intensity (a.u)
(14) We also adopt Gaussian spectrum in Eq. (10) and then obtain the intensity distributions by substituting Eq. (13) into Eqs. (8) and (9). The focused patterns of all frequency components get a transverse position shift after the laser beams pass through the dispersive wedge. Due to different spectral dispersion of the wedge to each frequency component, position shift of each pattern differs from each other and thus a relative position shift is generated between different patterns, as shown in Fig. 4. With small wedge angle, the relative position shift between frequency components with wavelength l1 and l2 is given by Dx ¼ zy½nðl2 Þ nðl1 Þ. (15)
597
802 nm 0.010
0.005
0 -90.62
-90.60
-90.58
-90.56
-90.54
Transverse position (µm)
Fig. 5. Side lobes of the frequency components with wavelengths 798, 800 and 802 nm at the focal plane of the central wave after inserting a wedge.
0.020 Δx Lens
0 nm 3 nm 6 nm
0.015
Laser 1
Intensity (a.u)
Aperture
0.010
0.005
0 2 Wedge
0 -90.62 F1
F0
F2
Fig. 4. A relative position shift between the focused patterns of different frequency components is generated at the focal plane of the central wave by inserting a wedge behind the lens.
-90.60
-90.58
-90.56
-90.54
Transverse position (µm)
Fig. 6. Side lobes of the monochromatic wave and the broadband laser with bandwidths 3 and 6 nm at the focal plane of the central wave after inserting a wedge.
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As compared with the case that the broadband laser only passes through a dispersive lens, better smoothing effect is obtained by inserting a dispersive wedge behind the lens. Especially, the side lobes can also be well smoothed even with small bandwidth, which has important significance in high power laser systems. Due to limitations of narrow gain bandwidth of the gain media and low efficiency of frequency multiplication, only several nm bandwidths is obtained after the amplifications and frequency multiplication of high power lasers. Thus, achieving well smoothing effect with small bandwidths is significant to smoothing the focused pattern in high power laser systems.
5. Conclusions In this paper, smoothing effects generated by spectral dispersion of the dispersive lens and the dispersive wedge on side lobes of focused pattern are investigated in detail. The numerical results demonstrate that the side lobes of the focused patterns can be smoothed by means of the spectral dispersion of the lens and the wedge. When the broadband laser passes through the dispersive lens, different patterns are generated for each frequency component at the focal plane of the central wave because of the spectral dispersion, and the overlapping of those patterns results in some smoothing effect on the side lobes of the focused patterns. Better smoothing effect is achieved by inserting a dispersive wedge behind the lens, where a relative position shift of different patterns is brought by the spectral dispersion of the wedge. Especially, well smoothing effect is achieved even with small bandwidths by using the dispersive wedge, which is a useful advantage in high power laser systems.
Acknowledgements This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 10576012 and 60538010), the National High Technology Research and Development Program of China (Grant No. 2004AA84ts12), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20040532005).
References [1] B. Schenkel, J. Biegert, U. Keller, Generation of 3.8-fs pulses from adaptive compression of a cashed hollow fiber supercontinuum, Opt. Lett. 28 (2003) 1987–1989. [2] M.A. Porras, Nonsinusoidal few-cycle pulsed light beams in free space, J. Opt. Soc. Am. B 16 (1999) 1468–1474. [3] S. Feng, H.G. Winful, Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses, Phys. Rev. E 61 (2000) 862–873. [4] S. Skupsky, R.W. Short, T. Kessler, R.S. Craxton, S. Letzring, J.M. Soures, Improved laser-beam uniformity using the angular dispersion of frequency-modulated light, J. Appl. Phys. 66 (1989) 3456–3462. [5] K.A. Bruecker, S. Jorna, Laser-driven fusion, Rev. Mod. Phys. 46 (1974) 325–367. [6] S. Skupsky, K. Lee, Uniformity of energy deposition for laser driven fusion, J Appl. Phys. 54 (1983) 3662–3671. [7] Two-dimensional SSD on OMEGA, Laboratory for Laser Energetics Review 69 (1996) 1–10. [8] J.E. Rothenberg, Comparison of beam-smoothing methods for direct-drive inertial confinement fusion, J. Opt. Soc. Am. B 14 (1997) 1664–1671. [9] G. Miyaji, N. Miyanaga, S. Urushihara, K. Suzuki, S. Matsuoka, M. Nakatsuka, A. Morimoto, T. Kobayashi, Three-directional spectral dispersion for smoothing of a laser irradiance profile, Opt. Lett. 27 (2002) 725–727. [10] Q.F. Tan, Q.S. He, Y.B. Yan, G.F. Jin, D.Y. Xu, Spatialfrequency spectrum analysis of the performance of diffractive optical element for beam smoothing, Optik 113 (2002) 163–166. [11] R.W. Peng, Y.X. Ye, Z.X. Tang, D.Y. Fan, Transverse intensity distributions of broadband laser modulated by a hard-edged aperture, J. Opt. Soc. Am. A 22 (2005) 1903–1908. [12] R.W. Peng, Z.X. Tang, Y.X. Ye, S.C. Wen, D.Y. Fan, Effects of the frequency chirp on the fields of a chirped Gaussian pulse passing through a hard-edged aperture, Opt. Common. 259 (2006) 474–478. [13] R.A. Haas, P.S. Banks, Generation of high intensity broad-bandwidth light for inertial confinement fusion, Opt. Commun. 107 (1994) 265–270. [14] M. Born, E. Wolf, Principles of Optics, seventh ed., Cambridge University, Cambridge, 1999, pp. 484–499. [15] I.H. Malitson, Interspecimen comparison of the refractive index of fused silica, J. Opt. Soc. Am. 55 (1965) 1205–1209. [16] A.E. Siegman, Lasers, University Science Books, California, 1986, pp. 706–711.
Optik
Optics
Optik 118 (2007) 594–598 www.elsevier.de/ijleo
Smoothing the side lobes of a focused pattern by spectral dispersion in the broadband laser Runwu Penga,b,, Zhixiang Tanga,b, Yunxia Yeb, Chujun Zhaob, Shuangchun Wena, Dianyuan Fanb a
School of Computer and Communication, Hunan University, Changsha 410082, China Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, P.O. Box 800-211, Shanghai 201800, China
b
Received 22 March 2006; accepted 1 June 2006
Abstract Starting from the Huygens–Fresnel diffraction integral, the propagation equations of a broadband laser passing through a dispersive lens and a dispersive wedge are derived. Smoothing effect on the side lobes of the focused pattern is achieved as the broadband laser passes through the lens because of the spectral dispersion of the lens. By inserting a dispersive wedge behind the lens, better smoothing effect is realized because a relative position shift between focused patterns of different frequency components is generated due to the spectral dispersion of the wedge. Smoothing effect on the side lobe is obtained even with small bandwidth of the broadband laser as the wedge is used. r 2006 Elsevier GmbH. All rights reserved. Keywords: Spectral dispersion; Smoothing effect; Dispersive wedge; Broadband laser
1. Introduction Following the rapid advance of laser techniques, the bandwidth of laser beam increases due to the increase of gain bandwidth of the gain media and the decrease of the pulse duration. For example, 3.8-fs pulse with a bandwidth of 270 THz was produced from adaptive compression of a cashed hollow fiber supercontinuum [1]. The properties of such beams became an interesting subject of lots of papers in recent decades [2,3]. Taking advantage of the laser beams with some bandwidths and using optical elements and techniques, beam smoothing could be achieved in high power laser systems [4]. It is well known that the uniform intensity is required in laser Corresponding author. Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, P.O. Box 800-211, Shanghai 201800, China. E-mail address: [email protected] (R. Peng).
0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.06.001
driven fusion [5,6] and thus the smoothing by spectral dispersion (SSD) technique plays an important role in reducing the intensity nonuniformity in high power laser systems [7]. Recently, the SSD technique has been paid further attention to and has been developed in many works [8–10]. Also, some investigations found that the beam smoothing could be realized only by increasing the bandwidth of the broadband laser [11,12]. However, it is difficult to obtain laser beams with broad bandwidth in high power laser systems due to some limitations of techniques. Fortunately, the bandwidth of laser beams in high power laser systems tends to broaden continuously following the development of the laser techniques [13]. Therefore, investigations about some new methods such as the combination of the broadband laser with some other techniques seem to be necessary. It is known that, when a monochromatic wave is focused by a lens, the focused pattern appears as a series of concentric bright and dark rings around a central disk
ARTICLE IN PRESS R. Peng et al. / Optik 118 (2007) 594–598
called the Airy disk. The series of concentric bright and dark rings can be regarded as nonuniform intensity and it is a disadvantage for the applications of lasers in high power laser systems. Therefore, some ways should be adopted to eliminate the diffraction rings to achieve smoothing effect in the focused patterns. In this paper, smoothing effects brought by spectral dispersion of a dispersive lens and a dispersive wedge on side lobes of focused pattern are investigated. Firstly, the propagation equation of a broadband laser passing through a dispersive lens is derived. Intensity distributions of the side lobes of the broadband laser with different bandwidths are studied in detail on the basis of the propagation equation and some smoothing effect on the side lobes is found. Then, the case that the broadband laser passes the dispersive lens followed by a dispersive wedge is analyzed and better smoothing effect is discovered. Finally, a brief summary concludes the paper.
2. Propagation of a broadband laser passing through an apertured dispersive lens
where E 0 ðx0 ; 0; oÞ is the incident field, k ¼ 2p=l is the wave number, and a is half width of the rectangular aperture. Assume that the space and spectrum fields of the initial beam can be separated, the transverse mode of the beam is Gaussian shape and the beam waist is located at the same plane of the lens, thus E 0 ðx0 ; 0; oÞ can be written as x20 E 0 ðx0 ; 0; oÞ ¼ A0 exp 2 SðoÞ, (2) w0 where w0 is the width of the beam waist and A0 is a complex constant. The focal length in Eq. (1) is different for each frequency component of the broadband laser and is given by f ¼
n0 1 f , nðlÞ 1 0
silica, whose refractive index is given by [15] n2 ðlÞ ¼ 1 þ
(3)
where n0 and f0 are the refractive index and the length of the central wave, respectively, and n(l) is the refractive index of the frequency component with wavelength l. Assume that the dispersive lens is made of the fused
3 X i¼1
Bi , 1 l2i =l2
(4)
where B1 ¼ 0:6961663, B2 ¼ 0:4079426, B3 ¼ 0:8974794, l1 ¼ 0:0684043 mm, l2 ¼ 0:1162414 mm, l2 ¼ 0:116241 mm and l3 ¼ 9:896161 mm. Substituting Eq. (2) into Eq. (1) and performing integral calculation, we obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A0 SðoÞ exp ðikzÞ izR Eðx; z; oÞ ¼ 2 z þ izR ðz f Þ=f 2 ikx ðf þ izR Þz=f exp ½erfðwþ Þ 2z z þ izR ðz f Þ=f þ erfðw Þ, where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ik z f z izR x þ a w ¼ , 2z f izR a þ izR ðz f Þ=f erfðyÞ ¼
Consider a broadband laser passes through an apertured dispersive lens. From the Huygens–Fresnel diffraction integral, field of each frequency component is obtained as [14] Z expðikzÞ a Eðx; z; oÞ ¼ E 0 ðx; 0; oÞ ðilzÞ1=2 a ik zf 2 x0 2x0 x þ x2 dx0 , exp 2z f ð1Þ
595
2 p1=2
Z
ð5Þ
(6)
y
exp ðx2 Þ dx.
(7)
0
erf() is error function and zR is the Rayleigh length. The fields is given by Z l0 þDl 1 Eðx; zÞ ¼ Eðx; z; oÞ dl; (8) Dl l0 Dl where l0 is central wavelength and Dl is bandwidth (FWHM) of the laser. Then, the intensity is obtained as 2 Iðx; zÞ ¼ Eðx; zÞ . (9)
3. Influence of the spectral dispersion of the lens on the side lobes Assume that the spectrum profile of the laser is Gaussian distribution as 2 2 ðo o0 Þ SðoÞ ¼ exp ag , (10) Do2 pffiffiffiffiffiffiffiffiffiffiffiffi where ag ¼ 2 ln 2 and Do is the spectrum width (FWHM) of the laser beams. On the basis of the above equations, the influences of the dispersive lens on the intensities of the side lobes of the focused pattern are analyzed by following numerical calculations, where the calculation parameters are l0 ¼ 800 nm, f0 ¼ 1 m, w0 ¼ 10 mm, a ¼ 10 mm. Fig. 1 gives the intensity distributions of the side lobes in the focused patterns of the monochromatic wave and the broadband laser with bandwidths 20 and 100 nm at the focal plane of the central wave as the beams pass through the dispersive
ARTICLE IN PRESS 596
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Intensity (a.u)
0.015 Aperture
0 nm 20 nm 100 nm
0.010
1 0 2
Laser 0.005 F1 0 40
F2
F0
Lens 60
80 100 120 140 Transverse position (µm)
160
Fig. 1. Side lobes of the monochromatic wave and the broadband laser with bandwidths 20 and 100 nm at the focal plane of the central wave.
Fig. 2. Each frequency component is focused at its own focal plane and different diffraction patterns are generated for the each frequency component at the focal plane of the central wave.
0.04
0.03 Intensity (a.u)
lens. It is seen from the figure that the intensity distribution of the side lobes in the broadband laser with bandwidth 20 nm differs only a little from that of the monochromatic wave and almost no smoothing effect is obtained. As the bandwidth increases to 100 nm, the amplitude of the side lobes reduces and some smoothing effect is achieved. The numerical results show that the difference between the third intensity peak and valley is 0.22 of that of the first as the bandwidth is 20 nm, whereas that of the third is only 0.08 of that of the first as the bandwidth is 100 nm. However, the intensity peaks and valleys are still exist obviously. Physically, the smoothing effect of the dispersive lens on side lobes results from different refractive index of dispersive media to each frequency component and thus it is known from Eq. (3) that all the components differ from each other in the focal length. As shown in Fig. 2, each frequency component is focused at its own focal plane, namely, the frequency components with l0, l1 (l1ol0) and l2 (l24l0) are focused at the corresponding focal plane F0, F1 and F2, respectively. Thus, the intensity distributions of the diffraction pattern of each frequency component differ from each other at the focal plane F0. The diffraction pattern of the frequency component with l0 is a typical focused pattern at focal plane, but those of the frequency components with l1 and l2 are not. The frequency component with l1 has been focused at its focal plane and is dispersing at F0; that with l2 has not arrived at its focal plane yet and is in course of focusing at F0. The side lobes of the frequency components with wavelength 750, 800 and 850 nm at F0 are depicted in Fig. 3, from which it is seen that, besides the patterns differ in the shape, the intensity peaks and valleys of the patterns are also not located at the same place. Therefore, the peaks of some patterns fill the valleys of others as the patterns are overlapped and thus smoothing effect is achieved.
800 nm 750 nm 850 nm
0.02
0.01
0 40
60
80
100
120
Transverse position (µm)
Fig. 3. Side lobes of the frequency components with wavelengths 750 nm, 800 nm and 850 nm at the focal plane of the central wave.
4. Smoothing effect brought by the dispersive wedge From the above discussions, we know that only some smoothing effect on side lobes is obtained as the broadband laser passes through the dispersive lens. To achieve better smoothing effect on the side lobes, a dispersive wedge is adopted and inserted following the lens. Assume that the wedge is also made of the fused silica. This, the field in Eq. (1) is written as Z exp ðikzÞ a Eðx; z; oÞ ¼ E 0 ðx; 0; oÞ ðilzÞ1=2 a ik zf 2 x0 2x0 x þ x2 exp 2z f þjðl; yÞ dx0 , ð11Þ
ARTICLE IN PRESS R. Peng et al. / Optik 118 (2007) 594–598
where jðl; yÞ is a phase perturbation brought by the dispersive wedge [16] and is given by d1 þ d2 x0 y . (12) jðl; yÞ ¼ ik½nðlÞ 1 2 In addition, d1 and d2 are widths of upside and underside of the dispersive wedge, respectively, y is the wedge, angle of the dispersive wedge. The integral calculation of Eq. (11) yields sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A0 SðoÞ exp ðikzÞ izR Eðx; z; oÞ ¼ 2 z þ izR ðz f Þ=f ik fx þ 2z½nðlÞ 1yg2 exp 2z ðz f Þ=f þ z=izR þx2 þ z½nðlÞ 1ðd 1 þ d 2 Þ ½erfðwþ Þ þ erfðw Þ,
ð13Þ
The relative position shift is more beneficial to the stagger of intensity peaks and valleys in different patterns, as shown in Fig. 5, where gives the side lobes of the frequency components with wavelength 798, 800 and 802 nm at F0 and the calculation parameters are the same as those in the Fig. 1 and the other parameters are d1 ¼ 3 mm, d2 ¼ 8 mm, y ¼ 0.1, additionally. As a result, the fill of the intensity peaks to valleys is also benefited from the relative position shift and better smoothing effect is obtained. Fig. 6 shows intensity distributions of the side lobes of the monochromatic wave and the broadband laser with bandwidths 3 nm and 6 nm passing through the dispersive lens and the dispersive wedge. From the figure it can be seen that changes of the amplitude of the side lobes in broadband laser with bandwidth 3 nm are smaller than those of monochromatic wave, and the peaks and valleys in broadband laser with bandwidth 6 nm are not obvious anymore, where better smoothing effect is achieved.
where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ik z f z izR fx þ 2z½nðlÞ 1yg w ¼ þ a . 2z f izR z þ izR ðz f Þ=f
0.020 800 nm 798 nm
0.015 Intensity (a.u)
(14) We also adopt Gaussian spectrum in Eq. (10) and then obtain the intensity distributions by substituting Eq. (13) into Eqs. (8) and (9). The focused patterns of all frequency components get a transverse position shift after the laser beams pass through the dispersive wedge. Due to different spectral dispersion of the wedge to each frequency component, position shift of each pattern differs from each other and thus a relative position shift is generated between different patterns, as shown in Fig. 4. With small wedge angle, the relative position shift between frequency components with wavelength l1 and l2 is given by Dx ¼ zy½nðl2 Þ nðl1 Þ. (15)
597
802 nm 0.010
0.005
0 -90.62
-90.60
-90.58
-90.56
-90.54
Transverse position (µm)
Fig. 5. Side lobes of the frequency components with wavelengths 798, 800 and 802 nm at the focal plane of the central wave after inserting a wedge.
0.020 Δx Lens
0 nm 3 nm 6 nm
0.015
Laser 1
Intensity (a.u)
Aperture
0.010
0.005
0 2 Wedge
0 -90.62 F1
F0
F2
Fig. 4. A relative position shift between the focused patterns of different frequency components is generated at the focal plane of the central wave by inserting a wedge behind the lens.
-90.60
-90.58
-90.56
-90.54
Transverse position (µm)
Fig. 6. Side lobes of the monochromatic wave and the broadband laser with bandwidths 3 and 6 nm at the focal plane of the central wave after inserting a wedge.
ARTICLE IN PRESS 598
R. Peng et al. / Optik 118 (2007) 594–598
As compared with the case that the broadband laser only passes through a dispersive lens, better smoothing effect is obtained by inserting a dispersive wedge behind the lens. Especially, the side lobes can also be well smoothed even with small bandwidth, which has important significance in high power laser systems. Due to limitations of narrow gain bandwidth of the gain media and low efficiency of frequency multiplication, only several nm bandwidths is obtained after the amplifications and frequency multiplication of high power lasers. Thus, achieving well smoothing effect with small bandwidths is significant to smoothing the focused pattern in high power laser systems.
5. Conclusions In this paper, smoothing effects generated by spectral dispersion of the dispersive lens and the dispersive wedge on side lobes of focused pattern are investigated in detail. The numerical results demonstrate that the side lobes of the focused patterns can be smoothed by means of the spectral dispersion of the lens and the wedge. When the broadband laser passes through the dispersive lens, different patterns are generated for each frequency component at the focal plane of the central wave because of the spectral dispersion, and the overlapping of those patterns results in some smoothing effect on the side lobes of the focused patterns. Better smoothing effect is achieved by inserting a dispersive wedge behind the lens, where a relative position shift of different patterns is brought by the spectral dispersion of the wedge. Especially, well smoothing effect is achieved even with small bandwidths by using the dispersive wedge, which is a useful advantage in high power laser systems.
Acknowledgements This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 10576012 and 60538010), the National High Technology Research and Development Program of China (Grant No. 2004AA84ts12), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20040532005).
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