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Design and properties analysis of total internal reflection gratings for pulse compressor at 1053 nm
Current Applied Physics 11 (2011) 21e27
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Current Applied Physics journal homepage: www.elsevier.com/locate/cap
Design and properties analysis of total internal reflection gratings for pulse compressor at 1053 nm Qunyu Bi*, Jiangjun Zheng, Ailin Guo, Meizhi Sun, Jianpeng Wang, Fuling Zhang, Qingwei Yang, Xinglong Xie, Zunqi Lin Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Science, No.390 Qinghe Road, Shanghai 201800, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 6 January 2010 Received in revised form 15 April 2010 Accepted 8 June 2010 Available online 14 June 2010
High-efficiency compression gratings based on total internal reflection (TIR) are promising alternatives of compressor gratings because of their high diffraction efficiency, potential high damage resistant ability, and compact structure. Dependence of the 1 order diffraction efficiencies on grating parameters is analyzed for TE- and TM-polarized incident light of 1053 nm at Littrow angle, which is calculated by using the rigorous coupled-wave analysis. A more intuitional view on the relation is offered through three-dimensional slicing figures instead of two-dimensional ones. The performances of high-efficiency gratings are compared and regarded as criteria for further choices, including spectral bandwidth, angle bandwidth, dispersion, and intensity distribution. For TE- and TM-irradiations, similar spectral bandwidth and angle bandwidth can be achieved by different grating parameters. However, the computer simulation result on the intensity distributions of the two polarized waves shows that such design should be used under the illumination of TE-polarized wave for lower intensity enhancement ratio, which is an important factor related to the gratings’ damage threshold. Ó 2010 Elsevier B.V. All rights reserved.
Keywords: Total internal reflection gratings Grating performance analysis Pulse compressor
1. Introduction The advent of the chirped-pulse amplification (CPA) technique has created a revolution in the production and use of high-power picosecond and femtosecond laser systems, which are essential tools for the research of lightematter interaction and inertial confinement fusion [1e7]. Although the stretch and compression stages can be implemented through fibers [1,8,9], most high-energy systems are using diffraction gratings to stretch or recompress the pulses so far. Various gratings are designed and fabricated to achieve high diffraction efficiency for the required order (often the -1st), high damage threshold, and large size with high surface flatness. Up to now, gratings working in the high-energy laser facilities are mainly Au-coated gratings [10,11] and multilayer dielectric gratings [12e15]. The two sorts of gratings fielded in CPA systems had obtained satisfied results to some extent [16,17]. Because the optical damage threshold of dielectric materials is considerably above that of metals, more attentions are paid to the multilayer
* Corresponding author. E-mail address: [email protected] (Q. Bi). 1567-1739/$ e see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cap.2010.06.005
dielectric gratings [12e15,18e20]. For example, at the wavelength of 1053 nm, the damage thresholds of gold-coated master gratings are e1 J/cm2 for nanosecond pulses and less than 0.4 J/cm2 for subpicosencond pulses [10,11,21]. For fused silica, the values can be up to 20 J/cm2 for nanosecond pulses, and over 2 J/cm2 for 400-fs pulses [12,22]. However, in the production process, disadvantages of multilayer gratings appear. Firstly, because they are formed by placing a grating structure on top of a multilayer structure of alternating index dielectric materials, such as fused silica and hafnium oxides, it is impossible to completely control the presence of mechanical constraints at the interfaces, due to different mechanical properties and the distortion of the spatial qualities of the laser beam. Secondly, the introduction of other oxides frequently used, such as HfO2, Al2O3, and TiO2, tends to reduce the damage threshold of the mirror, which might be applicable to multilayer gratings [23e26]. In 2004, Marciante et al. reported a new class of gratings based on the phenomenon of total internal reflection (TIR) regardless of the grating tooth shape. More important, they could be fabricated from a single dielectric material, requiring no metallic or layers, which could avoid the obstacles that multilayer dielectric gratings must face [27]. Later, Liu et al. firstly attempted to directly etch the gratings into the bottom side of a prism to form a compact pulse
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plots instead of the two-dimensional ones, and by analyzing on the performances of different grating parameters in detail, including the grating period, etching depth, and duty cycle. Two different points of view were obtained. One was that similar spectral bandwidth and angle bandwidth could be achieved by different grating parameters for both TE- and TM-waves; the other one was that TIR gratings were more favorably used under the illumination of TE-waves for lower intensity enhancement ratio.
2. Design and properties analysis
Fig. 1. Schematic of a TIR grating. q is the incidence angle, qi is the ist order diffraction angle, D is the depth of the substrate, d is the depth of the grating, p is the grating period, f is the duty cycle, and ni (i ¼ 1, 2) are refractive indices.
compressor [28]. However, there is little information about global extrema of diffraction efficiency and it is not conducive to the applications of the TIR gratings in high-energy laser systems. In this paper, we extend this work by offering a more intuitional view on the 1 order diffraction through three-dimensional slicing
Generally, the period of gratings used as compressors in highpower laser systems is close to the light wavelength in order to obtain large dispersion. So a vector theory has to be employed for accurate gratings designing. The rigorous coupled-wave analysis, which is one of the most well-developed and widely used tool for analysis of the gratings [29e31] is used in this paper. For applications in high-power lasers, a small decrease in grating efficiency would result in a large loss in the energy output, because the beam often diffracts from the grating four times. Therefore, high diffraction efficiency is a basic requirement and should be considered firstly. A scheme of a TIR grating is shown in Fig. 1. Because the diffraction efficiency is independent on the grating tooth shape [27], the surface-relief rectangular grating as a mode for numerical simulations is taken for simplicity. When
Fig. 2. Rigorous calculated -1st reflected diffraction efficiency of a TIR grating as a function of depth and period with different refractive indices: (a,c) n1 ¼ 1.3; (b,d) n1 ¼ 1.7. The angle of incidence is the Littrow angle varied with the refractive index and the period. The duty cycle is 0.5.
Q. Bi et al. / Current Applied Physics 11 (2011) 21e27
Fig. 3. Rigorous calculated -1st reflected diffraction efficiency of a TIR grating as a function of depth, period, and duty cycle. The angle of incidence is the Littrow angle varied with the period, the refractive index is 1.46, and the periods of the three slices from right to left are 380, 460, and 520 nm, respectively.
light of wavelength l incidents on the grating surface with Littrow angle, it will be diffracted back toward the direction from where it comes. When refractive index n1 of the grating is higher than refractive index n2 of the surrounding medium, high-efficiency diffraction is possible to be attained if the period (p) of gratings satisfies
n1 >
l 2p
> n2 :
(1)
Therefore, for specific wavelength and grating material, the period of gratings should be in a certain region. In our cases, the incident wavelength is 1053 nm and n2 is 1.0 (air), which means the longest available period is 527 nm and the shortest one can be calculated by l/(2n1). It is obvious that higher n1 can bring a larger available period region of high diffraction efficiency and helpful to reduce the etching depth. Fig. 2 shows the -1st diffraction efficiency as a function of grating depth and period for
23
TE- and TM-waves with n1 ¼ 1.3 and 1.7, where duty cycle is 0.5. It should be pointed out that the -1st diffraction efficiency depends on the grating parameters and incident ways considerably, including the refractive indices of boundary materials, period (p), etching depth (d), duty cycle (f), and incident angles (polar angle of q and azimuthal anlge of j [32], which is set to be zero in our case). Therefore, three-dimensional plots should be adopted to describe the relation between the diffraction efficiency and the grating parameters at Littrow angle. Because we are more interested in the gratings used in the high-energy laser systems and the fused silica owns a high damage threshold (about 2 J/cm2 for normal incidence 1053-nm radiation and 600 shots of 0.3-ps pulses with 500-mm diameter spots [33]), the refractive index n1 is directly set to 1.46, ignoring other possible influences of the refractive index n1 on the design and performances of gratings. As mentioned above, in order to clearly describe the relation between the diffraction efficiency and the constructive parameters of gratings (p, d, and f), Figs. 3(a) and 4(a) show the relation for TE-wave in two slicing ways, one is under specific wavelengths of 380, 460, and 520 nm, and the other one is under duty cycles of 0.1, 0.3, 0.5, 0.7 and 0.9. Similar analyses are undertaken for TM polarization and described in Figs. 3(b) and 4(b), respectively. It can be seen that there are numerous sets of grating parameters that can achieve diffraction efficiency over 0.95, or even 0.99 for both TE- and TM-waves, which means that different constructive parameters can exhibit the same high-efficiency. However, other aspects, such as the fabrication difficulty, intensity enhancement, spectral dependence, and incident angle dependence, which are also important criteria in design consideration, show quite different performances. In the view of fabrication process, the regions close to the left slices in Fig. 3 and the top and down ones in Fig. 4(a) and (b) are the last choices, because it is more difficult to fabricate gratings with deep groove and narrow duty cycle. In addition, the tolerance of fabrication error will vary with the designed parameters and the fabrication altitude can be estimated by the diffraction efficiency figures. For example, as shown in Fig. 4(c), a TIR grating with the depth of 393 nm can offer the latitude of period from 444 to 514 nm with diffraction efficiency over 0.99. However, tens-nm fabrication tolerance is limited and it is still far from practical implementation, especially for the fabrication of large gratings. So the improvement on the fabrication tolerance by selecting constructive parameters is quite limited and we pay more attention to the dependences of performances of gratings on the designed grating parameters in the following.
2.1. Spectral dependence, angle dependence, and dispersion Spectral and angle dependences of diffraction efficiency are of particular concern for the applications in pulse compression. In most cases, the relations between spectral or incident angle and diffraction efficiency are described only for a single grating structure, ignoring an important fact that they will vary with the grating parameters(p, d, and f). Because the deep groove depth is difficult to be fabricated, we tend to select the grating parameters with etching depth less than 1 mm, corresponding to the part contoured by the red curves in Fig. 4(c) and (d) for TE- and TM-waves. Their corresponding bandwidths are calculated and described in Fig. 5(a) and (b), respectively. As shown in Fig. 5(a), for TE-polarized incident light and duty cycle of 0.6, gratings with different depths and periods exhibit different spectral bandwidths, varying from 28 nm or less (within the range from 1040 to 1068 nm) to 51 nm(within the range from 1030 to 1081 nm). For
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Fig. 4. Rigorous calculated -1st reflected diffraction efficiency of a TIR grating as a function of depth, period, and duty cycle. The angle of incidence is the Littrow angle varied with the period, the refractive index is 1.46, and the duty cycle of the five slices from up to down are 0.9, 0.7, 0.5, 0.3, and 0.1, respectively; (c) and (d) are corresponding contours with duty cycle of 0.6 for TE and 0.5 for TM.
TM-polarized incident light and duty cycle of 0.5, the largest bandwidth with diffraction efficiency higher than 0.95 we can find is 52 nm (within the range of 1029e1081 nm) and the grating profile is 480-nm depth, 470-nm period, and duty cycle of 0.5. The results were chosen from at least fifty grating parameters and better solution might be found but the improvement would be slight. The dependence of the diffraction efficiency on the incident angle is also varied with grating parameters, as shown in Fig. 5 (c) and (d). Every grating exhibits the maximum diffraction efficiency at its corresponding Littrow angle under the wavelength of 1053 nm, which can be calculated through sin1 ð1053=2pn1 Þ. Comparing Fig. 5(a) with (c), and Fig. 5(b) with (d), we can see that the gratings with large spectral bandwidth may own a narrow angle bandwidth. This was also observed under other duty cycles, but it was not mean that the gratings with the largest spectral bandwidth had the smallest angle bandwidth.
In addition, the dispersion (K) is also an essential factor in the design of the gratings. For the 1 order under Littrow incidence, it can be calculated according to [27].
1 K ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: l2 p2 4n 2
(2)
1
For the specific wavelength of 1053 nm and n1 of 1.46, the dispersion decreases sharply from 17.73 to 5.78 mrad/nm when the period increases from 365 to 400 nm. However, it decreases slowly from 5.78 to 2.60 mrad/nm in the range from 400- to 527-nm period. The dispersion is irrelevant with the polarization states of illumination light. 2.2. Intensity distribution Laser induced damage threshold is one of the most important factors that influence the application of grating in high-energy
Q. Bi et al. / Current Applied Physics 11 (2011) 21e27
a
1
1 p=400 d=584 p=440 d=416 p=460 d=390 p=490 d=376
0.97 0.96 0.95 0.94 0.93
0.98 0.97 0.96 0.95 0.94 0.93
0.92
0.92
0.91
0.91 1030
1040
1050
1060
1070
1080
1090
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1100
Wavelength (nm)
0.97 0.96 0.95 0.94 0.93
1060
1070
3
4
d TM polarization
0.95 0.94 0.93
0.91 1050
2
0.96
0.92
1040
1
0.97
0.91 1030
0
0.98
0.92
0.9 1020
-1
0.99
Diffraction efficiency
Diffraction efficiency
0.98
-2
1 p=410 d=712 p=430 d=566 p=470 d=480 p=500 d=546
0.99
-3
Relative angle of incidence (degrees from Littrow)
b TM polarization
1
TE polarization
0.99
Diffraction efficiency
Diffraction efficiency
0.98
0.9 1020
c
TE polarization
0.99
25
1080
1090
1100
Wavelength (nm)
0.9 -4
-3
-2
-1
0
1
2
3
4
Relative angle of incidence (degrees from Littrow)
Fig. 5. Reflected diffraction efficiency of the 1 order as a function of wavelengths and incident angles under different grating parameters for TE (a, c) and TM (b, d). The angle of incidence is the Littrow angle under the central wavelength of 1053 nm, the groove duty cycles are 0.6 for (a, c), 0.5 for (b, d), and the refractive index is 1.46.
laser systems. Neauport et al. claimed and experimentally proved that the ratio of the damage threshold of two gratings was equal to the inverted ratio of their field enhancement [34]. Therefore, further insight into the near-field electric distribution is promising to improve the damage threshold of the gratings. As the spectral and angle bandwidth, the electric field enhancement ratio is also related to the grating parameters (p, d, and f) and illumination ways (polarization state of light and incident angle). As we know, there are no analytic expressions that can describe the relation between the ratio and its corresponding grating parameters. Therefore, computer simulation of the near-field optical distribution is essential in estimating the laser induced damage threshold of the gratings. The simulation results of TEand TM- radiation are different and we will discuss them separately. For TE-waves, the grating parameters that can achieve diffraction efficiency over 0.99 under different duty cycles are sampled and their corresponding maximum intensity enhancement ratios are calculated and described in Fig. 6(b). For example, the line of f ¼ 0.6 indicates the maximum intensity enhancement ratios of the sampled gratings parameters in Fig. 4 (c), marked by blue asterisks. The grating period that may attain the intensity enhancement ratio less than six is in the range of 380e440 nm, and the duty cycle should be selected from 0.6 to
0.8. Another point we should mention is that for the duty cycles of 0.8, 0.7, and 0.6, they all have a peak intensity enhancement ratio, which corresponds to the middle point in the vertical part of the figure describing the relation of efficiency and grating parameters. For instance, the grating parameters that fulfill the maximum intensity enhancement ratio of 16.96 presented by the line of f ¼ 0.6 in Fig. 6(b) are 517 nm(p) and 560 nm(d), which is the middle point of the vertical part of Fig. 4(c). Therefore, we should avoid selecting these grating parameters. Fig. 6(a) shows the simulation intensity distribution of a grating at Littrow angle of 55.04 . The period, depth, and duty cycle of the grating are 440 nm, 414 nm, and 0.6, respectively, which performs the spectral bandwidth of 47 nm (efficiency0.95), angular bandwidth of w4 (efficiency0.95), dispersion of 3.97 mrad/nm, and intensity enhancement ratio of 5.97. The maximum intensities usually locate in the gratings’ center or the ridge opposite to the incident direction. For the simulation of intensity distribution under TMpolarized illumination, there is a convergence problem for the rigorous coupled-wave analysis, although quick convergence is obtained in calculating diffraction efficiencies by reformulating the eigenproblem of the coupled-wave method [35]. Fig. 7(b) describes the dependence of the maximum intensity enhancement ratio on the number of space harmonics under three
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Fig. 6. (a) Intensity distribution of internal electric field of a TIR grating for the TEpolarized incident at Littrow angle (b) Maximum Intensity enhancement ratio under different cycle duties as function of depth and period with the diffraction efficiency higher than 0.999 and etching depth less than a micron.
Fig. 7. (a) Intensity distribution of internal electric field of a TIR grating for the TMpolarized incident at Littrow angle. (b) Maximum intensity enhancement ratio dependence on the number of harmonics under three gratings.
3. Conclusion
different grating parameters. It is obvious that although the number of space harmonics is as high as 1001, the maximum intensity ratio is still unstable. It also occurs for other grating parameters under illumination of TM-waves. Further increase of the number of space harmonics will easily lead to exceed the capacity of our personal computer’s memory (1 GB). Fig. 7(a) shows 1401 space harmonics result of the simulated intensity distribution of the grating denoted by the blue line in Fig. 7(b). Its maximum intensity ratio is 22.2, which is different from the result of 101 space harmonics (24.3). This might be caused by the existence of the evanescent waves or the different boundary condition with TE. More detailed discussion on this may be beyond the scope of this article. Therefore, considering the high intensity enhancement ratio and instability of simulation result, TIR gratings are more suitable for TE-illumination.
A high-efficiency diffraction grating used for pulse compressor at 1053 nm is designed. The grating parameters that can achieve high diffraction efficiency are calculated by the rigorous coupledwave analysis method and the results with etching depth less than 1 mm are described in two three-dimensional slicing plots, because the efficiency is a function of three grating parameters. There are numerous sets of grating parameters (p, d, and f) that can achieve high diffraction efficiency and other criteria should be considered for further optimization of gratings. Gratings’ performances on spectral and angle bandwidths, dispersion, and intensity distribution are all related to the grating parameters and become the criteria for grating optimization except high diffraction efficiency. Different grating parameters with diffraction efficiency higher than 0.999 are sampled and their corresponding performances are compared. The results show that, for TE- and TM-irradiation, they both own grating parameters that can achieve the similar maximum spectral bandwidths and angle bandwidths. However, the results of the numerical simulation on intensity distribution of
Q. Bi et al. / Current Applied Physics 11 (2011) 21e27
the two waves are quite different. For the simulation of TM-waves, there is a convergence problem and it displays some fluctuations in intensity distribution result. But the fluctuation becomes small enough and reliable to some extent with the increase of harmonic waves number. Because the maximum intensity enhancement ratios of TM-waves under most grating parameters are larger than fifteen and the ones of TE-waves are less than ten, TIR gratings are more appropriate for TE-polarized incident light. Although the use of the TIR gratings combined with prisms as compressors in laser systems requires considering other important factors, such as the group velocity dispersion and the third order dispersion [36e38], which would be the main point in our future work, with its compact setup and potential high damage threshold, the gratings could be a promising alternative of compression gratings. Acknowledgement This work was supported by the National High Technology Research and Development Program ("863"Program) of China References [1] D. Strickland, G. Mourou, Opt. Commun. 56 (1985) 219. [2] P. Maine, D. Strickland, P. Bado, M. Pessot, G. Mourou, J. Quantum Electron. 24 (1988) 398. [3] J.V. Rudd, G. Korn, S. Kane, J. Squier, G. Mourou, P. Bado, Opt. Lett. 18 (1993) 2044. [4] M.D. Perry, G. Mourou, Science 264 (1994) 917. [5] F. Tavella, A. Marcinkevicius, F. Krausz, Opt. Express 14 (2006) 12822. [6] D. Strickland, P. Maine, G. Mourou, J. Opt. Soc. Am. A 3 (1986) 97. [7] T. Yilmaz, L. Vaissie, M. Akbulut, D.M. Gaudiosi, L. Collura, T.J. Booth, J.C. Jasapara, M.J. Andrejco, A.D. Yablon, C. Headley, D.J. Digiovanni, In: Proc. SPIE 6873, 2008, pp. 68731I-1. [8] M. Nisoli, S. Desilvestri, O. Svelto, R. Szipocs, K. Ferencz, C. Spielmann, S. Sartania, F. Krausz, Opt. Lett. 22 (1997) 522. [9] S. Sartania, Z. Cheng, M. Lenzner, G. Tempea, C. Spielmann, F. Krausz, K. Ferencz, Opt. Lett. 22 (1997) 1562. [10] R.D. Boyd, J.A. Britten, D.E. Decker, B.W. Shore, B.C. Stuart, M.D. Perry, L. Li, Appl. Opt. 34 (1995) 1697. [11] J.A. Britten, M.D. Perry, B.W. Shore, R.D. Boyd, Opt. Lett. 21 (1996) 540.
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Contents lists available at ScienceDirect
Current Applied Physics journal homepage: www.elsevier.com/locate/cap
Design and properties analysis of total internal reflection gratings for pulse compressor at 1053 nm Qunyu Bi*, Jiangjun Zheng, Ailin Guo, Meizhi Sun, Jianpeng Wang, Fuling Zhang, Qingwei Yang, Xinglong Xie, Zunqi Lin Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Science, No.390 Qinghe Road, Shanghai 201800, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 6 January 2010 Received in revised form 15 April 2010 Accepted 8 June 2010 Available online 14 June 2010
High-efficiency compression gratings based on total internal reflection (TIR) are promising alternatives of compressor gratings because of their high diffraction efficiency, potential high damage resistant ability, and compact structure. Dependence of the 1 order diffraction efficiencies on grating parameters is analyzed for TE- and TM-polarized incident light of 1053 nm at Littrow angle, which is calculated by using the rigorous coupled-wave analysis. A more intuitional view on the relation is offered through three-dimensional slicing figures instead of two-dimensional ones. The performances of high-efficiency gratings are compared and regarded as criteria for further choices, including spectral bandwidth, angle bandwidth, dispersion, and intensity distribution. For TE- and TM-irradiations, similar spectral bandwidth and angle bandwidth can be achieved by different grating parameters. However, the computer simulation result on the intensity distributions of the two polarized waves shows that such design should be used under the illumination of TE-polarized wave for lower intensity enhancement ratio, which is an important factor related to the gratings’ damage threshold. Ó 2010 Elsevier B.V. All rights reserved.
Keywords: Total internal reflection gratings Grating performance analysis Pulse compressor
1. Introduction The advent of the chirped-pulse amplification (CPA) technique has created a revolution in the production and use of high-power picosecond and femtosecond laser systems, which are essential tools for the research of lightematter interaction and inertial confinement fusion [1e7]. Although the stretch and compression stages can be implemented through fibers [1,8,9], most high-energy systems are using diffraction gratings to stretch or recompress the pulses so far. Various gratings are designed and fabricated to achieve high diffraction efficiency for the required order (often the -1st), high damage threshold, and large size with high surface flatness. Up to now, gratings working in the high-energy laser facilities are mainly Au-coated gratings [10,11] and multilayer dielectric gratings [12e15]. The two sorts of gratings fielded in CPA systems had obtained satisfied results to some extent [16,17]. Because the optical damage threshold of dielectric materials is considerably above that of metals, more attentions are paid to the multilayer
* Corresponding author. E-mail address: [email protected] (Q. Bi). 1567-1739/$ e see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cap.2010.06.005
dielectric gratings [12e15,18e20]. For example, at the wavelength of 1053 nm, the damage thresholds of gold-coated master gratings are e1 J/cm2 for nanosecond pulses and less than 0.4 J/cm2 for subpicosencond pulses [10,11,21]. For fused silica, the values can be up to 20 J/cm2 for nanosecond pulses, and over 2 J/cm2 for 400-fs pulses [12,22]. However, in the production process, disadvantages of multilayer gratings appear. Firstly, because they are formed by placing a grating structure on top of a multilayer structure of alternating index dielectric materials, such as fused silica and hafnium oxides, it is impossible to completely control the presence of mechanical constraints at the interfaces, due to different mechanical properties and the distortion of the spatial qualities of the laser beam. Secondly, the introduction of other oxides frequently used, such as HfO2, Al2O3, and TiO2, tends to reduce the damage threshold of the mirror, which might be applicable to multilayer gratings [23e26]. In 2004, Marciante et al. reported a new class of gratings based on the phenomenon of total internal reflection (TIR) regardless of the grating tooth shape. More important, they could be fabricated from a single dielectric material, requiring no metallic or layers, which could avoid the obstacles that multilayer dielectric gratings must face [27]. Later, Liu et al. firstly attempted to directly etch the gratings into the bottom side of a prism to form a compact pulse
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plots instead of the two-dimensional ones, and by analyzing on the performances of different grating parameters in detail, including the grating period, etching depth, and duty cycle. Two different points of view were obtained. One was that similar spectral bandwidth and angle bandwidth could be achieved by different grating parameters for both TE- and TM-waves; the other one was that TIR gratings were more favorably used under the illumination of TE-waves for lower intensity enhancement ratio.
2. Design and properties analysis
Fig. 1. Schematic of a TIR grating. q is the incidence angle, qi is the ist order diffraction angle, D is the depth of the substrate, d is the depth of the grating, p is the grating period, f is the duty cycle, and ni (i ¼ 1, 2) are refractive indices.
compressor [28]. However, there is little information about global extrema of diffraction efficiency and it is not conducive to the applications of the TIR gratings in high-energy laser systems. In this paper, we extend this work by offering a more intuitional view on the 1 order diffraction through three-dimensional slicing
Generally, the period of gratings used as compressors in highpower laser systems is close to the light wavelength in order to obtain large dispersion. So a vector theory has to be employed for accurate gratings designing. The rigorous coupled-wave analysis, which is one of the most well-developed and widely used tool for analysis of the gratings [29e31] is used in this paper. For applications in high-power lasers, a small decrease in grating efficiency would result in a large loss in the energy output, because the beam often diffracts from the grating four times. Therefore, high diffraction efficiency is a basic requirement and should be considered firstly. A scheme of a TIR grating is shown in Fig. 1. Because the diffraction efficiency is independent on the grating tooth shape [27], the surface-relief rectangular grating as a mode for numerical simulations is taken for simplicity. When
Fig. 2. Rigorous calculated -1st reflected diffraction efficiency of a TIR grating as a function of depth and period with different refractive indices: (a,c) n1 ¼ 1.3; (b,d) n1 ¼ 1.7. The angle of incidence is the Littrow angle varied with the refractive index and the period. The duty cycle is 0.5.
Q. Bi et al. / Current Applied Physics 11 (2011) 21e27
Fig. 3. Rigorous calculated -1st reflected diffraction efficiency of a TIR grating as a function of depth, period, and duty cycle. The angle of incidence is the Littrow angle varied with the period, the refractive index is 1.46, and the periods of the three slices from right to left are 380, 460, and 520 nm, respectively.
light of wavelength l incidents on the grating surface with Littrow angle, it will be diffracted back toward the direction from where it comes. When refractive index n1 of the grating is higher than refractive index n2 of the surrounding medium, high-efficiency diffraction is possible to be attained if the period (p) of gratings satisfies
n1 >
l 2p
> n2 :
(1)
Therefore, for specific wavelength and grating material, the period of gratings should be in a certain region. In our cases, the incident wavelength is 1053 nm and n2 is 1.0 (air), which means the longest available period is 527 nm and the shortest one can be calculated by l/(2n1). It is obvious that higher n1 can bring a larger available period region of high diffraction efficiency and helpful to reduce the etching depth. Fig. 2 shows the -1st diffraction efficiency as a function of grating depth and period for
23
TE- and TM-waves with n1 ¼ 1.3 and 1.7, where duty cycle is 0.5. It should be pointed out that the -1st diffraction efficiency depends on the grating parameters and incident ways considerably, including the refractive indices of boundary materials, period (p), etching depth (d), duty cycle (f), and incident angles (polar angle of q and azimuthal anlge of j [32], which is set to be zero in our case). Therefore, three-dimensional plots should be adopted to describe the relation between the diffraction efficiency and the grating parameters at Littrow angle. Because we are more interested in the gratings used in the high-energy laser systems and the fused silica owns a high damage threshold (about 2 J/cm2 for normal incidence 1053-nm radiation and 600 shots of 0.3-ps pulses with 500-mm diameter spots [33]), the refractive index n1 is directly set to 1.46, ignoring other possible influences of the refractive index n1 on the design and performances of gratings. As mentioned above, in order to clearly describe the relation between the diffraction efficiency and the constructive parameters of gratings (p, d, and f), Figs. 3(a) and 4(a) show the relation for TE-wave in two slicing ways, one is under specific wavelengths of 380, 460, and 520 nm, and the other one is under duty cycles of 0.1, 0.3, 0.5, 0.7 and 0.9. Similar analyses are undertaken for TM polarization and described in Figs. 3(b) and 4(b), respectively. It can be seen that there are numerous sets of grating parameters that can achieve diffraction efficiency over 0.95, or even 0.99 for both TE- and TM-waves, which means that different constructive parameters can exhibit the same high-efficiency. However, other aspects, such as the fabrication difficulty, intensity enhancement, spectral dependence, and incident angle dependence, which are also important criteria in design consideration, show quite different performances. In the view of fabrication process, the regions close to the left slices in Fig. 3 and the top and down ones in Fig. 4(a) and (b) are the last choices, because it is more difficult to fabricate gratings with deep groove and narrow duty cycle. In addition, the tolerance of fabrication error will vary with the designed parameters and the fabrication altitude can be estimated by the diffraction efficiency figures. For example, as shown in Fig. 4(c), a TIR grating with the depth of 393 nm can offer the latitude of period from 444 to 514 nm with diffraction efficiency over 0.99. However, tens-nm fabrication tolerance is limited and it is still far from practical implementation, especially for the fabrication of large gratings. So the improvement on the fabrication tolerance by selecting constructive parameters is quite limited and we pay more attention to the dependences of performances of gratings on the designed grating parameters in the following.
2.1. Spectral dependence, angle dependence, and dispersion Spectral and angle dependences of diffraction efficiency are of particular concern for the applications in pulse compression. In most cases, the relations between spectral or incident angle and diffraction efficiency are described only for a single grating structure, ignoring an important fact that they will vary with the grating parameters(p, d, and f). Because the deep groove depth is difficult to be fabricated, we tend to select the grating parameters with etching depth less than 1 mm, corresponding to the part contoured by the red curves in Fig. 4(c) and (d) for TE- and TM-waves. Their corresponding bandwidths are calculated and described in Fig. 5(a) and (b), respectively. As shown in Fig. 5(a), for TE-polarized incident light and duty cycle of 0.6, gratings with different depths and periods exhibit different spectral bandwidths, varying from 28 nm or less (within the range from 1040 to 1068 nm) to 51 nm(within the range from 1030 to 1081 nm). For
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Q. Bi et al. / Current Applied Physics 11 (2011) 21e27
Fig. 4. Rigorous calculated -1st reflected diffraction efficiency of a TIR grating as a function of depth, period, and duty cycle. The angle of incidence is the Littrow angle varied with the period, the refractive index is 1.46, and the duty cycle of the five slices from up to down are 0.9, 0.7, 0.5, 0.3, and 0.1, respectively; (c) and (d) are corresponding contours with duty cycle of 0.6 for TE and 0.5 for TM.
TM-polarized incident light and duty cycle of 0.5, the largest bandwidth with diffraction efficiency higher than 0.95 we can find is 52 nm (within the range of 1029e1081 nm) and the grating profile is 480-nm depth, 470-nm period, and duty cycle of 0.5. The results were chosen from at least fifty grating parameters and better solution might be found but the improvement would be slight. The dependence of the diffraction efficiency on the incident angle is also varied with grating parameters, as shown in Fig. 5 (c) and (d). Every grating exhibits the maximum diffraction efficiency at its corresponding Littrow angle under the wavelength of 1053 nm, which can be calculated through sin1 ð1053=2pn1 Þ. Comparing Fig. 5(a) with (c), and Fig. 5(b) with (d), we can see that the gratings with large spectral bandwidth may own a narrow angle bandwidth. This was also observed under other duty cycles, but it was not mean that the gratings with the largest spectral bandwidth had the smallest angle bandwidth.
In addition, the dispersion (K) is also an essential factor in the design of the gratings. For the 1 order under Littrow incidence, it can be calculated according to [27].
1 K ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: l2 p2 4n 2
(2)
1
For the specific wavelength of 1053 nm and n1 of 1.46, the dispersion decreases sharply from 17.73 to 5.78 mrad/nm when the period increases from 365 to 400 nm. However, it decreases slowly from 5.78 to 2.60 mrad/nm in the range from 400- to 527-nm period. The dispersion is irrelevant with the polarization states of illumination light. 2.2. Intensity distribution Laser induced damage threshold is one of the most important factors that influence the application of grating in high-energy
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a
1
1 p=400 d=584 p=440 d=416 p=460 d=390 p=490 d=376
0.97 0.96 0.95 0.94 0.93
0.98 0.97 0.96 0.95 0.94 0.93
0.92
0.92
0.91
0.91 1030
1040
1050
1060
1070
1080
1090
0.9 -4
1100
Wavelength (nm)
0.97 0.96 0.95 0.94 0.93
1060
1070
3
4
d TM polarization
0.95 0.94 0.93
0.91 1050
2
0.96
0.92
1040
1
0.97
0.91 1030
0
0.98
0.92
0.9 1020
-1
0.99
Diffraction efficiency
Diffraction efficiency
0.98
-2
1 p=410 d=712 p=430 d=566 p=470 d=480 p=500 d=546
0.99
-3
Relative angle of incidence (degrees from Littrow)
b TM polarization
1
TE polarization
0.99
Diffraction efficiency
Diffraction efficiency
0.98
0.9 1020
c
TE polarization
0.99
25
1080
1090
1100
Wavelength (nm)
0.9 -4
-3
-2
-1
0
1
2
3
4
Relative angle of incidence (degrees from Littrow)
Fig. 5. Reflected diffraction efficiency of the 1 order as a function of wavelengths and incident angles under different grating parameters for TE (a, c) and TM (b, d). The angle of incidence is the Littrow angle under the central wavelength of 1053 nm, the groove duty cycles are 0.6 for (a, c), 0.5 for (b, d), and the refractive index is 1.46.
laser systems. Neauport et al. claimed and experimentally proved that the ratio of the damage threshold of two gratings was equal to the inverted ratio of their field enhancement [34]. Therefore, further insight into the near-field electric distribution is promising to improve the damage threshold of the gratings. As the spectral and angle bandwidth, the electric field enhancement ratio is also related to the grating parameters (p, d, and f) and illumination ways (polarization state of light and incident angle). As we know, there are no analytic expressions that can describe the relation between the ratio and its corresponding grating parameters. Therefore, computer simulation of the near-field optical distribution is essential in estimating the laser induced damage threshold of the gratings. The simulation results of TEand TM- radiation are different and we will discuss them separately. For TE-waves, the grating parameters that can achieve diffraction efficiency over 0.99 under different duty cycles are sampled and their corresponding maximum intensity enhancement ratios are calculated and described in Fig. 6(b). For example, the line of f ¼ 0.6 indicates the maximum intensity enhancement ratios of the sampled gratings parameters in Fig. 4 (c), marked by blue asterisks. The grating period that may attain the intensity enhancement ratio less than six is in the range of 380e440 nm, and the duty cycle should be selected from 0.6 to
0.8. Another point we should mention is that for the duty cycles of 0.8, 0.7, and 0.6, they all have a peak intensity enhancement ratio, which corresponds to the middle point in the vertical part of the figure describing the relation of efficiency and grating parameters. For instance, the grating parameters that fulfill the maximum intensity enhancement ratio of 16.96 presented by the line of f ¼ 0.6 in Fig. 6(b) are 517 nm(p) and 560 nm(d), which is the middle point of the vertical part of Fig. 4(c). Therefore, we should avoid selecting these grating parameters. Fig. 6(a) shows the simulation intensity distribution of a grating at Littrow angle of 55.04 . The period, depth, and duty cycle of the grating are 440 nm, 414 nm, and 0.6, respectively, which performs the spectral bandwidth of 47 nm (efficiency0.95), angular bandwidth of w4 (efficiency0.95), dispersion of 3.97 mrad/nm, and intensity enhancement ratio of 5.97. The maximum intensities usually locate in the gratings’ center or the ridge opposite to the incident direction. For the simulation of intensity distribution under TMpolarized illumination, there is a convergence problem for the rigorous coupled-wave analysis, although quick convergence is obtained in calculating diffraction efficiencies by reformulating the eigenproblem of the coupled-wave method [35]. Fig. 7(b) describes the dependence of the maximum intensity enhancement ratio on the number of space harmonics under three
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Fig. 6. (a) Intensity distribution of internal electric field of a TIR grating for the TEpolarized incident at Littrow angle (b) Maximum Intensity enhancement ratio under different cycle duties as function of depth and period with the diffraction efficiency higher than 0.999 and etching depth less than a micron.
Fig. 7. (a) Intensity distribution of internal electric field of a TIR grating for the TMpolarized incident at Littrow angle. (b) Maximum intensity enhancement ratio dependence on the number of harmonics under three gratings.
3. Conclusion
different grating parameters. It is obvious that although the number of space harmonics is as high as 1001, the maximum intensity ratio is still unstable. It also occurs for other grating parameters under illumination of TM-waves. Further increase of the number of space harmonics will easily lead to exceed the capacity of our personal computer’s memory (1 GB). Fig. 7(a) shows 1401 space harmonics result of the simulated intensity distribution of the grating denoted by the blue line in Fig. 7(b). Its maximum intensity ratio is 22.2, which is different from the result of 101 space harmonics (24.3). This might be caused by the existence of the evanescent waves or the different boundary condition with TE. More detailed discussion on this may be beyond the scope of this article. Therefore, considering the high intensity enhancement ratio and instability of simulation result, TIR gratings are more suitable for TE-illumination.
A high-efficiency diffraction grating used for pulse compressor at 1053 nm is designed. The grating parameters that can achieve high diffraction efficiency are calculated by the rigorous coupledwave analysis method and the results with etching depth less than 1 mm are described in two three-dimensional slicing plots, because the efficiency is a function of three grating parameters. There are numerous sets of grating parameters (p, d, and f) that can achieve high diffraction efficiency and other criteria should be considered for further optimization of gratings. Gratings’ performances on spectral and angle bandwidths, dispersion, and intensity distribution are all related to the grating parameters and become the criteria for grating optimization except high diffraction efficiency. Different grating parameters with diffraction efficiency higher than 0.999 are sampled and their corresponding performances are compared. The results show that, for TE- and TM-irradiation, they both own grating parameters that can achieve the similar maximum spectral bandwidths and angle bandwidths. However, the results of the numerical simulation on intensity distribution of
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the two waves are quite different. For the simulation of TM-waves, there is a convergence problem and it displays some fluctuations in intensity distribution result. But the fluctuation becomes small enough and reliable to some extent with the increase of harmonic waves number. Because the maximum intensity enhancement ratios of TM-waves under most grating parameters are larger than fifteen and the ones of TE-waves are less than ten, TIR gratings are more appropriate for TE-polarized incident light. Although the use of the TIR gratings combined with prisms as compressors in laser systems requires considering other important factors, such as the group velocity dispersion and the third order dispersion [36e38], which would be the main point in our future work, with its compact setup and potential high damage threshold, the gratings could be a promising alternative of compression gratings. Acknowledgement This work was supported by the National High Technology Research and Development Program ("863"Program) of China References [1] D. Strickland, G. Mourou, Opt. Commun. 56 (1985) 219. [2] P. Maine, D. Strickland, P. Bado, M. Pessot, G. Mourou, J. Quantum Electron. 24 (1988) 398. [3] J.V. Rudd, G. Korn, S. Kane, J. Squier, G. Mourou, P. Bado, Opt. Lett. 18 (1993) 2044. [4] M.D. Perry, G. Mourou, Science 264 (1994) 917. [5] F. Tavella, A. Marcinkevicius, F. Krausz, Opt. Express 14 (2006) 12822. [6] D. Strickland, P. Maine, G. Mourou, J. Opt. Soc. Am. A 3 (1986) 97. [7] T. Yilmaz, L. Vaissie, M. Akbulut, D.M. Gaudiosi, L. Collura, T.J. Booth, J.C. Jasapara, M.J. Andrejco, A.D. Yablon, C. Headley, D.J. Digiovanni, In: Proc. SPIE 6873, 2008, pp. 68731I-1. [8] M. Nisoli, S. Desilvestri, O. Svelto, R. Szipocs, K. Ferencz, C. Spielmann, S. Sartania, F. Krausz, Opt. Lett. 22 (1997) 522. [9] S. Sartania, Z. Cheng, M. Lenzner, G. Tempea, C. Spielmann, F. Krausz, K. Ferencz, Opt. Lett. 22 (1997) 1562. [10] R.D. Boyd, J.A. Britten, D.E. Decker, B.W. Shore, B.C. Stuart, M.D. Perry, L. Li, Appl. Opt. 34 (1995) 1697. [11] J.A. Britten, M.D. Perry, B.W. Shore, R.D. Boyd, Opt. Lett. 21 (1996) 540.
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