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Design of high-efficiency diffraction gratings based on total internal reflection for pulse compressor
Optics Communications 273 (2007) 290–295 www.elsevier.com/locate/optcom
Design of high-efficiency diffraction gratings based on total internal reflection for pulse compressor Shijie Liu
a,b,*
a
, Jianyong Ma a,b, Chaoyang Wei a,b, Zicai Shen a,b, Jianbin Huang a, Yunxia Jin a, Jianda Shao a, Zhengxiu Fan a
Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China b Graduate School of Chinese Academy of Sciences, Beijing 100039, China Received 14 August 2006; received in revised form 6 December 2006; accepted 27 December 2006
Abstract We present designs of high-efficiency compression grating based on total internal reflection (TIR) for picosecond pulse laser at 1053 nm. The setup is devised by directly etching gratings into the bottom side of a prism so that light can successfully enter (or exit) the compression grating. Dependence of the 1 order diffraction efficiencies on the constructive parameters is analyzed for TE- and TM-polarized incident light at Littrow angle by using Fourier modal method in order to obtain optimal grating structure. The electric field enhancement within the high-efficiency TIR gratings is regarded as another criterion to optimize the structure of the TIR gratings. With the criterion of high diffraction efficiency, low electric field enhancement and sufficient manufacturing latitude, TIR compression gratings with optimized constructive parameters are obtained for TE- and TM-polarized incident light, respectively. The grating for TE-polarized light exhibits diffraction efficiencies higher than 0.95 within 23 nm bandwidth and relatively low square of electric field enhancement ratio of 5.7. Regardless of the internal electric field enhancement, the grating for TM-polarized light provides diffraction efficiencies higher than 0.95 within 42 nm bandwidth. With compact structure, such TIR compression gratings made solely of fused silica should be of great interest for application to chirped pulse amplification (CPA) systems. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Diffraction gratings; Total internal reflection; Pulse compressor and laser damage
1. Introduction Since the beginning of 90s the development of highenergy laser has been achieved great progress owing to the application of the chirped pulse amplification (CPA) to solid-state lasers [1]. The concept behind CPA is a scheme to increase the energy of a short pulse, while avoiding very high peak powers in the laser amplification process itself. This is accomplished by lengthening the duration of the pulse being amplified with a pair of gratings and by *
Corresponding author. Address: Shanghai Institute of Optics and Fine Mechanics, Research and Development Center of Optical Thin Film, No. 390, Qinghe Road, Jiading, Shanghai 201800, China. Tel.: +86 021 69918492. E-mail address: [email protected] (S. Liu). 0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.12.038
recompressing the amplified pulse in a reversible manner with another pair of gratings. The grating pairs are typically planner reflection gratings used in the first order near the Littrow angle. To improve the output coefficient of laser energy, each grating should provide diffractive efficiency as high as possible. Additionally, laser induced damage thresholds (LIDT) of compression gratings should be compatible with the requirements for recompression of high-energy short pulses [2]. Multilayer dielectric (MLD) gratings [3–7] instead of metallic gratings [8] has been widely adopted in the CPA systems because of their high diffraction efficiency and great damage resistant ability. Generally, a MLD grating is obtained by etching grating structure into the top layer of a MLD coating consisting of alternating layers of highand low-refractive index materials, such as HfO2 and SiO2.
S. Liu et al. / Optics Communications 273 (2007) 290–295
Recently, the highest damage threshold of optimized MLD gratings design is 4.5 J/cm2 for 10 ps pulses at 1053 nm [9]. However, in addition to design of the grating itself, such MLD gratings require designing the properties of the MLD coating [10]. The uniformity and cleanness of the coating with tens of stacks must also be carefully controlled in the fabrication process. Therefore, diffraction grating instead formed in the dielectric bulk material is an alternative approach [11], which can provide high efficiencies in either transmission or reflection. Additionally, the LIDT of a bulk material is much higher than that of its coating. Thus gratings made of bulk fused silica will exhibit stronger damage resistance than the MLD grating with gratings etched into the top layer of a SiO2 or HfO2 film. Highly efficient transmission gratings in fused silica have been realized in the femtosecond fiber CPA system [12]. Recently, a new class of high-efficiency grating based on total internal reflection (TIR) has been invented, which is immersed in a single dielectric material and provides reflection efficiency of over 99% in the Littrow mount [13,14]. This invention has achieved many significant applications, such as polarization-insensitive components [15] and (de)multiplexers [16]. However, until now pulse compression grating based on TIR, to our knowledge, is seldom investigated. In this paper, we apply TIR-gratings to the pulse compressor in the CPA systems. Grating is directly etched into the bottom side of a prism so that the incident light can be successfully coupled into the substrate. Designs of the TIR-grating are presented for either TE (electric field perpendicular to the plane of incidence) or TM (magnetic field perpendicular to the plane of incidence) polarized picosecond pulse at 1053 nm. Besides the 1 order diffraction efficiency, the electric field enhancement ratio in the grating, which is closely related to damage resistance, is used as another criterion to optimize the grating structure. A set of optimum gratings parameters is obtained for TE- and TM-polarized incident light, respectively, which make the compression gratings achieve both high diffraction efficiency and minimal electric field enhancement. Finally, we also discuss the optical and angular spectrums of the optimized TIR compression gratings.
fluid. Alternatively, the grating can be directly etched into the bottom side of a prism, as shown in Fig. 1. Light incident upon the grating can be successfully coupled into the high-index material 1 at the Littrow angle of hi in spite of a small reflection loss at the slope interface. Given the highand low refractive indices, the coupling angle c defined as the angle of light incident on the slope interface can be determined by the angle of hi and the bottom angle of a, n1 c ¼ sin1 sinða hi Þ ð1Þ n2 where n1 and n2 ðn1 > n2 Þ are refractive indices of high and low dielectrics, respectively. Then no light escapes into material 2 regardless of the grating tooth shape with the grating period satisfying [13] n1 >
k > n2 2T
ð2Þ
where k is wavelength of incident light and T is period of grating along x-direction. In a Littrow mount there will be only a single diffracted order that will carry away most of the incident energy and travel back along the path of incident light. Without any other connection between grating and prism, this setup is made of solely bulk material and can be easier to be integrated in the CPA system. Since a prism can be conveniently devised, our attention is focused on optimization of the gratings etched into the bottom side of the prism. In our design, we consider compression gratings for a picosecond optical pulse at the central wavelength of 1053 nm, which are composed solely of fused silica ðn1 ¼ 1:46Þ. The outgoing layer is air ðn2 ¼ 1Þ. Because of narrow bandwidth of the short pulse, reflection performance of the grating at the central wavelength is concerned
2. Theory model In general, the TIR-grating is immersed in a single dielectric material, where light is incident upon the grating from the high-index dielectric to a low-index medium. Since the angle of incidence is required to satisfy the TIR condition at the grating, the light cannot enter (or likely exit) the high-index dielectric from an interface that is parallel to the grating. Therefore, how the incident light is coupled into the high-index material is an important issue for the practical applications of TIR-grating. Maciante et al. used a prism on the backside of the grating substrate, which has been connected by an index matching fluid [15]. However, this setup is probably not suitable for CPA systems due to absence of non-absorbing matching
291
Fig. 1. Structure of the TIR compression grating.
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in practice. With the grating period satisfying formula (2) and Littrow angle of incidence, no light escapes into air regardless of the grating tooth shape. The exact grating profile will determine the power distribution among the various reflected orders, which can be solved by Fourier modal method (FMM) [17,18]. The optimal design of compression gratings should provide high diffraction efficiency as well as sufficient manufacture latitudes. However, multiphoton ionization induced by strong electric field enhancement in the grating will result in grating breakdown [19]. Therefore, the electric field enhancement is used as another criterion to optimize the design of compression grating. 3. Numerical calculations and discussion In the Littrow mounting, the 1 order diffraction efficiency and internal electric field enhancement in the grating are closely related to the grating parameters, i.e. grating period, depth and duty cycle, and light polarization state. In this section, both the 1 order diffraction efficiency and the electric field enhancement as a function of grating parameters are discussed in detail for TE- and TM-polarized incident light, respectively, in order to obtain the optimal grating structures. 3.1. Diffraction efficiency analysis According to the formula (2), grating period is limited to be 361–527 nm long. The 1 order diffraction efficiency as a function of grating depth and period for TE-polarized wave is shown in Fig. 2a, where duty cycle is 0.5. It can be seen that there is more than one region where the 1 order diffraction efficiency is higher than 0.95 for grating depths up to 2 lm. However, gratings with deep grooves bring much more difficulties in the fabrication process [20]. In our design, grating depth is chosen to be less than a micron deep. For example, a TIR-grating with depth of 350 nm is preferred in Fig. 2a, which provides sufficient latitude for period ranging from 460 to 525 nm. Fig. 2b shows the relation between the 1 order diffraction efficiency, grating depth and duty cycle when grating period is 450 nm. It can be seen that the latitude of duty cycle becomes narrower with the increase of grating depth from 0 to 1 lm. Therefore, TIR grating with a relatively shallow depth is also beneficial to extend the latitude of its duty cycle. In addition, calculation results show that in these shallow-depth regions, the maximal grating depth and duty cycle along with minimal grating depth and duty cycle reduce with the increase of grating period from 361 to 527 lm, which results in the reduction of grating aspect ratio (groove depth divided by groove linewidth) and thus facilitates grating fabrication. The 1 order diffraction efficiency as a function of grating depth and period for TM-polarized wave is shown in Fig. 3a, where duty cycle is 0.5. Similarly, grating depth less than a micron deep is preferred in order to bring facilities
Fig. 2. (a) The 1 order diffraction efficiency of the TIR gratings with duty cycle of 0.5 as a function of grating depth and period. (b) The 1 order diffraction efficiency of the TIR gratings with period of 450 nm under the illumination of TE-polarized light at Littrow angle as a function of grating depth and duty cycle.
in the fabrication process. Fig. 3b shows the relation between the 1 order diffraction efficiency, grating depth and period when period is 450 nm. It can be seen that these band regions where 1 order diffraction efficiency higher than 0.95 show similarities to those for TE-polarized wave shown in Fig. 2b. However, for a same period, high-efficiency TIR gratings for TM-polarized wave require deeper values of grating depth, which also have bigger latitude. For example, a TIR grating with depth being 393– 540 nm in Fig. 3b achieves sufficient latitude of duty cycle to make the 1 order diffraction efficiency higher than 0.95. Also, the increase of grating period results in reduction of grating aspect ratio. 3.2. Near field analysis Further insight into the near-field optical distribution in the high-efficiency TIR-gratings, characterized by the distribution of the internal transverse electric field, is promising to improve the LIDT value of the gratings. Fig. 4a
S. Liu et al. / Optics Communications 273 (2007) 290–295
Fig. 3. (a) The 1 order diffraction efficiency of the TIR gratings with duty cycle of 0.5 as a function of grating depth and period. (b) The 1 order diffraction efficiency of the TIR gratings with period of 450 nm under the illumination of TM-polarized light at Littrow angle as a function of grating depth and duty cycle.
shows the amplitude of electric field in the TIR gratings with diffraction efficiency of 0.988 under the illumination of TE-polarized wave at Littrow angle of 53.26°. The grating period, depth and duty cycle are 450 nm, 390 nm and 0.5, respectively. Fig. 4b shows the amplitude of electric field in the TIR gratings with diffraction efficiency of 0.99 under the illumination of TM-polarized wave at Littrow angle of 53.26°. The grating period, depth and duty cycle are 450 nm, 450 lm and 0.5, respectively. It can be seen that most of the light for both TE- and TM-polarized states are reflected back into the fused silica by the TIR grating and only evanescent waves penetrate a small distance into air. Moreover, electric field enhancement for TE-polarized light over two times higher than the incident field appears in the gratings ridge and bulk fused silica. In contrast, electric field in the grating and bulk fused silica for TM-polarized wave only exhibits an enhancement less than two times smaller than the incident wave. The maximum amplitude is just localized in the air gap, as shown in Fig. 4b. The different transverse profiles between the two polarized waves can be understood by their different boundary conditions
293
Fig. 4. Amplitude distributions of internal electric field of a TIR grating for the TE-polarized incident at angle of 53.26° (a) and of another TIR grating for the TM-polarized incident at the same angle (b).
imposed on the near field by the gratings [21]. Multiphoton ionization induced by strong electric field enhancement in the fused silica will result in grating breakdown. Therefore, in order to improve the resistant ability, such high-efficiency gratings should be used under the illumination of TM-polarized wave or under the illumination of TE-polarized wave generating low electric field enhancement in the gratings. The grating parameters including period, depth and duty cycle are also responsible for the electric field enhancement in the grating under the illumination of TE-polarized light. Dependence of square of electric field enhancement ratio on grating depth, duty cycle and period is shown in Fig. 5a, b and c, respectively. From Fig. 5a and b, it is seen that electric field enhancement shows a slight dependence on both grating depth and duty cycle for different values of period. Whereas square of the electric field enhancement ratio increases rapidly with the increasing of the grating period from 380 nm to 520 nm, as shown in Fig. 5c. The maximal enhancement ratio is 9.3, which may bring catastrophic damage to the TIR gratings used in the highenergy laser system. Therefore, in order to reduce field enhancement within the grating, a short period should be
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Fig. 5. Square of electric field enhancement ratio as function of grating depth (a), duty cycle (b) and period (c).
preferred in the design of such compression gratings under illumination of TE-polarized wave. 3.3. Optimization results of TIR-gratings A TIR-grating used in the CPA system requires providing high diffraction efficiency, low field enhancement in the
grating and sufficient manufacturing latitude. From the numerical analysis in Sections 3.1 and 3.2, it is not difficult to find that a TIR-grating used under the illumination of TM-polarized wave can easily satisfy these requirements. An optimized TIR compression grating with period of 500 nm, depth of 450 nm and duty cycle of 0.35 is obtained for TM-polarized incident light, which achieves high diffraction efficiency of 0.999, aspect ratio of 1.38 and sufficient manufacturing latitude (depth of 450 ± 50 nm and duty cycle of 0.35 ± 0.1). However, for a TIR-grating under illumination of TE-polarized wave, selecting a short grating period effectively reduces its internal field enhancement while greatly increases the grating aspect ratio. For example, a TIR-grating with period of 380 nm, depth of 750 and duty cycle of 0.75 can achieve diffraction efficiency of 0.997 and the lowest square of electric field enhancement ratio being 4.3. But, to our knowledge, its aspect ratio being about 7.9 has already exceeded the maximum value of 3.37, which has been realized by Wang et al [20]. Therefore, a trade-off should be required between the selections of grating period and the realizable aspect ratio of grating. Given the aspect ratio of 3 in our design, an optimized grating with period of 410 nm, depth of 640 ± 20 nm and duty cycle of 0.47 ± 0.02 is obtained for TE-polarized incident light, which achieves high diffraction efficiency of 0.998 and square of field enhancement ratio of 5.7. For application in the pulse compressor, spectral dependence of the diffraction efficiency is of particular concern. The optical spectrum curves of the two optimized TIR compression gratings are shown in Fig. 6a, where the solid line represents the optical spectrum of the optimized grating for TE-polarized incident light at Littrow angle of 61.59° and the dashed line represents the optical spectrum of the optimized gratings for TM-polarized incident light at Littrow angle of 46.16°. It is found that the optimized TIR grating under illumination of TM-polarized light exhibits a diffraction efficiency of higher than 0.95 for 42 nm spectral bandwidth (within the range 1036– 1077 nm) around the central wavelength of 1053 nm. In contrast, the other optimized grating under illumination of TE-polarized light exhibits a diffraction efficiency of higher than 0.95 only for 23 nm bandwidth (within the range 1040–1063 nm), which is inferior to that of the former. The dependence of the diffraction efficiency on the incident angle is also shown in Fig. 6b, where the dashed line represents angular spectrum of the optimized gratings for TM-polarized incident light of 1053 nm and the solid line represents angular spectrum of the other optimized gratings for TE-polarized incident light of 1053 nm. It is seen that both gratings exhibit maxima of diffraction efficiency at Littrow angle being 46.16° and 61.59°, respectively. Moreover, the TIR grating for TE-polarized light exhibits diffraction efficiency higher than 0.95 for 5° bandwidth. The other TIR grating for TM-polarized light also exhibits diffraction efficiency higher than 0.95 for 2.4°
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4. Conclusion We have demonstrated the process of designing the compression gratings based on TIR for picosecond pulse laser at the central wavelength of 1053 nm. The setup is devised by directly etching gratings into the bottom side of a prism so that light can successfully enter (or exit) the compression grating. By using Fourier modal method, both the 1 order diffraction efficiency and internal electric field enhancement as functions of constructive parameters, including grating period, depth and duty cycle, are discussed in detail for both TE- and TM-polarized wave in order to obtain the optimal grating structure. With the criterion of high diffraction efficiency, low internal field enhancement and sufficient manufacturing latitude, TIR-gratings with optimized structure is obtained for two polarized waves, respectively. In comparison, such design is more favorably used for TM-polarized incident light. With compact setup, this TIR grating could be a good alternative of compression gratings. Acknowledgements We acknowledge the financial support from National Natural Science Foundation of China (Grant No. 10376040). References
Fig. 6. Optical spectrum in (a) and angular spectrum in (b) of two optimized grating respectively under the illumination of two-polarized light.
bandwidth. The angular dependence of diffraction efficiency will permit adjusting the incident angle of optical pulse in practice. Given the angle incident on the grating, the bottom angle a of a prism for grating fabrication can be conveniently evaluated, which is greater than the value of hi. The exact value of a can be adjusted in the real optical system. For example, a prism with bottom angle of 75° can be used to fabricate the above two optimized TIR-gratings. By using the formula (2), the coupling angle c is 19.79° for TE-polarized wave and 44.77° for TM-polarized wave, respectively. Unfortunately, energy losses by Fresnel reflection are generated at the prism–air-interface, which are 4.1% for TE incident wave and 0.7% for TM incident wave. These losses are unwanted in the CPA system, especially for TE incident wave, which can be avoided by adding an antireflection dielectric coating on the slope side of the prism.
[1] D. Strickland, G. Mourou, Opt. Commun. 56 (1985) 219. [2] Sterling Backus, Charles G. Durfee III, Margaret M. Murnane, et al., Rev. Sci. Instrum. 69 (3) (1998) 1207. [3] M.D. Perry, R.D. Boyd, J.A. Britten, et al., Opt. Lett. 20 (8) (1995) 940. [4] L. Li, J. Hirsh, Opt. Lett. 20 (11) (1995) 1349. [5] B.W. Shore, M.D. Perry, J.A. Britten, et al., J. Opt. Soc. Am. A 14 (5) (1997) 1124. [6] Karl Hehl, Joerg Bischoff, Appl. Opt. 38 (30) (1999) 6257. [7] A.J. Waddie, M.J. Thomson, M.R. Taghizadeh, Opt. Lett. 30 (9) (2005) 991. [8] R.D. Boyd, J.A. Britten, D.E. Decker, et al., Appl. Opt. 34 (10) (1995) 1697. [9] I. Jovanovic, C.G. Brown, B.C. Stuart, et al., SPIE 5647 (2005) 34. [10] S. Liu, J. Ma, Z. Shen, et al., Appl. Surf. Sci. 253 (7) (2006) 3642. [11] H.T. Nguyen, B.W. Shore, S.J. Bryan, et al., Opt. Lett. 22 (3) (1997) 142. [12] Tina Clausnitzer, Jens Limpert, Karsten Zollner, et al., Appl. Opt. 42 (34) (2003) 6934. [13] J.R. Marciante, D.H. Raguin, Opt. Lett. 29 (6) (2004) 542. [14] L. Eisen, M.A. Golub, A.A. Friesem, Opt. Commun. 261 (2006) 13. [15] J.R. Marciante, J.I. Hirsh, D.H. Raguin, et al., J. Opt. Soc. Am. A 22 (2) (2005) 299. [16] Y. Zhang, C. Zhou, J. Opt. Soc. Am. A 22 (2) (2005) 331. [17] L. Li, J. Opt. Soc. Am. A 10 (12) (1993) 2581. [18] S. Liu, Z. Shen, W. Kong, et al., Opt. Commun. 267 (2006) 50. [19] B.C. Stuart, M.D. Feit, S. Herman, et al., Phys. Rev. B 53 (4) (1996) 1749. [20] S. Wang, C. Zhou, Y. Zhang, et al., Appl. Opt. 45 (12) (2006) 2567. [21] W. Nakagawa, R. Tyan, P. Sun, et al., J. Opt. Soc. Am. A 18 (5) (2001) 1072.
Design of high-efficiency diffraction gratings based on total internal reflection for pulse compressor Shijie Liu
a,b,*
a
, Jianyong Ma a,b, Chaoyang Wei a,b, Zicai Shen a,b, Jianbin Huang a, Yunxia Jin a, Jianda Shao a, Zhengxiu Fan a
Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China b Graduate School of Chinese Academy of Sciences, Beijing 100039, China Received 14 August 2006; received in revised form 6 December 2006; accepted 27 December 2006
Abstract We present designs of high-efficiency compression grating based on total internal reflection (TIR) for picosecond pulse laser at 1053 nm. The setup is devised by directly etching gratings into the bottom side of a prism so that light can successfully enter (or exit) the compression grating. Dependence of the 1 order diffraction efficiencies on the constructive parameters is analyzed for TE- and TM-polarized incident light at Littrow angle by using Fourier modal method in order to obtain optimal grating structure. The electric field enhancement within the high-efficiency TIR gratings is regarded as another criterion to optimize the structure of the TIR gratings. With the criterion of high diffraction efficiency, low electric field enhancement and sufficient manufacturing latitude, TIR compression gratings with optimized constructive parameters are obtained for TE- and TM-polarized incident light, respectively. The grating for TE-polarized light exhibits diffraction efficiencies higher than 0.95 within 23 nm bandwidth and relatively low square of electric field enhancement ratio of 5.7. Regardless of the internal electric field enhancement, the grating for TM-polarized light provides diffraction efficiencies higher than 0.95 within 42 nm bandwidth. With compact structure, such TIR compression gratings made solely of fused silica should be of great interest for application to chirped pulse amplification (CPA) systems. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Diffraction gratings; Total internal reflection; Pulse compressor and laser damage
1. Introduction Since the beginning of 90s the development of highenergy laser has been achieved great progress owing to the application of the chirped pulse amplification (CPA) to solid-state lasers [1]. The concept behind CPA is a scheme to increase the energy of a short pulse, while avoiding very high peak powers in the laser amplification process itself. This is accomplished by lengthening the duration of the pulse being amplified with a pair of gratings and by *
Corresponding author. Address: Shanghai Institute of Optics and Fine Mechanics, Research and Development Center of Optical Thin Film, No. 390, Qinghe Road, Jiading, Shanghai 201800, China. Tel.: +86 021 69918492. E-mail address: [email protected] (S. Liu). 0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.12.038
recompressing the amplified pulse in a reversible manner with another pair of gratings. The grating pairs are typically planner reflection gratings used in the first order near the Littrow angle. To improve the output coefficient of laser energy, each grating should provide diffractive efficiency as high as possible. Additionally, laser induced damage thresholds (LIDT) of compression gratings should be compatible with the requirements for recompression of high-energy short pulses [2]. Multilayer dielectric (MLD) gratings [3–7] instead of metallic gratings [8] has been widely adopted in the CPA systems because of their high diffraction efficiency and great damage resistant ability. Generally, a MLD grating is obtained by etching grating structure into the top layer of a MLD coating consisting of alternating layers of highand low-refractive index materials, such as HfO2 and SiO2.
S. Liu et al. / Optics Communications 273 (2007) 290–295
Recently, the highest damage threshold of optimized MLD gratings design is 4.5 J/cm2 for 10 ps pulses at 1053 nm [9]. However, in addition to design of the grating itself, such MLD gratings require designing the properties of the MLD coating [10]. The uniformity and cleanness of the coating with tens of stacks must also be carefully controlled in the fabrication process. Therefore, diffraction grating instead formed in the dielectric bulk material is an alternative approach [11], which can provide high efficiencies in either transmission or reflection. Additionally, the LIDT of a bulk material is much higher than that of its coating. Thus gratings made of bulk fused silica will exhibit stronger damage resistance than the MLD grating with gratings etched into the top layer of a SiO2 or HfO2 film. Highly efficient transmission gratings in fused silica have been realized in the femtosecond fiber CPA system [12]. Recently, a new class of high-efficiency grating based on total internal reflection (TIR) has been invented, which is immersed in a single dielectric material and provides reflection efficiency of over 99% in the Littrow mount [13,14]. This invention has achieved many significant applications, such as polarization-insensitive components [15] and (de)multiplexers [16]. However, until now pulse compression grating based on TIR, to our knowledge, is seldom investigated. In this paper, we apply TIR-gratings to the pulse compressor in the CPA systems. Grating is directly etched into the bottom side of a prism so that the incident light can be successfully coupled into the substrate. Designs of the TIR-grating are presented for either TE (electric field perpendicular to the plane of incidence) or TM (magnetic field perpendicular to the plane of incidence) polarized picosecond pulse at 1053 nm. Besides the 1 order diffraction efficiency, the electric field enhancement ratio in the grating, which is closely related to damage resistance, is used as another criterion to optimize the grating structure. A set of optimum gratings parameters is obtained for TE- and TM-polarized incident light, respectively, which make the compression gratings achieve both high diffraction efficiency and minimal electric field enhancement. Finally, we also discuss the optical and angular spectrums of the optimized TIR compression gratings.
fluid. Alternatively, the grating can be directly etched into the bottom side of a prism, as shown in Fig. 1. Light incident upon the grating can be successfully coupled into the high-index material 1 at the Littrow angle of hi in spite of a small reflection loss at the slope interface. Given the highand low refractive indices, the coupling angle c defined as the angle of light incident on the slope interface can be determined by the angle of hi and the bottom angle of a, n1 c ¼ sin1 sinða hi Þ ð1Þ n2 where n1 and n2 ðn1 > n2 Þ are refractive indices of high and low dielectrics, respectively. Then no light escapes into material 2 regardless of the grating tooth shape with the grating period satisfying [13] n1 >
k > n2 2T
ð2Þ
where k is wavelength of incident light and T is period of grating along x-direction. In a Littrow mount there will be only a single diffracted order that will carry away most of the incident energy and travel back along the path of incident light. Without any other connection between grating and prism, this setup is made of solely bulk material and can be easier to be integrated in the CPA system. Since a prism can be conveniently devised, our attention is focused on optimization of the gratings etched into the bottom side of the prism. In our design, we consider compression gratings for a picosecond optical pulse at the central wavelength of 1053 nm, which are composed solely of fused silica ðn1 ¼ 1:46Þ. The outgoing layer is air ðn2 ¼ 1Þ. Because of narrow bandwidth of the short pulse, reflection performance of the grating at the central wavelength is concerned
2. Theory model In general, the TIR-grating is immersed in a single dielectric material, where light is incident upon the grating from the high-index dielectric to a low-index medium. Since the angle of incidence is required to satisfy the TIR condition at the grating, the light cannot enter (or likely exit) the high-index dielectric from an interface that is parallel to the grating. Therefore, how the incident light is coupled into the high-index material is an important issue for the practical applications of TIR-grating. Maciante et al. used a prism on the backside of the grating substrate, which has been connected by an index matching fluid [15]. However, this setup is probably not suitable for CPA systems due to absence of non-absorbing matching
291
Fig. 1. Structure of the TIR compression grating.
292
S. Liu et al. / Optics Communications 273 (2007) 290–295
in practice. With the grating period satisfying formula (2) and Littrow angle of incidence, no light escapes into air regardless of the grating tooth shape. The exact grating profile will determine the power distribution among the various reflected orders, which can be solved by Fourier modal method (FMM) [17,18]. The optimal design of compression gratings should provide high diffraction efficiency as well as sufficient manufacture latitudes. However, multiphoton ionization induced by strong electric field enhancement in the grating will result in grating breakdown [19]. Therefore, the electric field enhancement is used as another criterion to optimize the design of compression grating. 3. Numerical calculations and discussion In the Littrow mounting, the 1 order diffraction efficiency and internal electric field enhancement in the grating are closely related to the grating parameters, i.e. grating period, depth and duty cycle, and light polarization state. In this section, both the 1 order diffraction efficiency and the electric field enhancement as a function of grating parameters are discussed in detail for TE- and TM-polarized incident light, respectively, in order to obtain the optimal grating structures. 3.1. Diffraction efficiency analysis According to the formula (2), grating period is limited to be 361–527 nm long. The 1 order diffraction efficiency as a function of grating depth and period for TE-polarized wave is shown in Fig. 2a, where duty cycle is 0.5. It can be seen that there is more than one region where the 1 order diffraction efficiency is higher than 0.95 for grating depths up to 2 lm. However, gratings with deep grooves bring much more difficulties in the fabrication process [20]. In our design, grating depth is chosen to be less than a micron deep. For example, a TIR-grating with depth of 350 nm is preferred in Fig. 2a, which provides sufficient latitude for period ranging from 460 to 525 nm. Fig. 2b shows the relation between the 1 order diffraction efficiency, grating depth and duty cycle when grating period is 450 nm. It can be seen that the latitude of duty cycle becomes narrower with the increase of grating depth from 0 to 1 lm. Therefore, TIR grating with a relatively shallow depth is also beneficial to extend the latitude of its duty cycle. In addition, calculation results show that in these shallow-depth regions, the maximal grating depth and duty cycle along with minimal grating depth and duty cycle reduce with the increase of grating period from 361 to 527 lm, which results in the reduction of grating aspect ratio (groove depth divided by groove linewidth) and thus facilitates grating fabrication. The 1 order diffraction efficiency as a function of grating depth and period for TM-polarized wave is shown in Fig. 3a, where duty cycle is 0.5. Similarly, grating depth less than a micron deep is preferred in order to bring facilities
Fig. 2. (a) The 1 order diffraction efficiency of the TIR gratings with duty cycle of 0.5 as a function of grating depth and period. (b) The 1 order diffraction efficiency of the TIR gratings with period of 450 nm under the illumination of TE-polarized light at Littrow angle as a function of grating depth and duty cycle.
in the fabrication process. Fig. 3b shows the relation between the 1 order diffraction efficiency, grating depth and period when period is 450 nm. It can be seen that these band regions where 1 order diffraction efficiency higher than 0.95 show similarities to those for TE-polarized wave shown in Fig. 2b. However, for a same period, high-efficiency TIR gratings for TM-polarized wave require deeper values of grating depth, which also have bigger latitude. For example, a TIR grating with depth being 393– 540 nm in Fig. 3b achieves sufficient latitude of duty cycle to make the 1 order diffraction efficiency higher than 0.95. Also, the increase of grating period results in reduction of grating aspect ratio. 3.2. Near field analysis Further insight into the near-field optical distribution in the high-efficiency TIR-gratings, characterized by the distribution of the internal transverse electric field, is promising to improve the LIDT value of the gratings. Fig. 4a
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Fig. 3. (a) The 1 order diffraction efficiency of the TIR gratings with duty cycle of 0.5 as a function of grating depth and period. (b) The 1 order diffraction efficiency of the TIR gratings with period of 450 nm under the illumination of TM-polarized light at Littrow angle as a function of grating depth and duty cycle.
shows the amplitude of electric field in the TIR gratings with diffraction efficiency of 0.988 under the illumination of TE-polarized wave at Littrow angle of 53.26°. The grating period, depth and duty cycle are 450 nm, 390 nm and 0.5, respectively. Fig. 4b shows the amplitude of electric field in the TIR gratings with diffraction efficiency of 0.99 under the illumination of TM-polarized wave at Littrow angle of 53.26°. The grating period, depth and duty cycle are 450 nm, 450 lm and 0.5, respectively. It can be seen that most of the light for both TE- and TM-polarized states are reflected back into the fused silica by the TIR grating and only evanescent waves penetrate a small distance into air. Moreover, electric field enhancement for TE-polarized light over two times higher than the incident field appears in the gratings ridge and bulk fused silica. In contrast, electric field in the grating and bulk fused silica for TM-polarized wave only exhibits an enhancement less than two times smaller than the incident wave. The maximum amplitude is just localized in the air gap, as shown in Fig. 4b. The different transverse profiles between the two polarized waves can be understood by their different boundary conditions
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Fig. 4. Amplitude distributions of internal electric field of a TIR grating for the TE-polarized incident at angle of 53.26° (a) and of another TIR grating for the TM-polarized incident at the same angle (b).
imposed on the near field by the gratings [21]. Multiphoton ionization induced by strong electric field enhancement in the fused silica will result in grating breakdown. Therefore, in order to improve the resistant ability, such high-efficiency gratings should be used under the illumination of TM-polarized wave or under the illumination of TE-polarized wave generating low electric field enhancement in the gratings. The grating parameters including period, depth and duty cycle are also responsible for the electric field enhancement in the grating under the illumination of TE-polarized light. Dependence of square of electric field enhancement ratio on grating depth, duty cycle and period is shown in Fig. 5a, b and c, respectively. From Fig. 5a and b, it is seen that electric field enhancement shows a slight dependence on both grating depth and duty cycle for different values of period. Whereas square of the electric field enhancement ratio increases rapidly with the increasing of the grating period from 380 nm to 520 nm, as shown in Fig. 5c. The maximal enhancement ratio is 9.3, which may bring catastrophic damage to the TIR gratings used in the highenergy laser system. Therefore, in order to reduce field enhancement within the grating, a short period should be
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Fig. 5. Square of electric field enhancement ratio as function of grating depth (a), duty cycle (b) and period (c).
preferred in the design of such compression gratings under illumination of TE-polarized wave. 3.3. Optimization results of TIR-gratings A TIR-grating used in the CPA system requires providing high diffraction efficiency, low field enhancement in the
grating and sufficient manufacturing latitude. From the numerical analysis in Sections 3.1 and 3.2, it is not difficult to find that a TIR-grating used under the illumination of TM-polarized wave can easily satisfy these requirements. An optimized TIR compression grating with period of 500 nm, depth of 450 nm and duty cycle of 0.35 is obtained for TM-polarized incident light, which achieves high diffraction efficiency of 0.999, aspect ratio of 1.38 and sufficient manufacturing latitude (depth of 450 ± 50 nm and duty cycle of 0.35 ± 0.1). However, for a TIR-grating under illumination of TE-polarized wave, selecting a short grating period effectively reduces its internal field enhancement while greatly increases the grating aspect ratio. For example, a TIR-grating with period of 380 nm, depth of 750 and duty cycle of 0.75 can achieve diffraction efficiency of 0.997 and the lowest square of electric field enhancement ratio being 4.3. But, to our knowledge, its aspect ratio being about 7.9 has already exceeded the maximum value of 3.37, which has been realized by Wang et al [20]. Therefore, a trade-off should be required between the selections of grating period and the realizable aspect ratio of grating. Given the aspect ratio of 3 in our design, an optimized grating with period of 410 nm, depth of 640 ± 20 nm and duty cycle of 0.47 ± 0.02 is obtained for TE-polarized incident light, which achieves high diffraction efficiency of 0.998 and square of field enhancement ratio of 5.7. For application in the pulse compressor, spectral dependence of the diffraction efficiency is of particular concern. The optical spectrum curves of the two optimized TIR compression gratings are shown in Fig. 6a, where the solid line represents the optical spectrum of the optimized grating for TE-polarized incident light at Littrow angle of 61.59° and the dashed line represents the optical spectrum of the optimized gratings for TM-polarized incident light at Littrow angle of 46.16°. It is found that the optimized TIR grating under illumination of TM-polarized light exhibits a diffraction efficiency of higher than 0.95 for 42 nm spectral bandwidth (within the range 1036– 1077 nm) around the central wavelength of 1053 nm. In contrast, the other optimized grating under illumination of TE-polarized light exhibits a diffraction efficiency of higher than 0.95 only for 23 nm bandwidth (within the range 1040–1063 nm), which is inferior to that of the former. The dependence of the diffraction efficiency on the incident angle is also shown in Fig. 6b, where the dashed line represents angular spectrum of the optimized gratings for TM-polarized incident light of 1053 nm and the solid line represents angular spectrum of the other optimized gratings for TE-polarized incident light of 1053 nm. It is seen that both gratings exhibit maxima of diffraction efficiency at Littrow angle being 46.16° and 61.59°, respectively. Moreover, the TIR grating for TE-polarized light exhibits diffraction efficiency higher than 0.95 for 5° bandwidth. The other TIR grating for TM-polarized light also exhibits diffraction efficiency higher than 0.95 for 2.4°
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4. Conclusion We have demonstrated the process of designing the compression gratings based on TIR for picosecond pulse laser at the central wavelength of 1053 nm. The setup is devised by directly etching gratings into the bottom side of a prism so that light can successfully enter (or exit) the compression grating. By using Fourier modal method, both the 1 order diffraction efficiency and internal electric field enhancement as functions of constructive parameters, including grating period, depth and duty cycle, are discussed in detail for both TE- and TM-polarized wave in order to obtain the optimal grating structure. With the criterion of high diffraction efficiency, low internal field enhancement and sufficient manufacturing latitude, TIR-gratings with optimized structure is obtained for two polarized waves, respectively. In comparison, such design is more favorably used for TM-polarized incident light. With compact setup, this TIR grating could be a good alternative of compression gratings. Acknowledgements We acknowledge the financial support from National Natural Science Foundation of China (Grant No. 10376040). References
Fig. 6. Optical spectrum in (a) and angular spectrum in (b) of two optimized grating respectively under the illumination of two-polarized light.
bandwidth. The angular dependence of diffraction efficiency will permit adjusting the incident angle of optical pulse in practice. Given the angle incident on the grating, the bottom angle a of a prism for grating fabrication can be conveniently evaluated, which is greater than the value of hi. The exact value of a can be adjusted in the real optical system. For example, a prism with bottom angle of 75° can be used to fabricate the above two optimized TIR-gratings. By using the formula (2), the coupling angle c is 19.79° for TE-polarized wave and 44.77° for TM-polarized wave, respectively. Unfortunately, energy losses by Fresnel reflection are generated at the prism–air-interface, which are 4.1% for TE incident wave and 0.7% for TM incident wave. These losses are unwanted in the CPA system, especially for TE incident wave, which can be avoided by adding an antireflection dielectric coating on the slope side of the prism.
[1] D. Strickland, G. Mourou, Opt. Commun. 56 (1985) 219. [2] Sterling Backus, Charles G. Durfee III, Margaret M. Murnane, et al., Rev. Sci. Instrum. 69 (3) (1998) 1207. [3] M.D. Perry, R.D. Boyd, J.A. Britten, et al., Opt. Lett. 20 (8) (1995) 940. [4] L. Li, J. Hirsh, Opt. Lett. 20 (11) (1995) 1349. [5] B.W. Shore, M.D. Perry, J.A. Britten, et al., J. Opt. Soc. Am. A 14 (5) (1997) 1124. [6] Karl Hehl, Joerg Bischoff, Appl. Opt. 38 (30) (1999) 6257. [7] A.J. Waddie, M.J. Thomson, M.R. Taghizadeh, Opt. Lett. 30 (9) (2005) 991. [8] R.D. Boyd, J.A. Britten, D.E. Decker, et al., Appl. Opt. 34 (10) (1995) 1697. [9] I. Jovanovic, C.G. Brown, B.C. Stuart, et al., SPIE 5647 (2005) 34. [10] S. Liu, J. Ma, Z. Shen, et al., Appl. Surf. Sci. 253 (7) (2006) 3642. [11] H.T. Nguyen, B.W. Shore, S.J. Bryan, et al., Opt. Lett. 22 (3) (1997) 142. [12] Tina Clausnitzer, Jens Limpert, Karsten Zollner, et al., Appl. Opt. 42 (34) (2003) 6934. [13] J.R. Marciante, D.H. Raguin, Opt. Lett. 29 (6) (2004) 542. [14] L. Eisen, M.A. Golub, A.A. Friesem, Opt. Commun. 261 (2006) 13. [15] J.R. Marciante, J.I. Hirsh, D.H. Raguin, et al., J. Opt. Soc. Am. A 22 (2) (2005) 299. [16] Y. Zhang, C. Zhou, J. Opt. Soc. Am. A 22 (2) (2005) 331. [17] L. Li, J. Opt. Soc. Am. A 10 (12) (1993) 2581. [18] S. Liu, Z. Shen, W. Kong, et al., Opt. Commun. 267 (2006) 50. [19] B.C. Stuart, M.D. Feit, S. Herman, et al., Phys. Rev. B 53 (4) (1996) 1749. [20] S. Wang, C. Zhou, Y. Zhang, et al., Appl. Opt. 45 (12) (2006) 2567. [21] W. Nakagawa, R. Tyan, P. Sun, et al., J. Opt. Soc. Am. A 18 (5) (2001) 1072.