Abstract

We report on a kind of self-assembled volume grating in silica glass induced by tightly focused femtosecond laser pulses. The formation of the volume grating is attributed to the multiple microexplosion in the transparent materials induced by the femtosecond pulses. The first order diffractive efficiency is in dependence on the energy of the pulses and the scanning velocity of the laser greatly, and reaches as high as 30%. The diffraction pattern of the fabricated grating is numerically simulated and analyzed by a two dimensional FDTD method and the Fresnel Diffraction Integral. The numerical results proved our prediction on the formation of the volume grating, which agrees well with our experiment results.

© 2010 OSA

Femotosecond laser has attracted a lot of attentions in the past few years for its great potential application in damage-free processing, microfabrication and micromachining of multifunctional structures in dielectrics [15]. When a femtosecond laser beam is focused into transparent glasses, there would be a line of self-assembly periodic nanovoids generated spontaneously along the propagation direction of the laser beam. The entire length of the void structure and the shapes of the generated voids could be controlled by varying the laser parameters. Due to the potential application and simplicity for fabricating the three-dimensional photonic crystals in transparent materials by femtosecond laser pulses by one-step method, great efforts have been launched into this issue. So far, there are kinds of researches reported to form a submicrometer-sized void array in transparent samples, such as borosilicate glass [6], CaF2 crystals [7], SrTiO3 crystal [8], fused silica glass [9], PMMA [10,11] (polymethyl methacrylate) and so on.

It would be very significant to increase the diffraction efficiency of the diffractive optical elements. According to the grating parameterQ=2πλd/(nΛ2) defined by Kogelnik [12], where Λ is the period of the gratings, d is the thickness of the diffraction grating, n is the refractive index, λ is the wavelength of the incident beam, in order to obtain larger diffraction efficiency, it would be intuitional to find ways to enlarge the thickness of the volume grating or reduce its period [13]. In this paper, we report on self-assembled higher efficiency volume grating inside silica glass induced by tightly focused femtosecond laser pulses. The formation of the volume grating is attributed to the multiple microexplosion in the direction of the laser propagation in transparent materials induced by the femtosecond pulses. The measured first order diffractive efficiency (FODE) of the induced volume grating depends on the energy of the laser pulses and the scanning velocity heavily. We provide concrete physical analysis and numerical simulation through a two dimensional FDTD method to explain the transformation of the FODE of the gratings fabricated with different pulse energy and scanning speeds. The simulated results could verify our explanation on the experimental results very well.

In our experiments, a regeneratively amplified Ti:sapphire laser system (Coherent. Co.) was used, which delivered pulses with a duration of 120 fs, a center wavelength at 800 nm and a repetition rate of 1~1000 Hz. The glass sample was commercially available fused silica glass and was polished with size of 8mm×5mm×4mm. The sample was mounted on a computer-controlled XYZ translation micro-stage with a resolution of 0.1μm. The laser beam with a diameter of 6mm in Gaussian profile was focused inside the glass sample by a 50× objective lens with numerical aperture of 0.80. And the femtosecond laser beam is focused to 100μm beneath the surface of the sample. The energy of the pulse can be turned by a NDF (neutral density filter). A shutter system was used to control the number of the deposited pulses selectively. The process of the fabrication the experiments were carried out at room temperature and observed by a CCD camera in real-time. The schematic illustration is shown in Fig. 1(a) .

 

Fig. 1 (a) the sketch of the experimental setup for the generation of periodic voids, Side view of the voids array induced in silica glass by femtosecond laser pulses,(b) different number of with pulse energy of 20μJ, (c) single pulse with different energies, (d) double pulses with different energies. The scale bars in the Figure are 20μm. The arrows show the direction of the pulses propagation, and the lop lines express the focus place.

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As depicted in Fig. 1(b-d), line style structures induced by femtosecond laser pulses consisting of a group of voids and a filament could be observed along the incident direction (arrow direction) inside samples. It is observed that the induced structure is enlarged with the increase of the number of the pulses and the corresponding void group becomes more regular (see Fig. 1(b)). Similarly, with the increase of the pulse energy, the length of the line structure induced by both single pulse (see Fig. 1(c)) and the double pulses (see Fig. 1(d)) are also becoming longer. It is notable to mention that when the pulse energy is decreased to 3.80μJfor the single pulse interacting, the voids almost disappear, with a long sharp filament left, as is depicted in Fig. 1(c). The formation of the filament in the silica glass sample is one of the most fundamental nonlinear optical phenomena and results from the dynamic balance between self-focusing arising from an increase in the refractive index and self-defocusing arising from diffraction or plasma formation [14] [15].

To get a volume grating, we mount the sample onto a computer-controlled XYZ translation micro-stage. And the femtosecond laser beam is focused to 100μm beneath the surface of the sample by a 50× objective lens with numerical aperture of 0.80. The pulse energy could be turned by the ND shown in Fig. 1. By moving the computer-controlled XYZ translation micro-stage at different scanning speed and different pulse energy, a series of grating could be fabricated in the silica glass sample. The corresponding top view and side view of the grating are shown in Fig. 2(a) and (b) . The diffraction pattern of the fabricated grating could be obtained by a collimated He-Ne laser beam normally incident on the fabricated grating, and two typical pictures are shown in Fig. 2(c) and (d) for keeping the scanning speed at v = 1000μm/sbut different pulse powers E = 1.176μJ and E = 7.38μJ respectively. It is apparent that the first diffraction of the fabricated grating is more apparent for larger pulse energies.

 

Fig. 2 the fabricated grating: (a) top view, (b) side view; diffraction pattern of the grating obtained under different conditions: (c) E = 1.76μJ, v = 1000μm/s, (d) E = 7.38μJ, v = 1000 μm/s

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We also measure the transmittance efficiency and diffraction efficiency of the fabricated gratings by laying a power meter at the wavelength of 632.8nm behind the sample for detecting the diffractive He-Ne laser. Here, we defined the transmittance efficiency (TE) to be the ratio between the intensity of whole transmittance and that of incident beam. The variation tendency of the transmittance efficiency with the different pulse of energy and scanning velocity is shown in Fig. 3(a) . We found that the transmittance efficiency decreases for higher pulse energy and slower scanning speed, which can be explained by the fact that the absorption and scatterance are enhanced because of stronger microexplosion and the more generation of color centers. We also defined the first order diffraction efficiency (FODE) and normalized first order diffraction efficiency (NFODE) to be the ratio of the intensity of the first-order diffraction to the incident intensity and the transmitted intensity respectively. And the corresponding variation tendencies are shown in Fig. 3(b) and (c). With the increase of the pulse energy, the FODE would enlarge to a maximum firstly, and then decrease gradually depicted in Fig. 3(b). However, NFODE nearly increases all the time as shown in Fig. 3(c) expect for situations with too large energies. Hirono [10] and Sowa [11] have also fabricated similar thick gratings in the PMMA by femtosecond laser pulses; however, the first order diffractive efficiency (0.63%) is comparatively lower compared to our experimental results. Hirono [10] investigated that the diffraction efficiency of the fabricated gratings by femtosecond laser irradiation in PMMA samples was increased by more than an order of magnitude by subsequent heating of the samples at a temperature, which was attributed to an increase of refractive index changes (RIC) after heating because of the volume contraction of the irradiated area. The annealing treatment should be noneffective for increasing the RIC to the sample of the silica glass, because the formed color center is the main reason of the RIC of the irradiated dark region by the femtosecond laser pulses, and the color center would vanish after annealing treatment [16].

 

Fig. 3 (a) Variational tendency of the transmittance efficiency (TE) with the different pulse of energy and scanning velocity, (b) Variational tendency of the first order diffraction efficiency (FODE) with the different pulse of energy and scanning velocity, (c) Variational tendency of the normalized first order diffraction efficiency (NFODE) with the different pulse of energy and scanning velocity

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Now we give a physical description of the phenomenon observed above: according to Kogelnik’s equation, with the increase of the pulse energy, the depth of the fabricated gratings is increased, so does the first order diffraction [13]. However, the absorption and scatterance are also enhanced at the same time. When the former is dominated, the FODE would be increasing; while the posterior is dominated, the FODE would be decreasing, as shown in Fig. 4(b) . On the other hand, if we neglect the influence of the absorption and scatterance, the NFODE should be always increasing as is depicted in Fig. 4(c). For different scanning speed at the pulse energy of 1.760μJ, it is observed that the FODE decreases with increasing scanning speed, and the maximum is only 3.2% for v = 100μm/s. However, when the pulse energy is increased to 3.520μJ, the maximum of the FODE reaches 11.6% for v = 500μm/s, and the FODE at v = 100μm/s and v = 200μm/s are smaller compared to v = 500μm/s. For the pulse energy of 7.380μJ, the biggest diffraction efficiency reaches 15.1% for v = 1000μm/s, but the others decrease to a lower level compared to that of E = 3.520μJ. Going on increasing the pulse energy to 12.83μJ, all of the FODE decrease to a relative lower level compared to E = 7.380μJ.

 

Fig. 4 (a) A simple sketch of our numerical method, (b) A typical example of the distribution of the amplitude of the electric field inside the sample. (c) Distribution of the amplitude of the Electric filed in the silicate glass sample.

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We deem that when the pulse energy is 1.760μJ, the induced microexplosion and the darkening region is comparatively small, the fabricated grating is thicker for slower scanning speed and gets larger FODE. See Fig. 3(c), when the pulse energy is increased to 3.520μJ, the depth of the gratings is bigger and the FODE is increased at the same time. Meanwhile, the microexplosion and the darkening region are increased, which are bigger for slower scanning speed, so the FODE at v = 100μm/s is smaller than at v = 500μm/s and v = 200μm/s. For v = 1000μm/s, the resulted grating is not so compacted compared with other scanning speed, so the FODE is smaller. When the pulse energy is increased to 7.380μJ, the resulted gratings could be compacted enough, so the FODE reaches maximum too. The FODE at v = 500μm/s, v = 200μm/s, and v = 100μm/sis decreasing at E = 7.380μJ because of the enhancement of the microexplosion and the darkening region. When pulse energy reaches 12.83μJ, the resulted grating is too compacted that the absorption and scatterance dominate, so all of the FODE is decreased to a lower level. Moreover, the order of the FODE is in a reverse order compared to that of E = 1.760μJ, just because of the different dominant mechanism for lower and higher energy.

In order to verify our explanation on experimental results, we simulate the diffraction process of the induced gratings by the combined methods of two dimensional FDTD and Fresnel diffraction, and the corresponding sketch is shown in Fig. 4(a-c). In our model, we neglect the optical absorption and the scale of the simulation district is chosen to be a 200μm×70μmregion, the sample is assumed to be uniform along the direction of Y axes. The grating period is set to be dx=20μmaccording to our experimental reality, and the refractive indexes of silica glass and the filaments are ns=1.5and nf=n0+δn respectively, where δnis the refractive index change (RIC) induced by the laser. The refractive index of the background (air) isnb=1.0. According to our experimental results as depicted in Fig. 1(b-d), the filaments could be simplified as composite regions combined by a rectangle and followed by a triangle (see Fig. 4(b)), and the width of the rectangle region is drect. In our simulation, the sample is divided into four regions labeled A, B, C and D respectively, and the lengths (along Z axes) of regions A and B are set as hA=5μm and hB=10μm respectively. We could change the length of region C (hC), the RIC δnand the width of the rectangle regiondrect, so as to match different conditions in our experiment. We use a Gauss form continuous source in front of the sample with a twist ω0=40μmand wavelength λ0=0.6328μm. The(x,y) position of the source waist is marked with green line in Fig. 4(b).

To obtain the far field diffraction, we extract the amplitude and phase distribution of the electric field in the plane behind the silica sample to beE(x)=A(x)eiϕ(x), and assume the scale of the sample in the third dimension to be constant. Then the diffraction pattern of the far field in the (x',y') plane can be solved using the Fresnel Diffraction Integral as below:

E(x')=Cjλ+E(x)exp(ikr)rcosθ+12dx,
wherer=d2+(xx')2, θ=arccos(d/r), k is the wave-number and C is an constant introduced by the integration along the third dimension, respectively. The distance between the two planes is fixed to d=20cm, which is in consistent with our experiment. We observe the distribution of the electric field in the (x',y') plane along the x' axis.

The numerical results are shown in Fig. 5 . After more than 20000 iterations, we could attain stationary distribution of the electric field inside the silicate glass sample as depicted in Fig. 4(c) (athC=40μm,δn=0.002,drec=10μm). The far field diffraction patterns of the given volume grating at which drec=10μm,δn=0.007 with different depths (hc=5,20,40μm) are depicted in Fig. 5(a), which shows that with the increase of the depth of the grating, the diffraction efficiency is increased at the same time, which verifies our physical explanation and agrees with our experimental results very well as depicted in Fig. 3(c).

 

Fig. 5 The far field diffraction of the distribution of Electric filed along the x'direction, (a) drec=10μm,δn=0.007and the three curves correspond to different depths of the induced gratings: hc= 5(solid red), 20(dotted blue) and 40μm (dashed black) respectively; (b) hc=40μm, δn=0.007and the three curves correspond to different width of the filaments: drec= 5(solid red), 10(dotted blue) and 15μm (dashed black) respectively; (c) drec=10μm,hc=40μmand the three curves correspond to different RICs: δn=0.002 (solid red), 0.007 (dotted blue) and 0.012(dashed black) respectively.

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With the increase of the pulse energy or decrease of the scanning speed, the fabricated filament width (drec) of the induced grating is increased obviously as depicted in Fig. 1(b-d). So we also give a numerical simulation on the far field diffraction patterns of the volume grating by keepinghc=20μm and δn=0.007but with different filament widths (drec=5,10,15μm) are depicted in Fig. 5(b), which shows that the first order diffraction efficiency (FODE) is increased with the increase of the filament widths. In other words, increasing the incident pulse energy (leading to the increase of the filament width and the depth of the induced grating) would enhance the FODE, when taking no account of the optical absorption.

We also simulated FODE depending on the different RICs (δn=0.002,0.007,.0012) by keepingdrec=10μm,hc=40μm. From the simulated results as shown in Fig. 5(c), we can observe that the RIC would affect the FODE strongly. Especially, forδn=0.012, the FODE is higher than the zero order diffraction efficiency (ZODE) for the normally incident beam. In other words, the energy of the zero order diffraction could transfer to the first and higher order diffraction. Therefore, if we can accurately control the RIC and hc inside the glass sample induced by femtosecond laser, we could control the diffraction pattern very well.

In conclusion, we have fabricated the volume grating in silica glass by tightly focused femtosecond laser pulses. The femtosecond pulses induced multiple microexplosion in the transparent materials have been considered as the reasons for the formation of volume grating. The first order diffraction efficiency of the induced grating is measured by a collimated He-Ne laser and the highest NFODE reaches to nearly 30%. We have also constructed a grating model and attained the simulated results by using a two dimensional FDTD method and Fresnel Diffraction. The numerical simulated results verified that our physical viewpoint on the formation of the volume grating is correct and agree well with our experiment results.

Acknowledgement

This work was supported by the national science foundation of China under Grant No. 10904072, the National Basic Research Program of China under grant 2006CB302901 and the development program for outstanding young teachers in Harbin Institute of Technology, HITQNJS. 2009. 033.

References and links

1. Y. Li, Y. Dou, R. An, H. Yang, and Q. Gong, “Permanent computer-generated holograms embedded in silica glass by femtosecond laser pulses,” Opt. Express 13(7), 2433–2438 (2005). [CrossRef]   [PubMed]  

2. S. Qu, C. Zhao, Q. Zhao, J. Qiu, C. Zhu, and K. Hirao, “One-off writing of multimicrogratings on glass by two interfered femtosecond laser pulses,” Opt. Lett. 29(17), 2058–2060 (2004). [CrossRef]   [PubMed]  

3. Z. Guo, S. Qu, L. Ran, Y. Han, and S. Liu, “Formation of two-dimensional periodic microstructures by a single shot of three interfered femtosecond laser pulses on the surface of silica glass,” Opt. Lett. 33(20), 2383–2385 (2008). [CrossRef]   [PubMed]  

4. T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interference of femtosecond pulses,” Appl. Phys. Lett. 82(17), 2758–2760 (2003). [CrossRef]  

5. Z. Guo, S. Qu, and S. Liu, “Generating optical vortex with computer-generated hologram fabricated inside glass by femtosecond laser pulses,” Opt. Commun. 273(1), 286–289 (2007). [CrossRef]  

6. S. Kanehira, J. H. Si, J. R. Qiu, K. Fujita, and K. Hirao, “Periodic nanovoid structures via femtosecond laser irradiation,” Nano Lett. 5(8), 1591–1595 (2005). [CrossRef]   [PubMed]  

7. X. Hu, B. Qian, P. Zhang, X. Wang, L. Su, J. Qiu, and C. Zhu, “Self-organized microvoid array perpendicular to the femtosecond laser beam in CaF2 crystals,” Laser Phys. Lett. 5(5), 394–397 (2008). [CrossRef]  

8. J. Song, X. Wang, J. Xu, H. Sun, Z. Xu, and J. Qiu, “Microstructures induced in the bulk of SrTiO3 crystal by a femtosecond laser,” Opt. Express 15(5), 2341–2347 (2007). [CrossRef]   [PubMed]  

9. E. Toratani, M. Kamata, and M. Obara, “Self-fabrication of void array in fused silica by femtosecond laser processing,” Appl. Phys. Lett. 87(17), 171103 (2005). [CrossRef]  

10. S. Hirono, M. Kasuya, K. Matsuda, Y. Ozeki, K. Itoh, H. Mochizuki, and W. Watanabe, “Increasing diffraction efficiency by heating phase gratings formed by femtosecond laser irradiation in poly (methyl methacrylate),” Appl. Phys. Lett. 94(24), 241122 (2009). [CrossRef]  

11. S. Sowa, W. Watanabe, J. Nishii, and K. Itoh, “Filamentary cavity formation in poly (methyl methacrylate) by single femtosecond pulse,” Appl. Phys., A Mater. Sci. Process. 81(8), 1587–1590 (2005). [CrossRef]  

12. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

13. Q. Z. Zhao, J. R. Qiu, X. W. Jiang, E. W. Dai, C. H. Zhou, and C. S. Zhu, “Direct writing computer-generated holograms on metal film by an infrared femtosecond laser,” Opt. Express 13(6), 2089–2092 (2005). [CrossRef]   [PubMed]  

14. J. Song, X. Wang, X. Hu, Y. Dai, J. Qiu, Y. Cheng, and Z. Xu, “Formation mechanism of self-organized voids in dielectrics induced by tightly focused femtosecond laser pulses,” Appl. Phys. Lett. 92(9), 092904 (2008). [CrossRef]  

15. A. Brodeur, F. Ilkov, and S. Chin, “Beam filamentation and the white light continuum divergence,” Opt. Commun. 96(3-4), 193–198 (1996). [CrossRef]  

16. Q. Zhao, J. Qiu, X. Jiang, C. Zhao, and C. Zhu, “Mechanisms of the refractive index change in femtosecond laser-irradiated Au3+-doped silicate glasses,” J. Appl. Phys. 129(12), 7122 (2004). [CrossRef]  

References

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  1. Y. Li, Y. Dou, R. An, H. Yang, and Q. Gong, “Permanent computer-generated holograms embedded in silica glass by femtosecond laser pulses,” Opt. Express 13(7), 2433–2438 (2005).
    [CrossRef] [PubMed]
  2. S. Qu, C. Zhao, Q. Zhao, J. Qiu, C. Zhu, and K. Hirao, “One-off writing of multimicrogratings on glass by two interfered femtosecond laser pulses,” Opt. Lett. 29(17), 2058–2060 (2004).
    [CrossRef] [PubMed]
  3. Z. Guo, S. Qu, L. Ran, Y. Han, and S. Liu, “Formation of two-dimensional periodic microstructures by a single shot of three interfered femtosecond laser pulses on the surface of silica glass,” Opt. Lett. 33(20), 2383–2385 (2008).
    [CrossRef] [PubMed]
  4. T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interference of femtosecond pulses,” Appl. Phys. Lett. 82(17), 2758–2760 (2003).
    [CrossRef]
  5. Z. Guo, S. Qu, and S. Liu, “Generating optical vortex with computer-generated hologram fabricated inside glass by femtosecond laser pulses,” Opt. Commun. 273(1), 286–289 (2007).
    [CrossRef]
  6. S. Kanehira, J. H. Si, J. R. Qiu, K. Fujita, and K. Hirao, “Periodic nanovoid structures via femtosecond laser irradiation,” Nano Lett. 5(8), 1591–1595 (2005).
    [CrossRef] [PubMed]
  7. X. Hu, B. Qian, P. Zhang, X. Wang, L. Su, J. Qiu, and C. Zhu, “Self-organized microvoid array perpendicular to the femtosecond laser beam in CaF2 crystals,” Laser Phys. Lett. 5(5), 394–397 (2008).
    [CrossRef]
  8. J. Song, X. Wang, J. Xu, H. Sun, Z. Xu, and J. Qiu, “Microstructures induced in the bulk of SrTiO3 crystal by a femtosecond laser,” Opt. Express 15(5), 2341–2347 (2007).
    [CrossRef] [PubMed]
  9. E. Toratani, M. Kamata, and M. Obara, “Self-fabrication of void array in fused silica by femtosecond laser processing,” Appl. Phys. Lett. 87(17), 171103 (2005).
    [CrossRef]
  10. S. Hirono, M. Kasuya, K. Matsuda, Y. Ozeki, K. Itoh, H. Mochizuki, and W. Watanabe, “Increasing diffraction efficiency by heating phase gratings formed by femtosecond laser irradiation in poly (methyl methacrylate),” Appl. Phys. Lett. 94(24), 241122 (2009).
    [CrossRef]
  11. S. Sowa, W. Watanabe, J. Nishii, and K. Itoh, “Filamentary cavity formation in poly (methyl methacrylate) by single femtosecond pulse,” Appl. Phys., A Mater. Sci. Process. 81(8), 1587–1590 (2005).
    [CrossRef]
  12. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  13. Q. Z. Zhao, J. R. Qiu, X. W. Jiang, E. W. Dai, C. H. Zhou, and C. S. Zhu, “Direct writing computer-generated holograms on metal film by an infrared femtosecond laser,” Opt. Express 13(6), 2089–2092 (2005).
    [CrossRef] [PubMed]
  14. J. Song, X. Wang, X. Hu, Y. Dai, J. Qiu, Y. Cheng, and Z. Xu, “Formation mechanism of self-organized voids in dielectrics induced by tightly focused femtosecond laser pulses,” Appl. Phys. Lett. 92(9), 092904 (2008).
    [CrossRef]
  15. A. Brodeur, F. Ilkov, and S. Chin, “Beam filamentation and the white light continuum divergence,” Opt. Commun. 96(3-4), 193–198 (1996).
    [CrossRef]
  16. Q. Zhao, J. Qiu, X. Jiang, C. Zhao, and C. Zhu, “Mechanisms of the refractive index change in femtosecond laser-irradiated Au3+-doped silicate glasses,” J. Appl. Phys. 129(12), 7122 (2004).
    [CrossRef]

2009 (1)

S. Hirono, M. Kasuya, K. Matsuda, Y. Ozeki, K. Itoh, H. Mochizuki, and W. Watanabe, “Increasing diffraction efficiency by heating phase gratings formed by femtosecond laser irradiation in poly (methyl methacrylate),” Appl. Phys. Lett. 94(24), 241122 (2009).
[CrossRef]

2008 (3)

J. Song, X. Wang, X. Hu, Y. Dai, J. Qiu, Y. Cheng, and Z. Xu, “Formation mechanism of self-organized voids in dielectrics induced by tightly focused femtosecond laser pulses,” Appl. Phys. Lett. 92(9), 092904 (2008).
[CrossRef]

X. Hu, B. Qian, P. Zhang, X. Wang, L. Su, J. Qiu, and C. Zhu, “Self-organized microvoid array perpendicular to the femtosecond laser beam in CaF2 crystals,” Laser Phys. Lett. 5(5), 394–397 (2008).
[CrossRef]

Z. Guo, S. Qu, L. Ran, Y. Han, and S. Liu, “Formation of two-dimensional periodic microstructures by a single shot of three interfered femtosecond laser pulses on the surface of silica glass,” Opt. Lett. 33(20), 2383–2385 (2008).
[CrossRef] [PubMed]

2007 (2)

J. Song, X. Wang, J. Xu, H. Sun, Z. Xu, and J. Qiu, “Microstructures induced in the bulk of SrTiO3 crystal by a femtosecond laser,” Opt. Express 15(5), 2341–2347 (2007).
[CrossRef] [PubMed]

Z. Guo, S. Qu, and S. Liu, “Generating optical vortex with computer-generated hologram fabricated inside glass by femtosecond laser pulses,” Opt. Commun. 273(1), 286–289 (2007).
[CrossRef]

2005 (5)

S. Kanehira, J. H. Si, J. R. Qiu, K. Fujita, and K. Hirao, “Periodic nanovoid structures via femtosecond laser irradiation,” Nano Lett. 5(8), 1591–1595 (2005).
[CrossRef] [PubMed]

E. Toratani, M. Kamata, and M. Obara, “Self-fabrication of void array in fused silica by femtosecond laser processing,” Appl. Phys. Lett. 87(17), 171103 (2005).
[CrossRef]

S. Sowa, W. Watanabe, J. Nishii, and K. Itoh, “Filamentary cavity formation in poly (methyl methacrylate) by single femtosecond pulse,” Appl. Phys., A Mater. Sci. Process. 81(8), 1587–1590 (2005).
[CrossRef]

Q. Z. Zhao, J. R. Qiu, X. W. Jiang, E. W. Dai, C. H. Zhou, and C. S. Zhu, “Direct writing computer-generated holograms on metal film by an infrared femtosecond laser,” Opt. Express 13(6), 2089–2092 (2005).
[CrossRef] [PubMed]

Y. Li, Y. Dou, R. An, H. Yang, and Q. Gong, “Permanent computer-generated holograms embedded in silica glass by femtosecond laser pulses,” Opt. Express 13(7), 2433–2438 (2005).
[CrossRef] [PubMed]

2004 (2)

Q. Zhao, J. Qiu, X. Jiang, C. Zhao, and C. Zhu, “Mechanisms of the refractive index change in femtosecond laser-irradiated Au3+-doped silicate glasses,” J. Appl. Phys. 129(12), 7122 (2004).
[CrossRef]

S. Qu, C. Zhao, Q. Zhao, J. Qiu, C. Zhu, and K. Hirao, “One-off writing of multimicrogratings on glass by two interfered femtosecond laser pulses,” Opt. Lett. 29(17), 2058–2060 (2004).
[CrossRef] [PubMed]

2003 (1)

T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interference of femtosecond pulses,” Appl. Phys. Lett. 82(17), 2758–2760 (2003).
[CrossRef]

1996 (1)

A. Brodeur, F. Ilkov, and S. Chin, “Beam filamentation and the white light continuum divergence,” Opt. Commun. 96(3-4), 193–198 (1996).
[CrossRef]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

An, R.

Brodeur, A.

A. Brodeur, F. Ilkov, and S. Chin, “Beam filamentation and the white light continuum divergence,” Opt. Commun. 96(3-4), 193–198 (1996).
[CrossRef]

Cheng, Y.

J. Song, X. Wang, X. Hu, Y. Dai, J. Qiu, Y. Cheng, and Z. Xu, “Formation mechanism of self-organized voids in dielectrics induced by tightly focused femtosecond laser pulses,” Appl. Phys. Lett. 92(9), 092904 (2008).
[CrossRef]

Chin, S.

A. Brodeur, F. Ilkov, and S. Chin, “Beam filamentation and the white light continuum divergence,” Opt. Commun. 96(3-4), 193–198 (1996).
[CrossRef]

Dai, E. W.

Dai, Y.

J. Song, X. Wang, X. Hu, Y. Dai, J. Qiu, Y. Cheng, and Z. Xu, “Formation mechanism of self-organized voids in dielectrics induced by tightly focused femtosecond laser pulses,” Appl. Phys. Lett. 92(9), 092904 (2008).
[CrossRef]

Dou, Y.

Fujita, K.

S. Kanehira, J. H. Si, J. R. Qiu, K. Fujita, and K. Hirao, “Periodic nanovoid structures via femtosecond laser irradiation,” Nano Lett. 5(8), 1591–1595 (2005).
[CrossRef] [PubMed]

Gong, Q.

Guo, Z.

Z. Guo, S. Qu, L. Ran, Y. Han, and S. Liu, “Formation of two-dimensional periodic microstructures by a single shot of three interfered femtosecond laser pulses on the surface of silica glass,” Opt. Lett. 33(20), 2383–2385 (2008).
[CrossRef] [PubMed]

Z. Guo, S. Qu, and S. Liu, “Generating optical vortex with computer-generated hologram fabricated inside glass by femtosecond laser pulses,” Opt. Commun. 273(1), 286–289 (2007).
[CrossRef]

Han, Y.

Hirao, K.

S. Kanehira, J. H. Si, J. R. Qiu, K. Fujita, and K. Hirao, “Periodic nanovoid structures via femtosecond laser irradiation,” Nano Lett. 5(8), 1591–1595 (2005).
[CrossRef] [PubMed]

S. Qu, C. Zhao, Q. Zhao, J. Qiu, C. Zhu, and K. Hirao, “One-off writing of multimicrogratings on glass by two interfered femtosecond laser pulses,” Opt. Lett. 29(17), 2058–2060 (2004).
[CrossRef] [PubMed]

Hirono, S.

S. Hirono, M. Kasuya, K. Matsuda, Y. Ozeki, K. Itoh, H. Mochizuki, and W. Watanabe, “Increasing diffraction efficiency by heating phase gratings formed by femtosecond laser irradiation in poly (methyl methacrylate),” Appl. Phys. Lett. 94(24), 241122 (2009).
[CrossRef]

Hu, X.

J. Song, X. Wang, X. Hu, Y. Dai, J. Qiu, Y. Cheng, and Z. Xu, “Formation mechanism of self-organized voids in dielectrics induced by tightly focused femtosecond laser pulses,” Appl. Phys. Lett. 92(9), 092904 (2008).
[CrossRef]

X. Hu, B. Qian, P. Zhang, X. Wang, L. Su, J. Qiu, and C. Zhu, “Self-organized microvoid array perpendicular to the femtosecond laser beam in CaF2 crystals,” Laser Phys. Lett. 5(5), 394–397 (2008).
[CrossRef]

Ilkov, F.

A. Brodeur, F. Ilkov, and S. Chin, “Beam filamentation and the white light continuum divergence,” Opt. Commun. 96(3-4), 193–198 (1996).
[CrossRef]

Itoh, K.

S. Hirono, M. Kasuya, K. Matsuda, Y. Ozeki, K. Itoh, H. Mochizuki, and W. Watanabe, “Increasing diffraction efficiency by heating phase gratings formed by femtosecond laser irradiation in poly (methyl methacrylate),” Appl. Phys. Lett. 94(24), 241122 (2009).
[CrossRef]

S. Sowa, W. Watanabe, J. Nishii, and K. Itoh, “Filamentary cavity formation in poly (methyl methacrylate) by single femtosecond pulse,” Appl. Phys., A Mater. Sci. Process. 81(8), 1587–1590 (2005).
[CrossRef]

Jiang, X.

Q. Zhao, J. Qiu, X. Jiang, C. Zhao, and C. Zhu, “Mechanisms of the refractive index change in femtosecond laser-irradiated Au3+-doped silicate glasses,” J. Appl. Phys. 129(12), 7122 (2004).
[CrossRef]

Jiang, X. W.

Juodkazis, S.

T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interference of femtosecond pulses,” Appl. Phys. Lett. 82(17), 2758–2760 (2003).
[CrossRef]

Kamata, M.

E. Toratani, M. Kamata, and M. Obara, “Self-fabrication of void array in fused silica by femtosecond laser processing,” Appl. Phys. Lett. 87(17), 171103 (2005).
[CrossRef]

Kanehira, S.

S. Kanehira, J. H. Si, J. R. Qiu, K. Fujita, and K. Hirao, “Periodic nanovoid structures via femtosecond laser irradiation,” Nano Lett. 5(8), 1591–1595 (2005).
[CrossRef] [PubMed]

Kasuya, M.

S. Hirono, M. Kasuya, K. Matsuda, Y. Ozeki, K. Itoh, H. Mochizuki, and W. Watanabe, “Increasing diffraction efficiency by heating phase gratings formed by femtosecond laser irradiation in poly (methyl methacrylate),” Appl. Phys. Lett. 94(24), 241122 (2009).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Kondo, T.

T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interference of femtosecond pulses,” Appl. Phys. Lett. 82(17), 2758–2760 (2003).
[CrossRef]

Li, Y.

Liu, S.

Z. Guo, S. Qu, L. Ran, Y. Han, and S. Liu, “Formation of two-dimensional periodic microstructures by a single shot of three interfered femtosecond laser pulses on the surface of silica glass,” Opt. Lett. 33(20), 2383–2385 (2008).
[CrossRef] [PubMed]

Z. Guo, S. Qu, and S. Liu, “Generating optical vortex with computer-generated hologram fabricated inside glass by femtosecond laser pulses,” Opt. Commun. 273(1), 286–289 (2007).
[CrossRef]

Matsuda, K.

S. Hirono, M. Kasuya, K. Matsuda, Y. Ozeki, K. Itoh, H. Mochizuki, and W. Watanabe, “Increasing diffraction efficiency by heating phase gratings formed by femtosecond laser irradiation in poly (methyl methacrylate),” Appl. Phys. Lett. 94(24), 241122 (2009).
[CrossRef]

Matsuo, S.

T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interference of femtosecond pulses,” Appl. Phys. Lett. 82(17), 2758–2760 (2003).
[CrossRef]

Misawa, H.

T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interference of femtosecond pulses,” Appl. Phys. Lett. 82(17), 2758–2760 (2003).
[CrossRef]

Mizeikis, V.

T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interference of femtosecond pulses,” Appl. Phys. Lett. 82(17), 2758–2760 (2003).
[CrossRef]

Mochizuki, H.

S. Hirono, M. Kasuya, K. Matsuda, Y. Ozeki, K. Itoh, H. Mochizuki, and W. Watanabe, “Increasing diffraction efficiency by heating phase gratings formed by femtosecond laser irradiation in poly (methyl methacrylate),” Appl. Phys. Lett. 94(24), 241122 (2009).
[CrossRef]

Nishii, J.

S. Sowa, W. Watanabe, J. Nishii, and K. Itoh, “Filamentary cavity formation in poly (methyl methacrylate) by single femtosecond pulse,” Appl. Phys., A Mater. Sci. Process. 81(8), 1587–1590 (2005).
[CrossRef]

Obara, M.

E. Toratani, M. Kamata, and M. Obara, “Self-fabrication of void array in fused silica by femtosecond laser processing,” Appl. Phys. Lett. 87(17), 171103 (2005).
[CrossRef]

Ozeki, Y.

S. Hirono, M. Kasuya, K. Matsuda, Y. Ozeki, K. Itoh, H. Mochizuki, and W. Watanabe, “Increasing diffraction efficiency by heating phase gratings formed by femtosecond laser irradiation in poly (methyl methacrylate),” Appl. Phys. Lett. 94(24), 241122 (2009).
[CrossRef]

Qian, B.

X. Hu, B. Qian, P. Zhang, X. Wang, L. Su, J. Qiu, and C. Zhu, “Self-organized microvoid array perpendicular to the femtosecond laser beam in CaF2 crystals,” Laser Phys. Lett. 5(5), 394–397 (2008).
[CrossRef]

Qiu, J.

X. Hu, B. Qian, P. Zhang, X. Wang, L. Su, J. Qiu, and C. Zhu, “Self-organized microvoid array perpendicular to the femtosecond laser beam in CaF2 crystals,” Laser Phys. Lett. 5(5), 394–397 (2008).
[CrossRef]

J. Song, X. Wang, X. Hu, Y. Dai, J. Qiu, Y. Cheng, and Z. Xu, “Formation mechanism of self-organized voids in dielectrics induced by tightly focused femtosecond laser pulses,” Appl. Phys. Lett. 92(9), 092904 (2008).
[CrossRef]

J. Song, X. Wang, J. Xu, H. Sun, Z. Xu, and J. Qiu, “Microstructures induced in the bulk of SrTiO3 crystal by a femtosecond laser,” Opt. Express 15(5), 2341–2347 (2007).
[CrossRef] [PubMed]

S. Qu, C. Zhao, Q. Zhao, J. Qiu, C. Zhu, and K. Hirao, “One-off writing of multimicrogratings on glass by two interfered femtosecond laser pulses,” Opt. Lett. 29(17), 2058–2060 (2004).
[CrossRef] [PubMed]

Q. Zhao, J. Qiu, X. Jiang, C. Zhao, and C. Zhu, “Mechanisms of the refractive index change in femtosecond laser-irradiated Au3+-doped silicate glasses,” J. Appl. Phys. 129(12), 7122 (2004).
[CrossRef]

Qiu, J. R.

Qu, S.

Ran, L.

Si, J. H.

S. Kanehira, J. H. Si, J. R. Qiu, K. Fujita, and K. Hirao, “Periodic nanovoid structures via femtosecond laser irradiation,” Nano Lett. 5(8), 1591–1595 (2005).
[CrossRef] [PubMed]

Song, J.

J. Song, X. Wang, X. Hu, Y. Dai, J. Qiu, Y. Cheng, and Z. Xu, “Formation mechanism of self-organized voids in dielectrics induced by tightly focused femtosecond laser pulses,” Appl. Phys. Lett. 92(9), 092904 (2008).
[CrossRef]

J. Song, X. Wang, J. Xu, H. Sun, Z. Xu, and J. Qiu, “Microstructures induced in the bulk of SrTiO3 crystal by a femtosecond laser,” Opt. Express 15(5), 2341–2347 (2007).
[CrossRef] [PubMed]

Sowa, S.

S. Sowa, W. Watanabe, J. Nishii, and K. Itoh, “Filamentary cavity formation in poly (methyl methacrylate) by single femtosecond pulse,” Appl. Phys., A Mater. Sci. Process. 81(8), 1587–1590 (2005).
[CrossRef]

Su, L.

X. Hu, B. Qian, P. Zhang, X. Wang, L. Su, J. Qiu, and C. Zhu, “Self-organized microvoid array perpendicular to the femtosecond laser beam in CaF2 crystals,” Laser Phys. Lett. 5(5), 394–397 (2008).
[CrossRef]

Sun, H.

Toratani, E.

E. Toratani, M. Kamata, and M. Obara, “Self-fabrication of void array in fused silica by femtosecond laser processing,” Appl. Phys. Lett. 87(17), 171103 (2005).
[CrossRef]

Wang, X.

X. Hu, B. Qian, P. Zhang, X. Wang, L. Su, J. Qiu, and C. Zhu, “Self-organized microvoid array perpendicular to the femtosecond laser beam in CaF2 crystals,” Laser Phys. Lett. 5(5), 394–397 (2008).
[CrossRef]

J. Song, X. Wang, X. Hu, Y. Dai, J. Qiu, Y. Cheng, and Z. Xu, “Formation mechanism of self-organized voids in dielectrics induced by tightly focused femtosecond laser pulses,” Appl. Phys. Lett. 92(9), 092904 (2008).
[CrossRef]

J. Song, X. Wang, J. Xu, H. Sun, Z. Xu, and J. Qiu, “Microstructures induced in the bulk of SrTiO3 crystal by a femtosecond laser,” Opt. Express 15(5), 2341–2347 (2007).
[CrossRef] [PubMed]

Watanabe, W.

S. Hirono, M. Kasuya, K. Matsuda, Y. Ozeki, K. Itoh, H. Mochizuki, and W. Watanabe, “Increasing diffraction efficiency by heating phase gratings formed by femtosecond laser irradiation in poly (methyl methacrylate),” Appl. Phys. Lett. 94(24), 241122 (2009).
[CrossRef]

S. Sowa, W. Watanabe, J. Nishii, and K. Itoh, “Filamentary cavity formation in poly (methyl methacrylate) by single femtosecond pulse,” Appl. Phys., A Mater. Sci. Process. 81(8), 1587–1590 (2005).
[CrossRef]

Xu, J.

Xu, Z.

J. Song, X. Wang, X. Hu, Y. Dai, J. Qiu, Y. Cheng, and Z. Xu, “Formation mechanism of self-organized voids in dielectrics induced by tightly focused femtosecond laser pulses,” Appl. Phys. Lett. 92(9), 092904 (2008).
[CrossRef]

J. Song, X. Wang, J. Xu, H. Sun, Z. Xu, and J. Qiu, “Microstructures induced in the bulk of SrTiO3 crystal by a femtosecond laser,” Opt. Express 15(5), 2341–2347 (2007).
[CrossRef] [PubMed]

Yang, H.

Zhang, P.

X. Hu, B. Qian, P. Zhang, X. Wang, L. Su, J. Qiu, and C. Zhu, “Self-organized microvoid array perpendicular to the femtosecond laser beam in CaF2 crystals,” Laser Phys. Lett. 5(5), 394–397 (2008).
[CrossRef]

Zhao, C.

S. Qu, C. Zhao, Q. Zhao, J. Qiu, C. Zhu, and K. Hirao, “One-off writing of multimicrogratings on glass by two interfered femtosecond laser pulses,” Opt. Lett. 29(17), 2058–2060 (2004).
[CrossRef] [PubMed]

Q. Zhao, J. Qiu, X. Jiang, C. Zhao, and C. Zhu, “Mechanisms of the refractive index change in femtosecond laser-irradiated Au3+-doped silicate glasses,” J. Appl. Phys. 129(12), 7122 (2004).
[CrossRef]

Zhao, Q.

S. Qu, C. Zhao, Q. Zhao, J. Qiu, C. Zhu, and K. Hirao, “One-off writing of multimicrogratings on glass by two interfered femtosecond laser pulses,” Opt. Lett. 29(17), 2058–2060 (2004).
[CrossRef] [PubMed]

Q. Zhao, J. Qiu, X. Jiang, C. Zhao, and C. Zhu, “Mechanisms of the refractive index change in femtosecond laser-irradiated Au3+-doped silicate glasses,” J. Appl. Phys. 129(12), 7122 (2004).
[CrossRef]

Zhao, Q. Z.

Zhou, C. H.

Zhu, C.

X. Hu, B. Qian, P. Zhang, X. Wang, L. Su, J. Qiu, and C. Zhu, “Self-organized microvoid array perpendicular to the femtosecond laser beam in CaF2 crystals,” Laser Phys. Lett. 5(5), 394–397 (2008).
[CrossRef]

S. Qu, C. Zhao, Q. Zhao, J. Qiu, C. Zhu, and K. Hirao, “One-off writing of multimicrogratings on glass by two interfered femtosecond laser pulses,” Opt. Lett. 29(17), 2058–2060 (2004).
[CrossRef] [PubMed]

Q. Zhao, J. Qiu, X. Jiang, C. Zhao, and C. Zhu, “Mechanisms of the refractive index change in femtosecond laser-irradiated Au3+-doped silicate glasses,” J. Appl. Phys. 129(12), 7122 (2004).
[CrossRef]

Zhu, C. S.

Appl. Phys. Lett. (4)

T. Kondo, S. Matsuo, S. Juodkazis, V. Mizeikis, and H. Misawa, “Multiphoton fabrication of periodic structures by multibeam interference of femtosecond pulses,” Appl. Phys. Lett. 82(17), 2758–2760 (2003).
[CrossRef]

E. Toratani, M. Kamata, and M. Obara, “Self-fabrication of void array in fused silica by femtosecond laser processing,” Appl. Phys. Lett. 87(17), 171103 (2005).
[CrossRef]

S. Hirono, M. Kasuya, K. Matsuda, Y. Ozeki, K. Itoh, H. Mochizuki, and W. Watanabe, “Increasing diffraction efficiency by heating phase gratings formed by femtosecond laser irradiation in poly (methyl methacrylate),” Appl. Phys. Lett. 94(24), 241122 (2009).
[CrossRef]

J. Song, X. Wang, X. Hu, Y. Dai, J. Qiu, Y. Cheng, and Z. Xu, “Formation mechanism of self-organized voids in dielectrics induced by tightly focused femtosecond laser pulses,” Appl. Phys. Lett. 92(9), 092904 (2008).
[CrossRef]

Appl. Phys., A Mater. Sci. Process. (1)

S. Sowa, W. Watanabe, J. Nishii, and K. Itoh, “Filamentary cavity formation in poly (methyl methacrylate) by single femtosecond pulse,” Appl. Phys., A Mater. Sci. Process. 81(8), 1587–1590 (2005).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

J. Appl. Phys. (1)

Q. Zhao, J. Qiu, X. Jiang, C. Zhao, and C. Zhu, “Mechanisms of the refractive index change in femtosecond laser-irradiated Au3+-doped silicate glasses,” J. Appl. Phys. 129(12), 7122 (2004).
[CrossRef]

Laser Phys. Lett. (1)

X. Hu, B. Qian, P. Zhang, X. Wang, L. Su, J. Qiu, and C. Zhu, “Self-organized microvoid array perpendicular to the femtosecond laser beam in CaF2 crystals,” Laser Phys. Lett. 5(5), 394–397 (2008).
[CrossRef]

Nano Lett. (1)

S. Kanehira, J. H. Si, J. R. Qiu, K. Fujita, and K. Hirao, “Periodic nanovoid structures via femtosecond laser irradiation,” Nano Lett. 5(8), 1591–1595 (2005).
[CrossRef] [PubMed]

Opt. Commun. (2)

Z. Guo, S. Qu, and S. Liu, “Generating optical vortex with computer-generated hologram fabricated inside glass by femtosecond laser pulses,” Opt. Commun. 273(1), 286–289 (2007).
[CrossRef]

A. Brodeur, F. Ilkov, and S. Chin, “Beam filamentation and the white light continuum divergence,” Opt. Commun. 96(3-4), 193–198 (1996).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

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Figures (5)

Fig. 1
Fig. 1

(a) the sketch of the experimental setup for the generation of periodic voids, Side view of the voids array induced in silica glass by femtosecond laser pulses,(b) different number of with pulse energy of 20 μ J , (c) single pulse with different energies, (d) double pulses with different energies. The scale bars in the Figure are 20 μ m . The arrows show the direction of the pulses propagation, and the lop lines express the focus place.

Fig. 2
Fig. 2

the fabricated grating: (a) top view, (b) side view; diffraction pattern of the grating obtained under different conditions: (c) E = 1.76 μJ , v = 1000 μm/s , (d) E = 7.38 μJ , v = 1000 μm/s

Fig. 3
Fig. 3

(a) Variational tendency of the transmittance efficiency (TE) with the different pulse of energy and scanning velocity, (b) Variational tendency of the first order diffraction efficiency (FODE) with the different pulse of energy and scanning velocity, (c) Variational tendency of the normalized first order diffraction efficiency (NFODE) with the different pulse of energy and scanning velocity

Fig. 4
Fig. 4

(a) A simple sketch of our numerical method, (b) A typical example of the distribution of the amplitude of the electric field inside the sample. (c) Distribution of the amplitude of the Electric filed in the silicate glass sample.

Fig. 5
Fig. 5

The far field diffraction of the distribution of Electric filed along the x ' direction, (a) d r e c = 10 μm, δ n = 0.007 and the three curves correspond to different depths of the induced gratings: h c = 5(solid red), 20(dotted blue) and 40 μm (dashed black) respectively; (b) h c = 40 μm , δ n = 0.007 and the three curves correspond to different width of the filaments: d r e c = 5(solid red), 10(dotted blue) and 15 μm (dashed black) respectively; (c) d r e c = 10 μm , h c = 40 μm and the three curves correspond to different RICs: δ n = 0.002 (solid red), 0.007 (dotted blue) and 0.012(dashed black) respectively.

Equations (1)

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E ( x ' ) = C j λ + E ( x ) exp ( i k r ) r cos θ + 1 2 d x ,

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