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Quasi-fractional Talbot effect of resonant diffraction grating
Optics Communications 285 (2012) 4161–4165
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Quasi-fractional Talbot effect of resonant diffraction grating Yue Fang, Qiaofeng Tan n, Mingqian Zhang, Guofan Jin State Key Laboratory of Precision Measurement Technology and Instruments, Tsinghua University, Beijing 100084, China
a r t i c l e i n f o
abstract
Article history: Received 10 January 2012 Received in revised form 27 June 2012 Accepted 28 June 2012 Available online 13 July 2012
The Fresnel diffraction of resonant diffraction grating is studied on the basis of the modal method, and it shows that the conventional fractional Talbot effect can still occur by choosing appropriate parameters. The Fresnel diffraction distributions of the resonant diffraction gratings with periods between l and 4l are analyzed, and the diffraction fields exhibit periodicity in both grating direction and the propagating direction. The experiment is carried out to verify the simulation results of the quasi-fractional Talbot effect of the resonant diffraction grating. This effect has potential application in lithography for high-resolution patterning. & 2012 Elsevier B.V. All rights reserved.
Keywords: Resonant diffraction grating Quasi-fractional Talbot effect Modal method
1. Introduction The Talbot effect is a famous diffraction phenomenon in the Fresnel diffraction field, which the grating or other periodic structure will repeat itself at multiples of a certain distance along the propagating direction when illuminated by a monochromatic plane wave [1]. The certain distance is called the Talbot distance, and interesting effects can also be found at fractional Talbot distances, where the multiple frequency sub-self-imaging may occur. These self-imaging phenomena of grating have a very wide range of applications, such as array illumination [2], interferometry measurement [3–5], and lithography [6,7]. However, the Talbot effect and the fractional Talbot effect are observed when the period of the grating is much larger than the incident wavelength, for the case of resonant diffraction grating whose period is comparable to the wavelength, the field distribution at the outgoing plane of the grating will also revive periodically, known as the quasi-Talbot effect [8]. With the micro-fabrication technology, the quasi-Talbot effects of various kinds of gratings are received more and more concerns. The quasi-Talbot effects of a one-dimensional perfect conducting grating are analyzed on the basis of the finitedifference time domain technique [9] and the analysis of metal grating is also reported [10]. Lu et al. presented the polarization dependent Talbot effect [11] and this effect is explained by different theories [12,13]. The quasi-Talbot effect of a resonant diffraction grating under femtosecond laser illumination is studied with the rigorous electromagnetic theory [14]. However, to our knowledge, the quasi-fractional Talbot effect which has
n
Corresponding author. Tel.: þ86 10 62781187. E-mail address: [email protected] (Q. Tan).
0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.06.073
potential applications in many areas is often ignored in researches. In this paper, we analyze the quasi-fractional Talbot effect of the resonant diffraction grating. In Section 2, the numerical simulations are given with the gratings of different incident wavelengths and different periods. The quasi-fractional effect is discussed in detail in Section 3. The experimental results are shown in Section 4 and Section 5 is the conclusion of this paper.
2. Quasi-fractional Talbot effect The schematic of the grating is shown in Fig. 1. We discuss the rectangular dielectric grating with the monochromatic plane wave normally illuminating from the substrate side along the z-axis. The incident wave is in TE polarization, which the electric field oscillates perpendicular to the incident plane. The grating region occupies the volume hrz r0 (h is the depth of the grating) and consists of a periodic distribution of the fused-silica material. The refractive index n1 of the fused-silica grating layer and the substrate are chosen as 1.45, and the refractive index n0 of the air is 1.0. For conventional scalar grating whose period is much larger than the wavelength, the grating will produce smaller fractional revivals at the fractional Talbot distance where the period of the field distribution is a fraction of the period of the grating. For resonant grating, the fractional Talbot effect can still occur under certain conditions which is called as the quasi-fractional Talbot effect in this paper. The frequency-doubled effect which presented the revivals with half period of the grating is a basic fractional Talbot effect, and to realize this basic kind sub-selfimaging effect the period of the grating is at least between l and 2l to ensure three diffraction orders. Fig. 2 shows the power flow
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distributions of the gratings with the duty cycle 0.5 in space 0 rzr10 mm, and the periods of the gratings are 1.05l, 1.5l and 1.95l as instances for Fig. 2(a), (b) and (c) respectively, where l is 488 nm. We can see that the distributions at certain distances along the x direction have the same period as the grating. At the same time, the distribution along the z directions also shows a clear periodicity. For the grating period d between l and 2l, the diffraction field always has self-imaging with the field at the output plane rather than the grating structure, which is generally called the quasi-Talbot effect. The quasi-Talbot self-imaging distance can be calculated as [8] zT ¼
l : 1ð1ðl=dÞ2 Þ1=2
ð1Þ
The numerical simulation results of the resonant grating with period 1.95l at different distances are shown in Fig. 3(a)–(c) as an instance when illuminated by the incident wavelength at 488 nm. According to Eq. (1), the quasi-Talbot distance zT is 3.45 mm. The diffraction fields at zT and 2zT are exactly the same as shown in Fig. 3(a) and (b), respectively. Besides the quasi-Talbot effect, the quasi-fractional effect can also be observed at certain distances. As shown in Fig. 3(c), the period of the distribution is half of the period of the grating at z ¼4.87 mm. This frequency-doubled effect has potential application in lithography for a high-resolution patterning.
3. Modal explanation Since the scalar theory is no longer applicable in the case of resonant diffraction grating, the vector theory is used here to analyze the diffraction field. When illuminated by monochromatic plane wave, the transmission field of a periodic structure can be written in terms of plane wave which is called the Rayleigh expansion [15] þ1 X !d E ðx,y,zÞ ¼ y^ T m expðiam x þ ibm zÞ,
ðz 4 0Þ,
ð2Þ
m ¼ 1 þ1 X !d 1 ðx^ bm þ z^ am ÞT m expðiam x þibm zÞ H ðx,y,zÞ ¼ k0
ðz 4 0Þ,
m ¼ 1
ð3Þ where Tm is complex amplitude of the mth diffraction order, k0 is the magnitude of the vacuum wave-vector, am ¼2pm/d; bm ¼ ðk0 n20 a2m Þ1=2 , and d is the period of the grating. The zcomponent of the time-averaged Poynting vector which represents the energy propagating along the z direction is calculated as d ! !d ¼ ReðEx Hny Ey Hnx Þ: ð4Þ Sz ¼ E H z
So the power flow distribution for the resonant grating with period d between l and 2l can be written as Sz ¼ ReðEy Hn Þ ¼ k0 b0 E20 þ2k0 b1 E þ 1 2 þ 2k0 b1 E þ 1 2 cosð2KxÞ 1
1 þ2k0 ðb0 þ b1 ÞE0
1
1
E þ 1 cos½ðb þ 1 b0 Þz þ ðf þ 1 f0 Þ cosðKxÞ, ð5Þ
Fig. 1. Schematic of the grating.
where K is the grating vector, E0 and E þ 1 represent the amplitudes of the 0th and þ1st diffraction orders, f0 and f þ 1 represent the phases of the 0th and þ1st diffraction orders, and b0 and b þ 1 are the propagating constants of the 0th and þ1st diffraction orders. As shown in the equation above, the Fresnel diffraction distribution is related to the amplitudes and the phases of different orders and the propagating distance z. Therefore, the realization of the fractional Talbot effect divides naturally into two parts. First, the amplitudes and the phases of different orders which are only related to grating parameters should be decided. Secondly, the plane at the fractional Talbot distance z should be found. For the resonant grating we discussed in Fig. 1, the period, the duty cycle and the depth of the grating are the parameters that influence amplitudes and the phases of different orders. We choose the duty cycle to be 0.5 as an instance and the
Fig. 2. Power flow distributions of gratings with different periods for (a) 1.05l (b) 1.5l and (c) 1.95l.
Y. Fang et al. / Optics Communications 285 (2012) 4161–4165
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Fig. 3. One-dimensional power flow distributions of the grating with period 1.5l at different distances for (a) z ¼ 3.45 mm (b) z ¼6.9 mm and (c) z ¼4.87 mm.
appropriate grating period is only to ensure the necessary diffraction orders. However, the depth of the grating region has to be chosen properly because it determines the energy efficiency distributions of the diffraction orders. The modal method [16] is implemented to analyze the quasi-fractional effect of the resonant diffraction grating whose period is comparable to the wavelength of the incident wave, and to choose a proper depth of the grating. It is a simple method that can reveal the physical meaning when light propagates in the grating region. The electromagnetic field in the grating region is represented in terms of modes expansion as h i X !g E ðx,y,zÞ ¼ uq ðxÞ C q expðibq zÞ þDq expðibq zÞ
ðh r z r0Þ,
q
ð6Þ where uq(x) is the qth mode expression, bq is the propagation constant of the qth mode, and Cq and Dq are the coefficients that need to be determined. The propagating constants bq of the grating modes which is excited by the incident wave at the substrate–grating interface can be calculated by the dispersion equation for TE polarization [16]: 2
Fðn2ef f Þ ¼ cosðk1 f dÞ ¼ cosðkx dÞ,
2
k þ k2 cosðk2 ð1f ÞdÞ 1 sinðk1 f dÞ sinðk2 ð1f ÞdÞ 2k1 k2 ð7Þ
where k0 ¼ 2p=l; kx ¼ k0 n1 sin y; k1 ¼ k0 ðn21 n2ef f Þ1=2 ; k2 ¼ k0 ðn20 n2ef f Þ1=2 , y is the angle of the incident wave, and neff ¼ bqeff/k0 is the effective index of the grating mode. The right side is equal to 1 when the incident wave is normally incident. The diffraction process can be regarded as the propagating of modes and the interference at the interface. At the grating-air plane, the modes exchange the energy with the diffraction orders. For example, the energy exchanging between the incident wave and the modes propagating in the grating region can be calculated
Fig. 4. Diffraction efficiency ratio as a function of grating depth h.
by the overlap integral [16] t inq ¼ R
R 2 9 Ein ðx,0Þuq ðxÞdx9 , R 2 2 9Ein ðx,0Þ9 dx 9uq ðxÞ9 dx
ð8Þ
where Ein(x,0) represents the incident wave. In the case of normal incidence, only the even modes can be excited and the efficiencies of the diffraction orders can be expressed as functions of depth [17]: 2
jE0 j2 ¼ 9Aeik0 nef f 0 h þ ð1AÞeik0 nef f 2 h 9 ,
2
2
9E1 9 ¼ 4B sin
2
Dj 2
,
A¼
1 d
Z
d
t in0 u0 ðxÞdx,
ð9Þ
0
Z 1 d B¼ t in0 u0 ðxÞcos kx x dx d 0
ð10Þ
where Dj ¼k0(neff0 neff2)h, and u0(x) and u2(x) are the first two even modes.
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We choose the grating with period 1.95l as an example of the grating with period between l and 2l. As we can see from Eq. (5), to realize the quasi-fractional Talbot effect, we should reduce the impact of the fundamental frequency component cosðKxÞ. That is to say, the efficiency of the 0th diffraction order should be kept low. According to Eqs. (9) and (10), the diffraction efficiency ratio as a function of grating depth h (see Fig. 1) is plotted in Fig. 4. It is obvious that the frequency-doubled effect will possibly occur when the depth of the grating is larger than 0.3 mm and smaller than 0.8 mm. The design method of the frequency-doubled effect can be extended to the realization of the frequency-tripled effect by choosing the proper period and depth of the resonant diffraction grating. Firstly, the grating period should be increased to have more diffraction orders, and the grating with periods 2.5l and 3.5l which ensure five and seven diffraction orders respectively
are chosen as examples. The diffraction efficiencies of the grating with periods 2.5l and 3.5l are shown in Fig. 5(a) and (b) at wavelength 488 nm. When the depth of the grating is between 0.4 mm and 0.6 mm, the efficiency of the 0th order is kept low and there exists the possibility to realize the frequency-tripled effect. As illustrated in Fig. 6, the depth of the grating is chosen as 0.5 mm and the diffraction distributions of the gratings with periods 2.5l and 3.5l still show periodicity along the x and z directions. We calculate the diffraction fields at zT and 2zT for gratings with periods 2.5l and 3.5l respectively, and the results indicate that Eq. (1) cannot be applied to grating with a larger period. However, the quasi-fractional Talbot effect can still be realized by choosing appropriate grating parameters. The frequency-tripled effect appears with grating period 2.5l at z¼3.45 mm and period 3.5l at z ¼3.1 mm as shown in Fig. 7. For grating with period 2.5l, the distribution shows poor contrast
Fig. 5. The diffraction efficiencies of different orders of the gratings with different periods (a) 2.5l and (b) 3.5l.
Fig. 6. Power flow distributions of gratings with different periods for (a) 2.5l and (b) 3.5l.
Fig. 7. One-dimensional power flow distributions of the gratings for (a) period 2.5l at z¼ 3.45 mm and (b) period 3.5l at z¼ 3.1 mm.
Y. Fang et al. / Optics Communications 285 (2012) 4161–4165
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Fig. 8. Experimental results (a) two-dimensional scanning field distribution at z ¼4.87 mm and (b) one-dimensional scanning the field distribution at z¼ 4.87 mm.
because of the existence of the 0th order. With the increasing of the number of diffraction orders, the efficiency of the 0th order reduces and the contrast of the distribution is getting better as shown in Fig. 7(b). It is explicit that the possibility to realize the multiple frequency effect increases with the increasing of the period of the grating.
4. Experiment According to Fig. 4, the fabrication tolerance of the depth of the grating ranges from 0.3 mm to 0.8 mm. The depth is chosen as 0.37 mm taking the fabrication condition into account. The incident light source is a semiconductor laser at the wavelength 488 nm. The grating which fabricated by electron beam directwriting lithography was placed on the sample stage, and the transmission fields at different distances above the grating were scanned by the near-field scanning optical microscopy (NSOM) with the positioning accuracy in the order of 10 nm. A two-dimensional scanning was carried out and the experimental results are shown in Fig. 8. The field distribution and the cross-lines of the field distribution of the resonant diffraction grating with period 1.95l (0.95 mm) around z¼4.87 mm are shown in Fig. 8(a) and (b). The period of the distribution is 0.46 mm which is half of the grating period. The experimental result is consistent with the simulation result in Fig. 3(c).
5. Conclusion It is shown that the frequency-doubled and frequency-tripled effects which have already found for grating with period much larger than the incident wavelength can still occur for resonant diffraction grating. The diffracted fields of resonant diffraction gratings with different periods are studied, and the diffraction fields have periodicity in both grating direction and the propagating direction. The simulation results are verified by experiments in the
use of NSOM. The quasi-fractional Talbot effect exhibits promising application in lithography for high-resolution patterning.
Acknowledgments This work was supported by the National Natural Science Foundation of China (60978047, 60678033) and the National Basic Research Program of China (2007CB935303). The experiment was completed in the Nano Optics Laboratory directed by Prof. Jia Wang at Tsinghua University.
References [1] [2] [3] [4] [5] [6]
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
W.H.F. Talbot, Philosophical Magazine 9 (1836) 401. A.W. Lohmann, J.A. Thomas, Applied Optics 29 (1990) 4337. A.W. Lohmann, D.E. Silva, Optics Communications 4 (1972) 326. He Kaijun, J. Jahns, A.W. Lohmann, Optics Communications 45 (1983) 295. S.P. Trivedi, S. Prakash, Santosh Rana, Osami Sasaki, Applied Optics 49 (2010) 897. A. Isoyan, F. Jiang, Y.C. Cheng, F. Cerrina, P. Wachulak, L. Urbanski, J. Rocca, C. Menoni, M. Marconi, Journal of Vacuum Science and Technology B 27 (2009) 2931. L. Stuerzebecher, T. Harzendorf, U. Vogler, U. Zeitner, R. Voelkel, Optics Express 18 (2010) 19485. E. Noponen, J. Turunen, Optics Communications 98 (1993) 132. S. Teng, Y. Tan, C. Cheng, Journal of the Optical Society of America A—Optics and Image Science 25 (2008) 2945. X. Wan, Q. Wang, H. Tao, Optics Communications 283 (2010) 5231. Y. Lu, C. Zhou, S. Wang, B. Wang, Journal of the Optical Society of America A—Optics and Image Science (2006) 2154. J. Zheng, C. Zhou, B. Wang, Optics Communications 281 (2008) 3254. S. Teng, W. Guo, C. Cheng, Journal of the Optical Society of America A—Optics and Image Science 27 (2010) 366. J. Zheng, C. Zhou, E. Dai, B. Wang, W. Jia, Optics Communications 281 (2008) 3230. R. Petit (Ed.), Springer-Verlag, Berlin, 1980. L.C. Botten, M.S. Craig, R.C. McPhedran, J.L. Adams, J.R. Andrewartha, Journal of Modern Optics 28 (1981) 413. J. Feng, C. Zhou, B. Wang, J. Zheng, W. Jia, H. Cao, P. Lv, Applied Optics 47 (2008) 6638.
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Quasi-fractional Talbot effect of resonant diffraction grating Yue Fang, Qiaofeng Tan n, Mingqian Zhang, Guofan Jin State Key Laboratory of Precision Measurement Technology and Instruments, Tsinghua University, Beijing 100084, China
a r t i c l e i n f o
abstract
Article history: Received 10 January 2012 Received in revised form 27 June 2012 Accepted 28 June 2012 Available online 13 July 2012
The Fresnel diffraction of resonant diffraction grating is studied on the basis of the modal method, and it shows that the conventional fractional Talbot effect can still occur by choosing appropriate parameters. The Fresnel diffraction distributions of the resonant diffraction gratings with periods between l and 4l are analyzed, and the diffraction fields exhibit periodicity in both grating direction and the propagating direction. The experiment is carried out to verify the simulation results of the quasi-fractional Talbot effect of the resonant diffraction grating. This effect has potential application in lithography for high-resolution patterning. & 2012 Elsevier B.V. All rights reserved.
Keywords: Resonant diffraction grating Quasi-fractional Talbot effect Modal method
1. Introduction The Talbot effect is a famous diffraction phenomenon in the Fresnel diffraction field, which the grating or other periodic structure will repeat itself at multiples of a certain distance along the propagating direction when illuminated by a monochromatic plane wave [1]. The certain distance is called the Talbot distance, and interesting effects can also be found at fractional Talbot distances, where the multiple frequency sub-self-imaging may occur. These self-imaging phenomena of grating have a very wide range of applications, such as array illumination [2], interferometry measurement [3–5], and lithography [6,7]. However, the Talbot effect and the fractional Talbot effect are observed when the period of the grating is much larger than the incident wavelength, for the case of resonant diffraction grating whose period is comparable to the wavelength, the field distribution at the outgoing plane of the grating will also revive periodically, known as the quasi-Talbot effect [8]. With the micro-fabrication technology, the quasi-Talbot effects of various kinds of gratings are received more and more concerns. The quasi-Talbot effects of a one-dimensional perfect conducting grating are analyzed on the basis of the finitedifference time domain technique [9] and the analysis of metal grating is also reported [10]. Lu et al. presented the polarization dependent Talbot effect [11] and this effect is explained by different theories [12,13]. The quasi-Talbot effect of a resonant diffraction grating under femtosecond laser illumination is studied with the rigorous electromagnetic theory [14]. However, to our knowledge, the quasi-fractional Talbot effect which has
n
Corresponding author. Tel.: þ86 10 62781187. E-mail address: [email protected] (Q. Tan).
0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.06.073
potential applications in many areas is often ignored in researches. In this paper, we analyze the quasi-fractional Talbot effect of the resonant diffraction grating. In Section 2, the numerical simulations are given with the gratings of different incident wavelengths and different periods. The quasi-fractional effect is discussed in detail in Section 3. The experimental results are shown in Section 4 and Section 5 is the conclusion of this paper.
2. Quasi-fractional Talbot effect The schematic of the grating is shown in Fig. 1. We discuss the rectangular dielectric grating with the monochromatic plane wave normally illuminating from the substrate side along the z-axis. The incident wave is in TE polarization, which the electric field oscillates perpendicular to the incident plane. The grating region occupies the volume hrz r0 (h is the depth of the grating) and consists of a periodic distribution of the fused-silica material. The refractive index n1 of the fused-silica grating layer and the substrate are chosen as 1.45, and the refractive index n0 of the air is 1.0. For conventional scalar grating whose period is much larger than the wavelength, the grating will produce smaller fractional revivals at the fractional Talbot distance where the period of the field distribution is a fraction of the period of the grating. For resonant grating, the fractional Talbot effect can still occur under certain conditions which is called as the quasi-fractional Talbot effect in this paper. The frequency-doubled effect which presented the revivals with half period of the grating is a basic fractional Talbot effect, and to realize this basic kind sub-selfimaging effect the period of the grating is at least between l and 2l to ensure three diffraction orders. Fig. 2 shows the power flow
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distributions of the gratings with the duty cycle 0.5 in space 0 rzr10 mm, and the periods of the gratings are 1.05l, 1.5l and 1.95l as instances for Fig. 2(a), (b) and (c) respectively, where l is 488 nm. We can see that the distributions at certain distances along the x direction have the same period as the grating. At the same time, the distribution along the z directions also shows a clear periodicity. For the grating period d between l and 2l, the diffraction field always has self-imaging with the field at the output plane rather than the grating structure, which is generally called the quasi-Talbot effect. The quasi-Talbot self-imaging distance can be calculated as [8] zT ¼
l : 1ð1ðl=dÞ2 Þ1=2
ð1Þ
The numerical simulation results of the resonant grating with period 1.95l at different distances are shown in Fig. 3(a)–(c) as an instance when illuminated by the incident wavelength at 488 nm. According to Eq. (1), the quasi-Talbot distance zT is 3.45 mm. The diffraction fields at zT and 2zT are exactly the same as shown in Fig. 3(a) and (b), respectively. Besides the quasi-Talbot effect, the quasi-fractional effect can also be observed at certain distances. As shown in Fig. 3(c), the period of the distribution is half of the period of the grating at z ¼4.87 mm. This frequency-doubled effect has potential application in lithography for a high-resolution patterning.
3. Modal explanation Since the scalar theory is no longer applicable in the case of resonant diffraction grating, the vector theory is used here to analyze the diffraction field. When illuminated by monochromatic plane wave, the transmission field of a periodic structure can be written in terms of plane wave which is called the Rayleigh expansion [15] þ1 X !d E ðx,y,zÞ ¼ y^ T m expðiam x þ ibm zÞ,
ðz 4 0Þ,
ð2Þ
m ¼ 1 þ1 X !d 1 ðx^ bm þ z^ am ÞT m expðiam x þibm zÞ H ðx,y,zÞ ¼ k0
ðz 4 0Þ,
m ¼ 1
ð3Þ where Tm is complex amplitude of the mth diffraction order, k0 is the magnitude of the vacuum wave-vector, am ¼2pm/d; bm ¼ ðk0 n20 a2m Þ1=2 , and d is the period of the grating. The zcomponent of the time-averaged Poynting vector which represents the energy propagating along the z direction is calculated as d ! !d ¼ ReðEx Hny Ey Hnx Þ: ð4Þ Sz ¼ E H z
So the power flow distribution for the resonant grating with period d between l and 2l can be written as Sz ¼ ReðEy Hn Þ ¼ k0 b0 E20 þ2k0 b1 E þ 1 2 þ 2k0 b1 E þ 1 2 cosð2KxÞ 1
1 þ2k0 ðb0 þ b1 ÞE0
1
1
E þ 1 cos½ðb þ 1 b0 Þz þ ðf þ 1 f0 Þ cosðKxÞ, ð5Þ
Fig. 1. Schematic of the grating.
where K is the grating vector, E0 and E þ 1 represent the amplitudes of the 0th and þ1st diffraction orders, f0 and f þ 1 represent the phases of the 0th and þ1st diffraction orders, and b0 and b þ 1 are the propagating constants of the 0th and þ1st diffraction orders. As shown in the equation above, the Fresnel diffraction distribution is related to the amplitudes and the phases of different orders and the propagating distance z. Therefore, the realization of the fractional Talbot effect divides naturally into two parts. First, the amplitudes and the phases of different orders which are only related to grating parameters should be decided. Secondly, the plane at the fractional Talbot distance z should be found. For the resonant grating we discussed in Fig. 1, the period, the duty cycle and the depth of the grating are the parameters that influence amplitudes and the phases of different orders. We choose the duty cycle to be 0.5 as an instance and the
Fig. 2. Power flow distributions of gratings with different periods for (a) 1.05l (b) 1.5l and (c) 1.95l.
Y. Fang et al. / Optics Communications 285 (2012) 4161–4165
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Fig. 3. One-dimensional power flow distributions of the grating with period 1.5l at different distances for (a) z ¼ 3.45 mm (b) z ¼6.9 mm and (c) z ¼4.87 mm.
appropriate grating period is only to ensure the necessary diffraction orders. However, the depth of the grating region has to be chosen properly because it determines the energy efficiency distributions of the diffraction orders. The modal method [16] is implemented to analyze the quasi-fractional effect of the resonant diffraction grating whose period is comparable to the wavelength of the incident wave, and to choose a proper depth of the grating. It is a simple method that can reveal the physical meaning when light propagates in the grating region. The electromagnetic field in the grating region is represented in terms of modes expansion as h i X !g E ðx,y,zÞ ¼ uq ðxÞ C q expðibq zÞ þDq expðibq zÞ
ðh r z r0Þ,
q
ð6Þ where uq(x) is the qth mode expression, bq is the propagation constant of the qth mode, and Cq and Dq are the coefficients that need to be determined. The propagating constants bq of the grating modes which is excited by the incident wave at the substrate–grating interface can be calculated by the dispersion equation for TE polarization [16]: 2
Fðn2ef f Þ ¼ cosðk1 f dÞ ¼ cosðkx dÞ,
2
k þ k2 cosðk2 ð1f ÞdÞ 1 sinðk1 f dÞ sinðk2 ð1f ÞdÞ 2k1 k2 ð7Þ
where k0 ¼ 2p=l; kx ¼ k0 n1 sin y; k1 ¼ k0 ðn21 n2ef f Þ1=2 ; k2 ¼ k0 ðn20 n2ef f Þ1=2 , y is the angle of the incident wave, and neff ¼ bqeff/k0 is the effective index of the grating mode. The right side is equal to 1 when the incident wave is normally incident. The diffraction process can be regarded as the propagating of modes and the interference at the interface. At the grating-air plane, the modes exchange the energy with the diffraction orders. For example, the energy exchanging between the incident wave and the modes propagating in the grating region can be calculated
Fig. 4. Diffraction efficiency ratio as a function of grating depth h.
by the overlap integral [16] t inq ¼ R
R 2 9 Ein ðx,0Þuq ðxÞdx9 , R 2 2 9Ein ðx,0Þ9 dx 9uq ðxÞ9 dx
ð8Þ
where Ein(x,0) represents the incident wave. In the case of normal incidence, only the even modes can be excited and the efficiencies of the diffraction orders can be expressed as functions of depth [17]: 2
jE0 j2 ¼ 9Aeik0 nef f 0 h þ ð1AÞeik0 nef f 2 h 9 ,
2
2
9E1 9 ¼ 4B sin
2
Dj 2
,
A¼
1 d
Z
d
t in0 u0 ðxÞdx,
ð9Þ
0
Z 1 d B¼ t in0 u0 ðxÞcos kx x dx d 0
ð10Þ
where Dj ¼k0(neff0 neff2)h, and u0(x) and u2(x) are the first two even modes.
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We choose the grating with period 1.95l as an example of the grating with period between l and 2l. As we can see from Eq. (5), to realize the quasi-fractional Talbot effect, we should reduce the impact of the fundamental frequency component cosðKxÞ. That is to say, the efficiency of the 0th diffraction order should be kept low. According to Eqs. (9) and (10), the diffraction efficiency ratio as a function of grating depth h (see Fig. 1) is plotted in Fig. 4. It is obvious that the frequency-doubled effect will possibly occur when the depth of the grating is larger than 0.3 mm and smaller than 0.8 mm. The design method of the frequency-doubled effect can be extended to the realization of the frequency-tripled effect by choosing the proper period and depth of the resonant diffraction grating. Firstly, the grating period should be increased to have more diffraction orders, and the grating with periods 2.5l and 3.5l which ensure five and seven diffraction orders respectively
are chosen as examples. The diffraction efficiencies of the grating with periods 2.5l and 3.5l are shown in Fig. 5(a) and (b) at wavelength 488 nm. When the depth of the grating is between 0.4 mm and 0.6 mm, the efficiency of the 0th order is kept low and there exists the possibility to realize the frequency-tripled effect. As illustrated in Fig. 6, the depth of the grating is chosen as 0.5 mm and the diffraction distributions of the gratings with periods 2.5l and 3.5l still show periodicity along the x and z directions. We calculate the diffraction fields at zT and 2zT for gratings with periods 2.5l and 3.5l respectively, and the results indicate that Eq. (1) cannot be applied to grating with a larger period. However, the quasi-fractional Talbot effect can still be realized by choosing appropriate grating parameters. The frequency-tripled effect appears with grating period 2.5l at z¼3.45 mm and period 3.5l at z ¼3.1 mm as shown in Fig. 7. For grating with period 2.5l, the distribution shows poor contrast
Fig. 5. The diffraction efficiencies of different orders of the gratings with different periods (a) 2.5l and (b) 3.5l.
Fig. 6. Power flow distributions of gratings with different periods for (a) 2.5l and (b) 3.5l.
Fig. 7. One-dimensional power flow distributions of the gratings for (a) period 2.5l at z¼ 3.45 mm and (b) period 3.5l at z¼ 3.1 mm.
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Fig. 8. Experimental results (a) two-dimensional scanning field distribution at z ¼4.87 mm and (b) one-dimensional scanning the field distribution at z¼ 4.87 mm.
because of the existence of the 0th order. With the increasing of the number of diffraction orders, the efficiency of the 0th order reduces and the contrast of the distribution is getting better as shown in Fig. 7(b). It is explicit that the possibility to realize the multiple frequency effect increases with the increasing of the period of the grating.
4. Experiment According to Fig. 4, the fabrication tolerance of the depth of the grating ranges from 0.3 mm to 0.8 mm. The depth is chosen as 0.37 mm taking the fabrication condition into account. The incident light source is a semiconductor laser at the wavelength 488 nm. The grating which fabricated by electron beam directwriting lithography was placed on the sample stage, and the transmission fields at different distances above the grating were scanned by the near-field scanning optical microscopy (NSOM) with the positioning accuracy in the order of 10 nm. A two-dimensional scanning was carried out and the experimental results are shown in Fig. 8. The field distribution and the cross-lines of the field distribution of the resonant diffraction grating with period 1.95l (0.95 mm) around z¼4.87 mm are shown in Fig. 8(a) and (b). The period of the distribution is 0.46 mm which is half of the grating period. The experimental result is consistent with the simulation result in Fig. 3(c).
5. Conclusion It is shown that the frequency-doubled and frequency-tripled effects which have already found for grating with period much larger than the incident wavelength can still occur for resonant diffraction grating. The diffracted fields of resonant diffraction gratings with different periods are studied, and the diffraction fields have periodicity in both grating direction and the propagating direction. The simulation results are verified by experiments in the
use of NSOM. The quasi-fractional Talbot effect exhibits promising application in lithography for high-resolution patterning.
Acknowledgments This work was supported by the National Natural Science Foundation of China (60978047, 60678033) and the National Basic Research Program of China (2007CB935303). The experiment was completed in the Nano Optics Laboratory directed by Prof. Jia Wang at Tsinghua University.
References [1] [2] [3] [4] [5] [6]
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
W.H.F. Talbot, Philosophical Magazine 9 (1836) 401. A.W. Lohmann, J.A. Thomas, Applied Optics 29 (1990) 4337. A.W. Lohmann, D.E. Silva, Optics Communications 4 (1972) 326. He Kaijun, J. Jahns, A.W. Lohmann, Optics Communications 45 (1983) 295. S.P. Trivedi, S. Prakash, Santosh Rana, Osami Sasaki, Applied Optics 49 (2010) 897. A. Isoyan, F. Jiang, Y.C. Cheng, F. Cerrina, P. Wachulak, L. Urbanski, J. Rocca, C. Menoni, M. Marconi, Journal of Vacuum Science and Technology B 27 (2009) 2931. L. Stuerzebecher, T. Harzendorf, U. Vogler, U. Zeitner, R. Voelkel, Optics Express 18 (2010) 19485. E. Noponen, J. Turunen, Optics Communications 98 (1993) 132. S. Teng, Y. Tan, C. Cheng, Journal of the Optical Society of America A—Optics and Image Science 25 (2008) 2945. X. Wan, Q. Wang, H. Tao, Optics Communications 283 (2010) 5231. Y. Lu, C. Zhou, S. Wang, B. Wang, Journal of the Optical Society of America A—Optics and Image Science (2006) 2154. J. Zheng, C. Zhou, B. Wang, Optics Communications 281 (2008) 3254. S. Teng, W. Guo, C. Cheng, Journal of the Optical Society of America A—Optics and Image Science 27 (2010) 366. J. Zheng, C. Zhou, E. Dai, B. Wang, W. Jia, Optics Communications 281 (2008) 3230. R. Petit (Ed.), Springer-Verlag, Berlin, 1980. L.C. Botten, M.S. Craig, R.C. McPhedran, J.L. Adams, J.R. Andrewartha, Journal of Modern Optics 28 (1981) 413. J. Feng, C. Zhou, B. Wang, J. Zheng, W. Jia, H. Cao, P. Lv, Applied Optics 47 (2008) 6638.