- Email: [email protected]
Subwavelength focusing of micro grating-Fresnel lens
Optics Communications 298–299 (2013) 242–245
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Subwavelength focusing of micro grating-Fresnel lens Long Ma a, Jie Lin a, Yuan Ma b, Peng Jin a,n, Jiubin Tan a a b
Center of Ultra-precision Optoelectronic Instrument, Harbin Institute of Technology, Harbin 150080, China Department of Electrical and Computer Engineering, Dalhousie University, Halifax, B3H 4R2, Canada
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 November 2012 Received in revised form 4 February 2013 Accepted 18 February 2013 Available online 13 March 2013
The focusing performance of micro grating-Fresnel (G-Fresnel) diffractive optical element is investigated, for the first time, by finite-difference time-domain method based on rigorous electromagnetic theory. The focusing of G-Fresnel lens with different incident wavelength, focal length, and period of grating are explored. Numerical result shows that a subwavelength focusing can be achieved by GFresnel lens. Meanwhile, for incident beam with wavelength larger than the period of grating, the transmission of light decreases abruptly. The optical performance is governed by scattering effects from micro G-Fresnel lens. Our results provide very useful information in the field of optical imaging with high resolution and optical precision measurement. & 2013 Elsevier B.V. All rights reserved.
Keywords: Diffractive optical elements Grating-Fresnel lens Focusing performance
1. Introduction Recently, diffractive optical elements (DOEs) with a very small feature size can be fabricated using methods of microphotolithography technology [1–9]. DOEs have attracted extensive research interest considering its merits such as high power efficiency, easy fabrication, small dimensions, and light weight [10,11]. For DOEs with small feature size, the diffractive field is governed by light scattering near the fine structure. Therefore, the optical performance of DOE should be modeled accurately based on the rigorous electromagnetic theory [12,13]. In previous literatures, some numerical methods, such as boundary element method (BEM), finite element method (FEM) and finite-difference timedomain method (FDTD), have been developed to solve the Maxwell’s electromagnetic [14–19]. As we know, performance analysis of micro lens, diffractive micro lens and other DOEs are based on FDTD, in which the micro lens and diffractive micro lens are designed using geometrical optical theory [20]. With the advance of micro fabrication technology, complex micro optical devices are proposed by combining various diffractive optical elements. For example, dispersion and focusing can be simultaneously achieved by combining a Fresnel diffractive lens and a finite length grating, which is named hybrid grating-Fresnel (G-Fresnel) lens or GFresnel diffractive optical element. The G-Fresnel device of large size can be fabricated using soft lithography [21]. Their optical performance is analyzed by Fresnel diffractive integral and verified by experiments [22]. However, with the decrease of the size of Fresnel lens and the grating period, the focusing performance is
n
Corresponding author. Tel.: þ86 451 86412041; fax: þ86 451 86412698. E-mail address: [email protected] (P. Jin).
0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.02.041
governed by scattering and diffraction. The validity of the result based on scalar diffractive integral is limited. Therefore, accurate optical performance analysis of G-Fresnel lens requires rigorous solution to electromagnetic Maxwell’s equations. In this paper, we devote ourselves to investigate the optical performance of the two-dimensional (2D) G-Fresnel lens by rigorous electromagnetic theory and FDTD method. The focusing characteristics, such as the region field distribution and the intensity distribution along optical axis, are calculated in details. In Section 2, G-Fresnel lens and FDTD method are described. In Section 3, we present the numerical results together with detailed analysis. A conclusion is given in Section 4.
2. Description of G-Fresnel and FDTD method The schematic diagram of the two dimensional (2D) micro GFresnel diffractive optical element is shown in Fig. 1. The upper surface of the micro G-Fresnel lens is shaped by grating with finite length and the lower surface by Fresnel diffractive lens. The materials of grating and Fresnel diffractive lens are silicon and silica, respectively. The xy-plane is the incident plane and the collimated monochromatic beam with wavelength l is incident along the negative direction of y-axis. The 2D G-Fresnel lens is infinite along z-axis. As shown in Fig. 1, the period of grating is d, the total thickness of grating is h, height of bench is h2, and the substrate thickness of Fresnel diffractive lens is h1. If the transmittance of constituent grating and Fresnel diffractive lens are TG(x) and TF(x), the transmittance of G-Fresnel lens is governed by TðxÞ ¼ T G ðxÞT F ðxÞ [22]. The transmittances are determined by their surface relief, i.e., the thickness function h(x) of effective structure.
L. Ma et al. / Optics Communications 298–299 (2013) 242–245
The thickness function h(x) can be obtained in Refs. [21,22]. In general, the distribution of transmitted field of G-Fresnel lens can be solved by applying Huygens–Fresnel integral. However, as the device’s size decreases, the focusing performance of micro G-Fresnel lens is governed by scattering effects and its field distribution should be solved by rigorous electromagnetic theory. For our non-magnetic 2D G-Fresnel lens, in a Cartesian coordinate system for TE polarizations, Maxwell’s equations can be written as [23] @Ex 1 @Hy @Hx ¼ sEx @t e @x @y @Hx 1 @Ez ¼ þ sn Hx @t m @y @Hy 1 @Ez ¼ sn Hy ð1Þ @t m @x where Ex is the electric component and Hx, Hy are the magnetic components. Obviously, Ex, Hx and Hy are the functions of space coordinates (x, y) and time t. E, s, sn , and m are dielectric coefficient, electric conductivity, magnetic conductivity and permeability, respectively. By using Yee’s grids, Eq. (1) can be discretely solved by the FDTD method [24]. In the paper, the FDTD method is performed by Lumerical Solutions’ FDTD solutions [25].
3. Numerical results and analysis 3.1. Design of micro G-Fresnel lens As shown in Fig. 1, the period d of grating is assumed as 1:0 mm and the parameter s is 0:5 mm, i.e., duty cycle of grating is 1:1. The material of grating is silicon. In order to control the ability of phase modulation, the height h2 is chosen to be 0:125 mm. The substrate thickness of grating and Fresnel lens are h ¼ 0:25 mm and h1 ¼ 0:375 mm, respectively. The transverse size is 50:0 mm. The preset focal length is f ¼ 15:0 mm and the glass is chosen as the material in FDTD solutions. The incident beam with wavelength l can be chosen freely. For normal incident light, the grating equation can be expressed as d sin ym ¼ ml [26]. m is the order of diffraction and ym is the corresponding diffractive angle. For larger wavelength, the maximum order of diffraction decreases. For example, for l ¼ 1:0 mm, y 7 1 ¼ 7 901, it means that only zero order diffracted light is transmitted through grating and can be focused by Fresnel lens. Therefore, once the period d of grating is smaller than the wavelength of incident light, the phase modulation is ineffective for phase grating. Meanwhile, owing to the incident light being modulated by phase grating, the focusing performance of
d
s
243
G-Fresnel lens is different with that of lens illuminated by parallel plane light wave.
3.2. Sub-wavelength focusing of G-Fresnel lens For the aforementioned G-Fresnel lens with its parameters given in Section 3.1, the focusing performance is analyzed for normal TE polarized incident light with wavelength l ¼ 0:5 mm. The 2D electric field intensity distribution is displayed as Fig. 2. One can easily know that the focusing spot of zero order diffraction is located at y-axis. Based on the theory of geometric optics, the preset focus position of zero order diffraction is (0, f). The diffractive angle y of 71 order is determined by the equation sin y 7 m ¼ ml=d and y 7 1 ¼ 301 for incident light with wavelength 0:5 mm [26]. Therefore, the position of 7 1 order is focused at (f tan y 7 1 , f), i.e., ( 7 8.66, f). Numerical results also show that most of incident energy is focused at zero order focusing spot. Therefore, only the focusing performance of zero order focal spot is investigated and analyzed in details. The simulated results indicate that the real position of the zero order diffraction is (0, 14:57 mm). It means that focus shift appears in the hybrid device. That can be considered as the effect of finite size of grating and scattering of micro structure of G-Fresnel lens. In order to further elaborate the focusing characteristics of the designed G-Fresnel lens, the axial intensity distribution and the lateral intensity at real focal plane are investigated. Therefore, the axial intensity distribution is displayed in Fig. 3a, while the intensity distribution along the axis of y ¼ 14:57 mm is shown in Fig. 3b. Obviously, the maximum of intensity appears at optical axis as shown in Fig. 3a. However, due to the introduction of grating, the axial intensity distribution is a little different with that of a single lens. It is seen from Fig. 3b that most of the incident power is concentrated in the main lobe. The full-width at half maximum (FWHM) is 0:22 mm and the range of FWHM is marked by dashed line in Fig. 3b. The FWHM is approximately 0:44l and the subwavelength focusing is achieved. Focal distance of a G-Fresnel lens is larger than that of a single Fresnel lens. The electric intensity of the main lobe is lower than that of a single Fresnel lens for diffraction of grating. We can also see that the lateral size at the focal plane of a Fresnel lens is a little different than that of a G-Fresnel lens.
y h2 h1
h o
x
Fig. 1. A schematic diagram of a micro Grating-Fresnel lens (G-Fresnel).
Fig. 2. 2D electric intensity distribution for the G-Fresnel lens illuminated by TE polarized collimated beam with wavelength l ¼ 0:5 mm. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
L. Ma et al. / Optics Communications 298–299 (2013) 242–245
25
25
20
20
E Intensity(a.u.)
E Intensity(a.u.)
244
15
10
5
0
15
10
5
0
−2
−4
−6
−8
−10 −12 −14 −16 −18
0 −2
−1.5
−1
Y (microns)
−0.5
0
0.5
1
1.5
2
X (microns)
Fig. 3. For incident beam with wavelength l ¼ 0:5 mm, (a) and (b) display the electric field-intensity distributions of G-Fresnel lens on the axial plane and on the real focal plane, respectively.
Further, we investigated the focusing performance of G-Fresnel lens for different incident wavelengths. The parameters of G-Fresnel lens remain unchanged. For simplicity, the wavelength l of incident beam is chosen in the range of 0:4720:53 mm. The focusing characteristics are given in Table 1. The numerical results indicate that G-Fresnel lens has excellent performance for wavelength smaller than the period of grating. Obviously, most of the incident power is concentrated in the main lobe of focusing spot. From the table, it is apparent that Focal distance decreases as the wavelength increases, consistent with the grating equation. Depth of focus (DOF) also decreases as the wavelength increases. According to Table 1, tendency of FWHM is increased as wavelength is increased, except for the case of l ¼ 470 nm. The ratio of FWHM to l decreases from 0.47 to 0.44 as wavelength increases from 0.47 to 0:53 mm, and all cases wavelengths near l ¼ 500 nm break the Rayleigh diffraction limit. As is shown, it is steady around the target wavelength of 500 nm, for the ratio of FWHM to wavelength is generally invariant, while aberration still exists. Incident light with wavelength smaller than target wavelength is more sensitive than that with larger, therefore the focusing performance latter is not as good as the former, especially wavelength of 470 nm. The focusing performance is significantly different when the incident wavelength l is greater than or equal to the period d of grating. The grating equation d sin y ¼ ml can be rewritten as sin y ¼ m in the case of l ¼ d. Here m is an integer. Obviously, only zero order diffractive beams exist. In addition, the width of grating slit is half of the wavelength, resulting in energy of light transmitting through the slit decay fast. Therefore, power of the focusing spot is decreased sharply. The 2D intensity distribution for this case is shown in Fig. 4 as the incident wavelength l ¼ 1:0 mm. The light has been scattered into space, and no obvious focal spot can be observed behind the hybrid lens. The optical performance is governed by scattering effect of silicon grating for the micro hybrid G-Fresnel lens. In the case where incident light wavelength is shorter than the object wavelength, a focal spot can be obtained. The electric intensity distribution is consistent with what the diffraction theory indicates. But, by changing the light wavelength to l ¼ 0:4 mm, the maximal intensity is 2 orders smaller than the case as the light wavelength l ¼ 0:5 mm. It is mainly due to the absorb effect of silica. The absorb effect rises as the wavelength decreases from 500 nm. The absorption is referred to handbook of optical constants of solids, and a bigger imaginary part of refractive index indicates a higher loss of energy [27].
Table 1 Focal performance of different wavelength. Wavelength (nm)
Maximal Intensity
Focal distance ðmmÞ
DOF ðmmÞ
FWHM (nm)
FWHM=l
470 480 490 500 510 520 530
15.28 13.95 19.24 26.36 34.35 24.92 14.55
16.26 15.66 15.10 14.57 14.08 13.59 13.12
0.86 0.81 0.79 0.78 0.78 0.78 0.78
221 215 217 220 225 230 233
0.47 0.45 0.44 0.44 0.44 0.44 0.44
Fig. 4. Electric intensity distribution at TE polarized collimated incident beam with wavelength l ¼ 1:0 mm. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
4. Conclusion In conclusion, we used finite-difference time-domain method to investigate the optical performance of micro G-Fresnel lens, including the focal distance, maximum intensity, depth of focus, and the full-width at half maximum. There are some differences between the simulation results and what geometrical optics determine, implying that scattering effects of micro lens dominate
L. Ma et al. / Optics Communications 298–299 (2013) 242–245
the optical performance. The hybrid of grating and Fresnel lens combines the function of band-pass filter and focusing. In the case where wavelength equals to or is greater than the period of the grating, the light has been scattered into space. For ultraviolet light, the absorbing effect acts as the main function of the GratingFresnel lens. These properties are desirable in many technology fields, such as optical confocal microscopic imaging, subwavelength focusing and facet fabrication of laser.
Acknowledgments This work is funded by National Natural Science Foundation of China (Grant nos. 60978044, 61008039 and 51275111), the Program for New Century Excellent Talents in University (Grant NECT100059), the Research Fund for the Doctoral Program of Higher Education of China (Grant nos. 20102304110006 and 20102302120008) and the Fundamental Research Funds for the Central Universities (Grant no. HIT.NSRIF.2012017). References [1] H.P. Herzig, Micro-Optics: Elements, Systems and Applications, Taylor & Francis, London, 1997. [2] Michael T. Gale, Markus Rossi, Joern Pedersen, Helmut Schuetz, Optical Engineering 33 (1994) 3556. [3] T. Fujita, H. Nishihara, J. Koyama, Optics Letters 6 (1981) 613. [4] Gregory P. Behrmann, Michael T. Duignan, Applied Optics 36 (1997) 4666. [5] Madanagopal V. Kunnavakkam, F.M. Houlihan, M. Schlax, J.A. Liddle, P. Kolodner, O. Nalamasu, J.A. Rogers, Applied Physics Letters 82 (2003) 1152.
245
[6] C.G. Blough, M. Rossi, S.K. Mack, R.L. Michaels, Applied Optics 36 (1997) 4648. [7] H.J. Jiang, X.C. Yuan, Y. Zhou, Y.C. Chan, Y.L. Lam, Optics Communications 185 (2000) 19. [8] R. Huszank, S.Z. Szilasi, I. Rajta, A. Csik, Optics Communications 283 (2010) 176. [9] Yuchao Zhang, Nan Gao, Changqing Xie, Applied Physics Letters 100 (2012) 041107. [10] Jani Tervo, Jari Turunen, Optics Letters 25 (2000) 785. [11] Jennifer E. Curtis, Brian A. Koss, David G. Grier, Optics Communications 207 (2002) 169. [12] V.A. Soifer, Methods for Computer Design of Diffractive Optical Elements, John Wiley & Sons, Inc., New York, 2002. [13] D. Maystre, Progress in Optics 21 (1984) 1. [14] K. Hirayama, E.N. Glytsis, T.K. Gaylord, D.W. Wilson, Journal of the Optical Society of America A 13 (1996) 2219. [15] Jie Lin, Jiubin Tan, Jian Liu, Shutian Liu, Optics Express 17 (2009) 1466. [16] J.S. Ye, Y. Zhang, Optics Communications 283 (2010) 1661. [17] Bernd Lichtenberg, Neal C. Gallagher Jr., Optical Engineering 33 (1994) 3518. [18] Dennis W. Prather, Shouyuan Shi, Journal of the Optical Society of America A 16 (1999) 1131. [19] Mark S. Mirotznik, Dennis W. Prather, Joseph N. Mait, William A. Beck, Shouyuan Shi, Xiang Gao, Applied Optics 39 (2000) 2871. [20] J.W. Goodman, Introduction to Fourier Optics, McGraw-Hill, San Francisco, 1968. [21] Chuan Yang, Kebin Shi, Perry Edwards, Zhiwen Liu, Optics Express 18 (2010) 23529. [22] Chuan Yang, Perry Edwards, Kebin Shi, Zhiwen Liu, Optics Letters 36 (2011) 2023. [23] A. Taflove, Computational Electrodynamics, Artech House, Boston, 1995. [24] Kane Yee, IEEE Transactions on Antennas and Propagation 17 (1966) 302. [25] /http://www.lumerical.com/tcad-products/fdtd/S. [26] M. Born, E. Wolf, Principles of Optics, Pergamon, New York, 1980. [27] E.D. Palik, Handbook of Optical Constants of Solids, Academic Press, Orlando, 1998.
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Subwavelength focusing of micro grating-Fresnel lens Long Ma a, Jie Lin a, Yuan Ma b, Peng Jin a,n, Jiubin Tan a a b
Center of Ultra-precision Optoelectronic Instrument, Harbin Institute of Technology, Harbin 150080, China Department of Electrical and Computer Engineering, Dalhousie University, Halifax, B3H 4R2, Canada
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 November 2012 Received in revised form 4 February 2013 Accepted 18 February 2013 Available online 13 March 2013
The focusing performance of micro grating-Fresnel (G-Fresnel) diffractive optical element is investigated, for the first time, by finite-difference time-domain method based on rigorous electromagnetic theory. The focusing of G-Fresnel lens with different incident wavelength, focal length, and period of grating are explored. Numerical result shows that a subwavelength focusing can be achieved by GFresnel lens. Meanwhile, for incident beam with wavelength larger than the period of grating, the transmission of light decreases abruptly. The optical performance is governed by scattering effects from micro G-Fresnel lens. Our results provide very useful information in the field of optical imaging with high resolution and optical precision measurement. & 2013 Elsevier B.V. All rights reserved.
Keywords: Diffractive optical elements Grating-Fresnel lens Focusing performance
1. Introduction Recently, diffractive optical elements (DOEs) with a very small feature size can be fabricated using methods of microphotolithography technology [1–9]. DOEs have attracted extensive research interest considering its merits such as high power efficiency, easy fabrication, small dimensions, and light weight [10,11]. For DOEs with small feature size, the diffractive field is governed by light scattering near the fine structure. Therefore, the optical performance of DOE should be modeled accurately based on the rigorous electromagnetic theory [12,13]. In previous literatures, some numerical methods, such as boundary element method (BEM), finite element method (FEM) and finite-difference timedomain method (FDTD), have been developed to solve the Maxwell’s electromagnetic [14–19]. As we know, performance analysis of micro lens, diffractive micro lens and other DOEs are based on FDTD, in which the micro lens and diffractive micro lens are designed using geometrical optical theory [20]. With the advance of micro fabrication technology, complex micro optical devices are proposed by combining various diffractive optical elements. For example, dispersion and focusing can be simultaneously achieved by combining a Fresnel diffractive lens and a finite length grating, which is named hybrid grating-Fresnel (G-Fresnel) lens or GFresnel diffractive optical element. The G-Fresnel device of large size can be fabricated using soft lithography [21]. Their optical performance is analyzed by Fresnel diffractive integral and verified by experiments [22]. However, with the decrease of the size of Fresnel lens and the grating period, the focusing performance is
n
Corresponding author. Tel.: þ86 451 86412041; fax: þ86 451 86412698. E-mail address: [email protected] (P. Jin).
0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.02.041
governed by scattering and diffraction. The validity of the result based on scalar diffractive integral is limited. Therefore, accurate optical performance analysis of G-Fresnel lens requires rigorous solution to electromagnetic Maxwell’s equations. In this paper, we devote ourselves to investigate the optical performance of the two-dimensional (2D) G-Fresnel lens by rigorous electromagnetic theory and FDTD method. The focusing characteristics, such as the region field distribution and the intensity distribution along optical axis, are calculated in details. In Section 2, G-Fresnel lens and FDTD method are described. In Section 3, we present the numerical results together with detailed analysis. A conclusion is given in Section 4.
2. Description of G-Fresnel and FDTD method The schematic diagram of the two dimensional (2D) micro GFresnel diffractive optical element is shown in Fig. 1. The upper surface of the micro G-Fresnel lens is shaped by grating with finite length and the lower surface by Fresnel diffractive lens. The materials of grating and Fresnel diffractive lens are silicon and silica, respectively. The xy-plane is the incident plane and the collimated monochromatic beam with wavelength l is incident along the negative direction of y-axis. The 2D G-Fresnel lens is infinite along z-axis. As shown in Fig. 1, the period of grating is d, the total thickness of grating is h, height of bench is h2, and the substrate thickness of Fresnel diffractive lens is h1. If the transmittance of constituent grating and Fresnel diffractive lens are TG(x) and TF(x), the transmittance of G-Fresnel lens is governed by TðxÞ ¼ T G ðxÞT F ðxÞ [22]. The transmittances are determined by their surface relief, i.e., the thickness function h(x) of effective structure.
L. Ma et al. / Optics Communications 298–299 (2013) 242–245
The thickness function h(x) can be obtained in Refs. [21,22]. In general, the distribution of transmitted field of G-Fresnel lens can be solved by applying Huygens–Fresnel integral. However, as the device’s size decreases, the focusing performance of micro G-Fresnel lens is governed by scattering effects and its field distribution should be solved by rigorous electromagnetic theory. For our non-magnetic 2D G-Fresnel lens, in a Cartesian coordinate system for TE polarizations, Maxwell’s equations can be written as [23] @Ex 1 @Hy @Hx ¼ sEx @t e @x @y @Hx 1 @Ez ¼ þ sn Hx @t m @y @Hy 1 @Ez ¼ sn Hy ð1Þ @t m @x where Ex is the electric component and Hx, Hy are the magnetic components. Obviously, Ex, Hx and Hy are the functions of space coordinates (x, y) and time t. E, s, sn , and m are dielectric coefficient, electric conductivity, magnetic conductivity and permeability, respectively. By using Yee’s grids, Eq. (1) can be discretely solved by the FDTD method [24]. In the paper, the FDTD method is performed by Lumerical Solutions’ FDTD solutions [25].
3. Numerical results and analysis 3.1. Design of micro G-Fresnel lens As shown in Fig. 1, the period d of grating is assumed as 1:0 mm and the parameter s is 0:5 mm, i.e., duty cycle of grating is 1:1. The material of grating is silicon. In order to control the ability of phase modulation, the height h2 is chosen to be 0:125 mm. The substrate thickness of grating and Fresnel lens are h ¼ 0:25 mm and h1 ¼ 0:375 mm, respectively. The transverse size is 50:0 mm. The preset focal length is f ¼ 15:0 mm and the glass is chosen as the material in FDTD solutions. The incident beam with wavelength l can be chosen freely. For normal incident light, the grating equation can be expressed as d sin ym ¼ ml [26]. m is the order of diffraction and ym is the corresponding diffractive angle. For larger wavelength, the maximum order of diffraction decreases. For example, for l ¼ 1:0 mm, y 7 1 ¼ 7 901, it means that only zero order diffracted light is transmitted through grating and can be focused by Fresnel lens. Therefore, once the period d of grating is smaller than the wavelength of incident light, the phase modulation is ineffective for phase grating. Meanwhile, owing to the incident light being modulated by phase grating, the focusing performance of
d
s
243
G-Fresnel lens is different with that of lens illuminated by parallel plane light wave.
3.2. Sub-wavelength focusing of G-Fresnel lens For the aforementioned G-Fresnel lens with its parameters given in Section 3.1, the focusing performance is analyzed for normal TE polarized incident light with wavelength l ¼ 0:5 mm. The 2D electric field intensity distribution is displayed as Fig. 2. One can easily know that the focusing spot of zero order diffraction is located at y-axis. Based on the theory of geometric optics, the preset focus position of zero order diffraction is (0, f). The diffractive angle y of 71 order is determined by the equation sin y 7 m ¼ ml=d and y 7 1 ¼ 301 for incident light with wavelength 0:5 mm [26]. Therefore, the position of 7 1 order is focused at (f tan y 7 1 , f), i.e., ( 7 8.66, f). Numerical results also show that most of incident energy is focused at zero order focusing spot. Therefore, only the focusing performance of zero order focal spot is investigated and analyzed in details. The simulated results indicate that the real position of the zero order diffraction is (0, 14:57 mm). It means that focus shift appears in the hybrid device. That can be considered as the effect of finite size of grating and scattering of micro structure of G-Fresnel lens. In order to further elaborate the focusing characteristics of the designed G-Fresnel lens, the axial intensity distribution and the lateral intensity at real focal plane are investigated. Therefore, the axial intensity distribution is displayed in Fig. 3a, while the intensity distribution along the axis of y ¼ 14:57 mm is shown in Fig. 3b. Obviously, the maximum of intensity appears at optical axis as shown in Fig. 3a. However, due to the introduction of grating, the axial intensity distribution is a little different with that of a single lens. It is seen from Fig. 3b that most of the incident power is concentrated in the main lobe. The full-width at half maximum (FWHM) is 0:22 mm and the range of FWHM is marked by dashed line in Fig. 3b. The FWHM is approximately 0:44l and the subwavelength focusing is achieved. Focal distance of a G-Fresnel lens is larger than that of a single Fresnel lens. The electric intensity of the main lobe is lower than that of a single Fresnel lens for diffraction of grating. We can also see that the lateral size at the focal plane of a Fresnel lens is a little different than that of a G-Fresnel lens.
y h2 h1
h o
x
Fig. 1. A schematic diagram of a micro Grating-Fresnel lens (G-Fresnel).
Fig. 2. 2D electric intensity distribution for the G-Fresnel lens illuminated by TE polarized collimated beam with wavelength l ¼ 0:5 mm. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
L. Ma et al. / Optics Communications 298–299 (2013) 242–245
25
25
20
20
E Intensity(a.u.)
E Intensity(a.u.)
244
15
10
5
0
15
10
5
0
−2
−4
−6
−8
−10 −12 −14 −16 −18
0 −2
−1.5
−1
Y (microns)
−0.5
0
0.5
1
1.5
2
X (microns)
Fig. 3. For incident beam with wavelength l ¼ 0:5 mm, (a) and (b) display the electric field-intensity distributions of G-Fresnel lens on the axial plane and on the real focal plane, respectively.
Further, we investigated the focusing performance of G-Fresnel lens for different incident wavelengths. The parameters of G-Fresnel lens remain unchanged. For simplicity, the wavelength l of incident beam is chosen in the range of 0:4720:53 mm. The focusing characteristics are given in Table 1. The numerical results indicate that G-Fresnel lens has excellent performance for wavelength smaller than the period of grating. Obviously, most of the incident power is concentrated in the main lobe of focusing spot. From the table, it is apparent that Focal distance decreases as the wavelength increases, consistent with the grating equation. Depth of focus (DOF) also decreases as the wavelength increases. According to Table 1, tendency of FWHM is increased as wavelength is increased, except for the case of l ¼ 470 nm. The ratio of FWHM to l decreases from 0.47 to 0.44 as wavelength increases from 0.47 to 0:53 mm, and all cases wavelengths near l ¼ 500 nm break the Rayleigh diffraction limit. As is shown, it is steady around the target wavelength of 500 nm, for the ratio of FWHM to wavelength is generally invariant, while aberration still exists. Incident light with wavelength smaller than target wavelength is more sensitive than that with larger, therefore the focusing performance latter is not as good as the former, especially wavelength of 470 nm. The focusing performance is significantly different when the incident wavelength l is greater than or equal to the period d of grating. The grating equation d sin y ¼ ml can be rewritten as sin y ¼ m in the case of l ¼ d. Here m is an integer. Obviously, only zero order diffractive beams exist. In addition, the width of grating slit is half of the wavelength, resulting in energy of light transmitting through the slit decay fast. Therefore, power of the focusing spot is decreased sharply. The 2D intensity distribution for this case is shown in Fig. 4 as the incident wavelength l ¼ 1:0 mm. The light has been scattered into space, and no obvious focal spot can be observed behind the hybrid lens. The optical performance is governed by scattering effect of silicon grating for the micro hybrid G-Fresnel lens. In the case where incident light wavelength is shorter than the object wavelength, a focal spot can be obtained. The electric intensity distribution is consistent with what the diffraction theory indicates. But, by changing the light wavelength to l ¼ 0:4 mm, the maximal intensity is 2 orders smaller than the case as the light wavelength l ¼ 0:5 mm. It is mainly due to the absorb effect of silica. The absorb effect rises as the wavelength decreases from 500 nm. The absorption is referred to handbook of optical constants of solids, and a bigger imaginary part of refractive index indicates a higher loss of energy [27].
Table 1 Focal performance of different wavelength. Wavelength (nm)
Maximal Intensity
Focal distance ðmmÞ
DOF ðmmÞ
FWHM (nm)
FWHM=l
470 480 490 500 510 520 530
15.28 13.95 19.24 26.36 34.35 24.92 14.55
16.26 15.66 15.10 14.57 14.08 13.59 13.12
0.86 0.81 0.79 0.78 0.78 0.78 0.78
221 215 217 220 225 230 233
0.47 0.45 0.44 0.44 0.44 0.44 0.44
Fig. 4. Electric intensity distribution at TE polarized collimated incident beam with wavelength l ¼ 1:0 mm. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
4. Conclusion In conclusion, we used finite-difference time-domain method to investigate the optical performance of micro G-Fresnel lens, including the focal distance, maximum intensity, depth of focus, and the full-width at half maximum. There are some differences between the simulation results and what geometrical optics determine, implying that scattering effects of micro lens dominate
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the optical performance. The hybrid of grating and Fresnel lens combines the function of band-pass filter and focusing. In the case where wavelength equals to or is greater than the period of the grating, the light has been scattered into space. For ultraviolet light, the absorbing effect acts as the main function of the GratingFresnel lens. These properties are desirable in many technology fields, such as optical confocal microscopic imaging, subwavelength focusing and facet fabrication of laser.
Acknowledgments This work is funded by National Natural Science Foundation of China (Grant nos. 60978044, 61008039 and 51275111), the Program for New Century Excellent Talents in University (Grant NECT100059), the Research Fund for the Doctoral Program of Higher Education of China (Grant nos. 20102304110006 and 20102302120008) and the Fundamental Research Funds for the Central Universities (Grant no. HIT.NSRIF.2012017). References [1] H.P. Herzig, Micro-Optics: Elements, Systems and Applications, Taylor & Francis, London, 1997. [2] Michael T. Gale, Markus Rossi, Joern Pedersen, Helmut Schuetz, Optical Engineering 33 (1994) 3556. [3] T. Fujita, H. Nishihara, J. Koyama, Optics Letters 6 (1981) 613. [4] Gregory P. Behrmann, Michael T. Duignan, Applied Optics 36 (1997) 4666. [5] Madanagopal V. Kunnavakkam, F.M. Houlihan, M. Schlax, J.A. Liddle, P. Kolodner, O. Nalamasu, J.A. Rogers, Applied Physics Letters 82 (2003) 1152.
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[6] C.G. Blough, M. Rossi, S.K. Mack, R.L. Michaels, Applied Optics 36 (1997) 4648. [7] H.J. Jiang, X.C. Yuan, Y. Zhou, Y.C. Chan, Y.L. Lam, Optics Communications 185 (2000) 19. [8] R. Huszank, S.Z. Szilasi, I. Rajta, A. Csik, Optics Communications 283 (2010) 176. [9] Yuchao Zhang, Nan Gao, Changqing Xie, Applied Physics Letters 100 (2012) 041107. [10] Jani Tervo, Jari Turunen, Optics Letters 25 (2000) 785. [11] Jennifer E. Curtis, Brian A. Koss, David G. Grier, Optics Communications 207 (2002) 169. [12] V.A. Soifer, Methods for Computer Design of Diffractive Optical Elements, John Wiley & Sons, Inc., New York, 2002. [13] D. Maystre, Progress in Optics 21 (1984) 1. [14] K. Hirayama, E.N. Glytsis, T.K. Gaylord, D.W. Wilson, Journal of the Optical Society of America A 13 (1996) 2219. [15] Jie Lin, Jiubin Tan, Jian Liu, Shutian Liu, Optics Express 17 (2009) 1466. [16] J.S. Ye, Y. Zhang, Optics Communications 283 (2010) 1661. [17] Bernd Lichtenberg, Neal C. Gallagher Jr., Optical Engineering 33 (1994) 3518. [18] Dennis W. Prather, Shouyuan Shi, Journal of the Optical Society of America A 16 (1999) 1131. [19] Mark S. Mirotznik, Dennis W. Prather, Joseph N. Mait, William A. Beck, Shouyuan Shi, Xiang Gao, Applied Optics 39 (2000) 2871. [20] J.W. Goodman, Introduction to Fourier Optics, McGraw-Hill, San Francisco, 1968. [21] Chuan Yang, Kebin Shi, Perry Edwards, Zhiwen Liu, Optics Express 18 (2010) 23529. [22] Chuan Yang, Perry Edwards, Kebin Shi, Zhiwen Liu, Optics Letters 36 (2011) 2023. [23] A. Taflove, Computational Electrodynamics, Artech House, Boston, 1995. [24] Kane Yee, IEEE Transactions on Antennas and Propagation 17 (1966) 302. [25] /http://www.lumerical.com/tcad-products/fdtd/S. [26] M. Born, E. Wolf, Principles of Optics, Pergamon, New York, 1980. [27] E.D. Palik, Handbook of Optical Constants of Solids, Academic Press, Orlando, 1998.