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By investigating the difference between the analytic solutions obtained from commonly used two-dimensional effective medium theory and the numerical solutions, we found that any analytical solution was quite accurate only at its right normalized cycle, determined by its own effective range. Thus, one should solve the problem that there was no closed-form solution for the effective permittivity of a two-dimensional zero-order grating, and expand the applied scope of the effective medium theory to the boundary of zero-order diffraction. Secondly, by using the two-dimensional analytical solution, we have designed a nanowires anti-reflection layer in silicon, which fully meet the needs of the design that reach zero reflectance at 650 nm; and the spectrum averaged reflection from 310-1120 nm is 8%, lower than silicon nitride anti-reflection layer 9.9%. Stavenga formula can be used to design a large normalized period antireflective microstructure, while the Maxwell-Garnett formula can be used to design a small normalized cycle antireflective microstructure. Design of antireflection structure by two-dimensional closed form solution directly is viable, which have huge potential application value.
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Keywords:
- zero-order diffraction grating /
- effective medium theory /
- anti-reflection for solar cell /
- nanowires array
[1] Kelzenberg M D, Boettcher S W, Petykiewicz J A, Turner-Evans D B, Putnam M C, Warren E L, Spurgeon J M, Briggs R M, Lewis N S, Atwater H A 2010 Nat. Mater. 9 239
[2] Polman A, Atwater H A 2012 Nat. Mater. 11 174
[3] Kuo M L, Poxson D J, Kim Y S, Mont F W, Kim J K, Schubert E F, Lin S Y 2008 Opt. Lett. 11 174
[4] Prevo B G, Hon E W, Velev O D 2007 J. Mater. Chem. 17 791
[5] Her T H, Finlay R J, Wu C, Deliwala S, Mazur E 1998 Appl. Phys. Lett. 73 1673
[6] Strehlke S, Bastide S, Guillet J, Le’vy-Cle’ment C 2000 Mater. Sci. Eng. B 69 81
[7] Xi J Q, Schubert M F, Kim J K, Schubert E F, Chen M, Lin S Y, Liu W, Smart J A 2007 Nat. Photonics 1 176
[8] Grann E B, Moharam M G, Pommet D A 1994 J. Opt. Soc. Am. A 11 2695
[9] Green M A, Pillai S 2012 Nat. Photonics 6 130
[10] Spinelli P, Verschuuren M A, Polman A 2012 Nat. Comms. 3:692 1
[11] Raguin D H, Morris G M 1993 Appl. Opt. 32 1154
[12] Raguin D H, Morris G M 1993 Appl. Opt. 32 2582
[13] Scheller M, Wietzke S, Jansen C, Koch M 2009 J. Phys. D: Appl. Phys. 42 065415
[14] Sancho-Parramon J, Janicki V 2008 J. Phys. D: Appl. Phys. 41 215304
[15] Chen S H, Wang H W, Chang T W 2012 Opt. Express 20 A197
[16] Aspnes D E 2011 Thin Solid Films 519 2571
[17] Grann E B, Moharam M G, Pommet D A 1995 J. Opt. Soc. Am. A 12 333
[18] Sun C H, Jiang P, Jiang B 2008 Appl. Phys. Lett. 92 061112
[19] Huang Y F, Chattopadhyay S, Jen Y J, Peng C Y, Liu T, Hsu Y K, Pan C L, Lo H C, Hsu C H, Chang Y H, Lee C S, Chen K H, Chen L C 2007 Nat. Nanotech. 2 770
[20] Stavenga D G, Foletti S, Palasantzas G, Arikawa K 2006 Proc. R. Soc. B 273 661
[21] Pei T H, Thiyagu S, Pei Z 2011 Appl. Phys. Lett. 99 153108
[22] Zhu J, Yu Z F, Burkhard G F, Hsu C M, Connor S T, Xu Y Q, Wang Q, McGehee M, Fan S H, Cui Y 2009 Nano Lett. 9 279
[23] Motamedi M E, Southwell W H, Gunning W J 1992 Appl. Opt. 31 4371
[24] Hu L, Chen L 2007 Nano Lett. 7 3250
[25] Xiong Z Q, Zhao F Y, Yang J, Hu X H 2010 Appl. Phys. Lett. 96 181903
[26] Heine C, Morf R H 1995 Appl. Opt. 34 2476
[27] Kikuta H, Toyota H, Yu W J 2003 Opt. Rev. 10 63
[28] Gaylord T K, Baird W E, Moharam M G 1986 Appl. Opt. 25 4562
[29] Diedenhofen S L, Janssen O T A, Grzela G, Bakkers E P A M, Rivas J G 2011 Acs Nano 5 2316
[30] Southwell W H 1991 J. Opt. Soc. Am. A 8 549
[31] Merrilll W M, Diaz R E, LoRe M M, Squires M C, Alexopoulos N G 1999 IEEE Trans. Antennas Propag. 47 142
[32] Chen F T, Craighead H G 1995 Opt. Lett. 20 121
[33] Pe’ rez R 2009 J. Appl. Polym. Sci. 113 2264
[34] Zhou J, Sun Y T, Sun T T, Liu X, Song W J 2011 Acta Phys. Sin. 60 088802 (in Chinese) [周骏, 孙永堂, 孙铁囤, 刘晓, 宋伟杰 2011 60 088802]
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[1] Kelzenberg M D, Boettcher S W, Petykiewicz J A, Turner-Evans D B, Putnam M C, Warren E L, Spurgeon J M, Briggs R M, Lewis N S, Atwater H A 2010 Nat. Mater. 9 239
[2] Polman A, Atwater H A 2012 Nat. Mater. 11 174
[3] Kuo M L, Poxson D J, Kim Y S, Mont F W, Kim J K, Schubert E F, Lin S Y 2008 Opt. Lett. 11 174
[4] Prevo B G, Hon E W, Velev O D 2007 J. Mater. Chem. 17 791
[5] Her T H, Finlay R J, Wu C, Deliwala S, Mazur E 1998 Appl. Phys. Lett. 73 1673
[6] Strehlke S, Bastide S, Guillet J, Le’vy-Cle’ment C 2000 Mater. Sci. Eng. B 69 81
[7] Xi J Q, Schubert M F, Kim J K, Schubert E F, Chen M, Lin S Y, Liu W, Smart J A 2007 Nat. Photonics 1 176
[8] Grann E B, Moharam M G, Pommet D A 1994 J. Opt. Soc. Am. A 11 2695
[9] Green M A, Pillai S 2012 Nat. Photonics 6 130
[10] Spinelli P, Verschuuren M A, Polman A 2012 Nat. Comms. 3:692 1
[11] Raguin D H, Morris G M 1993 Appl. Opt. 32 1154
[12] Raguin D H, Morris G M 1993 Appl. Opt. 32 2582
[13] Scheller M, Wietzke S, Jansen C, Koch M 2009 J. Phys. D: Appl. Phys. 42 065415
[14] Sancho-Parramon J, Janicki V 2008 J. Phys. D: Appl. Phys. 41 215304
[15] Chen S H, Wang H W, Chang T W 2012 Opt. Express 20 A197
[16] Aspnes D E 2011 Thin Solid Films 519 2571
[17] Grann E B, Moharam M G, Pommet D A 1995 J. Opt. Soc. Am. A 12 333
[18] Sun C H, Jiang P, Jiang B 2008 Appl. Phys. Lett. 92 061112
[19] Huang Y F, Chattopadhyay S, Jen Y J, Peng C Y, Liu T, Hsu Y K, Pan C L, Lo H C, Hsu C H, Chang Y H, Lee C S, Chen K H, Chen L C 2007 Nat. Nanotech. 2 770
[20] Stavenga D G, Foletti S, Palasantzas G, Arikawa K 2006 Proc. R. Soc. B 273 661
[21] Pei T H, Thiyagu S, Pei Z 2011 Appl. Phys. Lett. 99 153108
[22] Zhu J, Yu Z F, Burkhard G F, Hsu C M, Connor S T, Xu Y Q, Wang Q, McGehee M, Fan S H, Cui Y 2009 Nano Lett. 9 279
[23] Motamedi M E, Southwell W H, Gunning W J 1992 Appl. Opt. 31 4371
[24] Hu L, Chen L 2007 Nano Lett. 7 3250
[25] Xiong Z Q, Zhao F Y, Yang J, Hu X H 2010 Appl. Phys. Lett. 96 181903
[26] Heine C, Morf R H 1995 Appl. Opt. 34 2476
[27] Kikuta H, Toyota H, Yu W J 2003 Opt. Rev. 10 63
[28] Gaylord T K, Baird W E, Moharam M G 1986 Appl. Opt. 25 4562
[29] Diedenhofen S L, Janssen O T A, Grzela G, Bakkers E P A M, Rivas J G 2011 Acs Nano 5 2316
[30] Southwell W H 1991 J. Opt. Soc. Am. A 8 549
[31] Merrilll W M, Diaz R E, LoRe M M, Squires M C, Alexopoulos N G 1999 IEEE Trans. Antennas Propag. 47 142
[32] Chen F T, Craighead H G 1995 Opt. Lett. 20 121
[33] Pe’ rez R 2009 J. Appl. Polym. Sci. 113 2264
[34] Zhou J, Sun Y T, Sun T T, Liu X, Song W J 2011 Acta Phys. Sin. 60 088802 (in Chinese) [周骏, 孙永堂, 孙铁囤, 刘晓, 宋伟杰 2011 60 088802]
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