光学变换:从量子到经典
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摘要
概率论是研究大量随机现象的统计规律的学科。用数学语言来说，就是研究对随机现象的观察次数趋于无穷时，“它的极限”呈现出的某种规律性。因此强极限理论在概率论中占有重要地位。二十世纪六十年代以来，继独立随机变量和序列的极限理论获得完善发展之后，各种混合随机变量序列、相伴随机变量序列及鞅的强极限理论又有很大发展，我国学者在这方面做出了许多出色的工作，在国际上也有一定的影响(参见[43，76，108，81，77，82])。强极限理论在国际上的文献浩如烟海。关于强极限理论的经典结果可参见专著[14，13，70，28，79]，而最近的文献可参见[31，4，32，69，16，11]。信息论的熵定理也称Shannon—McMillan定理或信源的渐进均分割性(AEP)，是信息论的基本定理，是各种编码定理的基础。关于熵定理的最新发展可参考文献[26]。
设{X_n，n≥0}为随机变量序列，如果
E[f(X_(n+1))_X_0，…，X_n]=E[f(X_(n+1))|X_N]a.s. (-1.0.1)其中f为有界函数。则称{X_n，n≥0}为马氏链。如果E[f(X_(n+1))|X_n]与n有关，则称{X_n，n≥0}为非齐次马氏链，如果E[f(X_(n+1))|X_n]与n无关，则称{X_n，n≥0}为齐次马氏链。如果{X_n，n≥0}在有限或可列状态空间取值，则称之为有限或可列马氏链，如果{X_n，n≥0}在一般状态空间取值，则称之为在一般状态空间取值的马氏链。如果
E[f(X_(n+1))|X_0，…，X_n]=E[f(X_(n+1))|X_n，…，X_(n－k+1)]a.s. (-1.0.2)且{X_n，n≥0}与n有关，则称{X_n，n≥0}为非齐次K阶马氏链，
马氏随机场是马氏过程推广到多维指标情形。由于有广泛的应用前景而受到物理学、概率论、信息论界的广泛兴趣。由于马氏随机场具有相变现象，其研究内容更加深刻且具有很大的难度。马氏随机场理论是近年来发展起来的概率论重要分支之一，而马氏随机场极限理论又是其中重要的研究内容，其中关于马氏随机场强极限定理的研究目前尚无系统和深刻的结果。本博士论文将推进这方面的研究。
By virtue of the technique of integration within an ordered product (IWOP) of operators we find a one-to-one correspondence between the unitary operators in quantum optics and the diffraction integral transforms in classical optics. This is achieved by setting up new quantum mechanical representations that represent various photon field distributions and constructing new unitary operators that describe the transformations between those vectors. By this means, not only can the theory of representation and transformation be efficiently applied to the study of classical optics, but also the optical transform theory can be adopted to develop the quantum optical theory. The main content includes:
The IWOP technique efficiently generalize the Newton-Leibniz integration formula which only applies to commuting functions of continuum variables to the noncommutative operators made of Dirac's symbols and greatly simplifies the calculation of operators. By using the IWOP technique we set up several new quantum mechanical representations, including deduced entangled state representation, intermediate coordinate-momentum state representation, intermediate entangled state representation, etc. We prove that the transformation between two conjugate deduced entangled state representations just corresponds to the famous Hankel transform in classical optics. The Hatikel transform is a most fundamental optical transform and is as important as the Fourier transform, however, people only know the transformation between the coordinate eigenstate and the momentum eigenstate corresponds to the Fourier transform before. The discover of the above-mentioned relation reveal the more general relation between two research regions, and now the optical transformations which abide by cylindrical symmetiy (including the Hankel transformation) can be studied by means of the deduced entangled representations. In addition, we find the results of measuring quantum states in the intermediate states correspond to the Radon transformations of Wigner functions of corresponding classical optical fields, which manifestly shows that the
设{X_n，n≥0}为随机变量序列，如果
E[f(X_(n+1))_X_0，…，X_n]=E[f(X_(n+1))|X_N]a.s. (-1.0.1)其中f为有界函数。则称{X_n，n≥0}为马氏链。如果E[f(X_(n+1))|X_n]与n有关，则称{X_n，n≥0}为非齐次马氏链，如果E[f(X_(n+1))|X_n]与n无关，则称{X_n，n≥0}为齐次马氏链。如果{X_n，n≥0}在有限或可列状态空间取值，则称之为有限或可列马氏链，如果{X_n，n≥0}在一般状态空间取值，则称之为在一般状态空间取值的马氏链。如果
E[f(X_(n+1))|X_0，…，X_n]=E[f(X_(n+1))|X_n，…，X_(n－k+1)]a.s. (-1.0.2)且{X_n，n≥0}与n有关，则称{X_n，n≥0}为非齐次K阶马氏链，
马氏随机场是马氏过程推广到多维指标情形。由于有广泛的应用前景而受到物理学、概率论、信息论界的广泛兴趣。由于马氏随机场具有相变现象，其研究内容更加深刻且具有很大的难度。马氏随机场理论是近年来发展起来的概率论重要分支之一，而马氏随机场极限理论又是其中重要的研究内容，其中关于马氏随机场强极限定理的研究目前尚无系统和深刻的结果。本博士论文将推进这方面的研究。
By virtue of the technique of integration within an ordered product (IWOP) of operators we find a one-to-one correspondence between the unitary operators in quantum optics and the diffraction integral transforms in classical optics. This is achieved by setting up new quantum mechanical representations that represent various photon field distributions and constructing new unitary operators that describe the transformations between those vectors. By this means, not only can the theory of representation and transformation be efficiently applied to the study of classical optics, but also the optical transform theory can be adopted to develop the quantum optical theory. The main content includes:
The IWOP technique efficiently generalize the Newton-Leibniz integration formula which only applies to commuting functions of continuum variables to the noncommutative operators made of Dirac's symbols and greatly simplifies the calculation of operators. By using the IWOP technique we set up several new quantum mechanical representations, including deduced entangled state representation, intermediate coordinate-momentum state representation, intermediate entangled state representation, etc. We prove that the transformation between two conjugate deduced entangled state representations just corresponds to the famous Hankel transform in classical optics. The Hatikel transform is a most fundamental optical transform and is as important as the Fourier transform, however, people only know the transformation between the coordinate eigenstate and the momentum eigenstate corresponds to the Fourier transform before. The discover of the above-mentioned relation reveal the more general relation between two research regions, and now the optical transformations which abide by cylindrical symmetiy (including the Hankel transformation) can be studied by means of the deduced entangled representations. In addition, we find the results of measuring quantum states in the intermediate states correspond to the Radon transformations of Wigner functions of corresponding classical optical fields, which manifestly shows that the
引文
[1] M. Planck, Ann. d. Physik 4 (1901) 553
[2] M. Planck, Ann. d. Physik 4 (1901) 564
[3] A. Einstein, Ann. d. Physik 17 (1905) 132 [English translation in Collected Papers of Albert Einstein, Vol. 2 (1900-1909), Princton University Press (Pfincton, NJ)]
[4] R. H. Brown and R. Q. Twiss, Nature 177 (1956) 27
[5] R. H. Brown and R. Q. Twiss, Nature 178 (1956) 1046
[6] R. J. Glauber, Phys. Rev. 130 (1963) 2529
[7] R. J. Glauber, Phys. Rev. 131 (1963) 2766
[8] W. Heitler, The Quantum Theory of Radiation, (3rd edition), Oxford Clarendon Press, London, 1954
[9] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge, 1996
[10] M. O. Scully and m. S. Zubairy, Quantum Optics, Cambridge University Press, 1997
[11] D. F. Walls and G. J. Milburn, Quantum Optics, Springer-Verlag, 1994
[12] W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH, 2001
[13] M. Born and E. Wolf, Principles of Optics, Cambridge University Press, 1999 (seventh edition)
[14] V. Namias, J. Inst. Maths. Applics 25 (1980) 241
[15] A. W. Lohmann, J. Opt. Soc. Am. A 10 (1993) 2181
[16] D. Dragoman, PROGRESS IN OPTICS, 42 (2002) 424 and references therein
[17] Dragoman, D., and M. Dragoman, Prog. Quantum Electron. 23 1999 131
[18] O. Crasser, H. Mack and W. P. Schleich, quant-ph/0402115
[19] G. Nienhuis and L. Allen, Phys. Rev. A 48 (1993) 656
[20] Wolf, K. B., and E. V. Kurmyshev, Phys. Rev. A 47 1993 3365
[21] 范洪义，量子力学表象与变换论，上海科技出版社，1997
[22] 范洪义，量子力学纠缠态及其应用，上海交通大学出版社，2001
[23] Hong-yi Fan, Hai-liang Lu and Yue Fan, Annals of Physics 321 (2006) 480-494
[24] A. Sommerfeld, Optics, volume Ⅳ of Lectures on Theoretical Physics, Academic Press New York, NY, 1954
[25] J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill 1968
[26] D. Mendlovic, H. M. Ozakatas. J. Opt. Soc. Am. A 10 (1993) 1875
[27] D. Mendlovic, H. M. Ozaktas and A. W. Lohmann, Appl. Opt. 33 (1994)6188;
[28] H. M. Ozakatas, D. Mendlovic. J. Opt. Soc. Am. A 10 (1993) 2522
[29] J. Hua, L. Liu and G. Li, Opt. Comm. 137 (1997) 11
[30] Shaomin Wang and Daomu Zhao, Matrix Optics, China Higher Education Press, Beijing and Springer-Verlag, Berlin Heidelberg, 2000
[31] A. Gerrard and J. M. Burch Introduction to Matrix Methods in Optics John Wiley&Sons, Ltd., London, 1975
[32] H. Kogelnik, Applied Optics 4 (1965) 1562
[33] M. J. Bastiaans, in: Laser Propagation, eds. P. M. Meijas, H. Weber, R. Martinez-Herrero and A. Gonzalez-Urena (S. E. D. O., Madrid, 1993) pp. 65-87
[34] M. J. Bastiaans, Opt. Comm. 25 (1978) 26
[35] S. A. Collins, J. Opt. Soc. Am. 60 (1970) 1168
[36] J. A. Arnaud, J. Opt. Soc. Am. 61 (1971) 751
[37] C. Gomez-Reino, Int. J. Optoelectron. 7 (1992) 607
[38] O. Seger, Ph. D. Dissertation 301 (Linkoping University, 1993)
[39] T. Alieva and F. Agullo-Lopez, Opt. Comm., 114 (1995) 161; 125 (1996) 267
[40] D. F. V. James and G. S. Agarwal, Opt. Comm. 126 (1996) 207
[41] J. A. Arnaud, Beam andFiber Optics Academic Press, 1976
[42] P. A. M. Dirac, The Principles of Quantum Mechanics, 3rd, Clarendon Press, Oxford (1930)
[43] J. R. Klauder, Annals Phys. (New York) 11 (1960) 123
[44] J. R. Klauder and B.-S. Skagerstam, Coherent States, World Scientific, Singapore, 1985
[45] Fan Hong-yi, H. R. Zaidi and J. R. Klauder, Phys. Rev. D (1987) 1831; Fan Hongyi and H. R. Zaidi, Phys. Rev. A 37 (1988)2985; Hongyi Fan and J. R. Klauder, J. Phys. A 21 (1988)L715; Hongyi Fan and Hui Zou, Phys. Lett. A 252 (1999) 281; Hong-yi Fan and A. Wǖnsche, J. Opt. B: Quantum & Semiclass. Opt. 2 (2000) 464
[46] Weyl, H., Z. Phys. 46 (1927) 1-46.
[47] Fan Hong-yi, Phys. Lett. A 161 (1991)1
[48] Hong-yi Fan and Vanderlinde J., J. Phys. A 23 1990 L1113
[49] Louisell, W. H. Quantum Statistical Properties of Radiation (Wiley, New York, 1973).
[50] Hong-yi Fan, J. Opt. B: Quantum & Semiclass. Opt. 5 (2003) R147; Inter. J. Mod. Phys. 18 (2004) 1387
[51] H. R Yuen, Phys. Rev. A 13 (1976)2226
[52] R. Loudon and P. L. Knight, J. Mod. Opt. 34 (1987) 709
[53] D. F. Walls, Nature 306 (1983) 141
[54] J. Mehra and H. Rechenberg, The Historical Development of Quantum Theory, Vol. 4: The Fundamental Equations of Quantum Mechanics 1925-1926. The Reception of the New Quantum Mechanics (Springer-Verlag, New York, 1982)
[55] E. Wigner, Phys. Rev. 40 (1932) 749
[56] Hong-yi Fan and H. R. Zaidi, Phys. Lett. A 124 (1987) 303
[57] A. Einstein, B. Podolsky & N. Rosen, Phys. Rev. 47 (1935) 777
[58] Hong-yi Fan and Klauder, J. R., Phys. Rev. A 49 (1994) 704
[59] H.-Y. Fan, H. Zou and Y. Fan, Phys. Lett. A 254 (1999) 137
[60] Hong-yi Fan, Phys. Lett. A 313 (2003) 343
[61] Hong-yi Fan, Phys. Rev. A 65 (2002) 064102
[62] Hong-yi Fan, J. Phys. A 25 (1992) 3443
[63] Hong-yi Fan and Yue Fan, Int. J. Mod. Phys. A 17 (2002) 701
[64] Vogel K and Risken H, Phys. Rev. A 40 (1989) 2847.
[65] Smithey D T, Beck M, Raymer M G and Faridani A, Phys. Rev. Lett. 70 (1993) 1244.
[66] Hong-yi Fan and Yue Fan, Euro. Phys. J. D 21 (2002) 233
[67] A. Erdelyi, Higher Transcendental Functions: The Batemann Manuscript Project (McGraw-Hill, New York, 1953)
[68] L. Yu et al., Opt. Lett. 23 (1998) 1158
[69] Hong-yi Fan and Hai-liang Lu, Opt. Lett. 28 (2003) 680
[70] 特殊函数概论，王竹溪，郭敦仁，北京大学出版社，2000
[71] Hai-liang Lu and Hong-yi Fan, Comm. Theor. Phys., in press
[72] D. Gabor, J. IEE, 93 (1946)429
[73] J. Preskill, Lecture Notes for Physics 229: Quantum Information and Computation (Pasadena, CA: California Institute of Technology)
[74] Hong-yi Fan, Phys. Lett. A 294 (2002) 253
[75] Hong-yi Fan and Hai-liang Lu and Hao Ren, Int. J. Mod. Phys. B., in press
[76] Luis B. Almeida, IEEE Singal Processing Letters 4 (1997) 15
[77] Ahmed I. Zayed, IEEE Singal Processing Letters 5 (1998) 101-103
[78] H. Goldstein, Classical Mechanics, the second edition, Addisson-Wesley, Massachusetts (1980)
[79] Fan Hong-yi and Chen Jun-hua, Commun. Theor. Phys. 38 (2002) 147; 40 (2003) 589; Fan Hongyi and Xu Zhi-hua Phys. Rev. A 50 (1994) 2921; Hong-Yi, F., and J. VanderLinde, Phys. Rev. A 39 (1989) 2987
[80] Hong-yi Fan and Hai-liang Lu, International Journal of Theoretical Physics, in press
[81] A. Wǖnsche, J. Opt. B: Quantum Semiclass. Opt. 1 (1999) R11
[82] T. Alieva and F. Agullo-Lopez, Opt. Comm., 125 (1996) 267
[83] Hong-yi Fan and Hai-liang Lu, Optics Comm., 258 (2006) 51-58
[84] Pang Qian-Jun, Chinese Physics Letters, 22 (2005) 2469
[85] Pang Qian-Jun, Chinese Phys., 14 (2005) 1307
[86] Glasser and M. Lawrence, Montaldi and Emilio, eprint arXiv: math/9307213
[87] Hong-yi Fan and Hai-liang Lu, Optics Letters, 31 (2006) 407
[88] Daubechies, I. Ten Lectures on Wavelets (Philadelphia, PA: Society for Industrial and Applied Mathematics, 1992).
[89] Kaiser, G. A Friendly Guide to Wavelets (Cambridge, MA: Birkhauser, 1994)
[90] Chui, C. K. An Introduction to Wavelets (San Diego, CA: Academic Press, 1992)
[91] Hong-yi Fan and Hai-liang Lu, Physics Letters A, 334 (2005) 132-139
[92] D. Bhaumik, et al, Jour. Phys. A 9 (1976) 1291
[93] G. S. Agarwal and K. Tara, Phys. Rev. A 43 (1991) 492
[94] Hong-yi Fan and Hai-liang Lu, J. Phys. A: Math. Gen. 37 (2004) 10993-11001
[95] A. Furusawa et al, Science 282 (1998) 706
[96] Hong-Yi Fan, Hai-liang Lu and Xiao-Min Qian, Il Nuovo Cimento B 119 (2004) 167
[97] Hong-yi Fan, Xu-bing Tang and Hai-liang Lu, Phys. Lett. A, in press
[2] M. Planck, Ann. d. Physik 4 (1901) 564
[3] A. Einstein, Ann. d. Physik 17 (1905) 132 [English translation in Collected Papers of Albert Einstein, Vol. 2 (1900-1909), Princton University Press (Pfincton, NJ)]
[4] R. H. Brown and R. Q. Twiss, Nature 177 (1956) 27
[5] R. H. Brown and R. Q. Twiss, Nature 178 (1956) 1046
[6] R. J. Glauber, Phys. Rev. 130 (1963) 2529
[7] R. J. Glauber, Phys. Rev. 131 (1963) 2766
[8] W. Heitler, The Quantum Theory of Radiation, (3rd edition), Oxford Clarendon Press, London, 1954
[9] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge, 1996
[10] M. O. Scully and m. S. Zubairy, Quantum Optics, Cambridge University Press, 1997
[11] D. F. Walls and G. J. Milburn, Quantum Optics, Springer-Verlag, 1994
[12] W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH, 2001
[13] M. Born and E. Wolf, Principles of Optics, Cambridge University Press, 1999 (seventh edition)
[14] V. Namias, J. Inst. Maths. Applics 25 (1980) 241
[15] A. W. Lohmann, J. Opt. Soc. Am. A 10 (1993) 2181
[16] D. Dragoman, PROGRESS IN OPTICS, 42 (2002) 424 and references therein
[17] Dragoman, D., and M. Dragoman, Prog. Quantum Electron. 23 1999 131
[18] O. Crasser, H. Mack and W. P. Schleich, quant-ph/0402115
[19] G. Nienhuis and L. Allen, Phys. Rev. A 48 (1993) 656
[20] Wolf, K. B., and E. V. Kurmyshev, Phys. Rev. A 47 1993 3365
[21] 范洪义，量子力学表象与变换论，上海科技出版社，1997
[22] 范洪义，量子力学纠缠态及其应用，上海交通大学出版社，2001
[23] Hong-yi Fan, Hai-liang Lu and Yue Fan, Annals of Physics 321 (2006) 480-494
[24] A. Sommerfeld, Optics, volume Ⅳ of Lectures on Theoretical Physics, Academic Press New York, NY, 1954
[25] J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill 1968
[26] D. Mendlovic, H. M. Ozakatas. J. Opt. Soc. Am. A 10 (1993) 1875
[27] D. Mendlovic, H. M. Ozaktas and A. W. Lohmann, Appl. Opt. 33 (1994)6188;
[28] H. M. Ozakatas, D. Mendlovic. J. Opt. Soc. Am. A 10 (1993) 2522
[29] J. Hua, L. Liu and G. Li, Opt. Comm. 137 (1997) 11
[30] Shaomin Wang and Daomu Zhao, Matrix Optics, China Higher Education Press, Beijing and Springer-Verlag, Berlin Heidelberg, 2000
[31] A. Gerrard and J. M. Burch Introduction to Matrix Methods in Optics John Wiley&Sons, Ltd., London, 1975
[32] H. Kogelnik, Applied Optics 4 (1965) 1562
[33] M. J. Bastiaans, in: Laser Propagation, eds. P. M. Meijas, H. Weber, R. Martinez-Herrero and A. Gonzalez-Urena (S. E. D. O., Madrid, 1993) pp. 65-87
[34] M. J. Bastiaans, Opt. Comm. 25 (1978) 26
[35] S. A. Collins, J. Opt. Soc. Am. 60 (1970) 1168
[36] J. A. Arnaud, J. Opt. Soc. Am. 61 (1971) 751
[37] C. Gomez-Reino, Int. J. Optoelectron. 7 (1992) 607
[38] O. Seger, Ph. D. Dissertation 301 (Linkoping University, 1993)
[39] T. Alieva and F. Agullo-Lopez, Opt. Comm., 114 (1995) 161; 125 (1996) 267
[40] D. F. V. James and G. S. Agarwal, Opt. Comm. 126 (1996) 207
[41] J. A. Arnaud, Beam andFiber Optics Academic Press, 1976
[42] P. A. M. Dirac, The Principles of Quantum Mechanics, 3rd, Clarendon Press, Oxford (1930)
[43] J. R. Klauder, Annals Phys. (New York) 11 (1960) 123
[44] J. R. Klauder and B.-S. Skagerstam, Coherent States, World Scientific, Singapore, 1985
[45] Fan Hong-yi, H. R. Zaidi and J. R. Klauder, Phys. Rev. D (1987) 1831; Fan Hongyi and H. R. Zaidi, Phys. Rev. A 37 (1988)2985; Hongyi Fan and J. R. Klauder, J. Phys. A 21 (1988)L715; Hongyi Fan and Hui Zou, Phys. Lett. A 252 (1999) 281; Hong-yi Fan and A. Wǖnsche, J. Opt. B: Quantum & Semiclass. Opt. 2 (2000) 464
[46] Weyl, H., Z. Phys. 46 (1927) 1-46.
[47] Fan Hong-yi, Phys. Lett. A 161 (1991)1
[48] Hong-yi Fan and Vanderlinde J., J. Phys. A 23 1990 L1113
[49] Louisell, W. H. Quantum Statistical Properties of Radiation (Wiley, New York, 1973).
[50] Hong-yi Fan, J. Opt. B: Quantum & Semiclass. Opt. 5 (2003) R147; Inter. J. Mod. Phys. 18 (2004) 1387
[51] H. R Yuen, Phys. Rev. A 13 (1976)2226
[52] R. Loudon and P. L. Knight, J. Mod. Opt. 34 (1987) 709
[53] D. F. Walls, Nature 306 (1983) 141
[54] J. Mehra and H. Rechenberg, The Historical Development of Quantum Theory, Vol. 4: The Fundamental Equations of Quantum Mechanics 1925-1926. The Reception of the New Quantum Mechanics (Springer-Verlag, New York, 1982)
[55] E. Wigner, Phys. Rev. 40 (1932) 749
[56] Hong-yi Fan and H. R. Zaidi, Phys. Lett. A 124 (1987) 303
[57] A. Einstein, B. Podolsky & N. Rosen, Phys. Rev. 47 (1935) 777
[58] Hong-yi Fan and Klauder, J. R., Phys. Rev. A 49 (1994) 704
[59] H.-Y. Fan, H. Zou and Y. Fan, Phys. Lett. A 254 (1999) 137
[60] Hong-yi Fan, Phys. Lett. A 313 (2003) 343
[61] Hong-yi Fan, Phys. Rev. A 65 (2002) 064102
[62] Hong-yi Fan, J. Phys. A 25 (1992) 3443
[63] Hong-yi Fan and Yue Fan, Int. J. Mod. Phys. A 17 (2002) 701
[64] Vogel K and Risken H, Phys. Rev. A 40 (1989) 2847.
[65] Smithey D T, Beck M, Raymer M G and Faridani A, Phys. Rev. Lett. 70 (1993) 1244.
[66] Hong-yi Fan and Yue Fan, Euro. Phys. J. D 21 (2002) 233
[67] A. Erdelyi, Higher Transcendental Functions: The Batemann Manuscript Project (McGraw-Hill, New York, 1953)
[68] L. Yu et al., Opt. Lett. 23 (1998) 1158
[69] Hong-yi Fan and Hai-liang Lu, Opt. Lett. 28 (2003) 680
[70] 特殊函数概论，王竹溪，郭敦仁，北京大学出版社，2000
[71] Hai-liang Lu and Hong-yi Fan, Comm. Theor. Phys., in press
[72] D. Gabor, J. IEE, 93 (1946)429
[73] J. Preskill, Lecture Notes for Physics 229: Quantum Information and Computation (Pasadena, CA: California Institute of Technology)
[74] Hong-yi Fan, Phys. Lett. A 294 (2002) 253
[75] Hong-yi Fan and Hai-liang Lu and Hao Ren, Int. J. Mod. Phys. B., in press
[76] Luis B. Almeida, IEEE Singal Processing Letters 4 (1997) 15
[77] Ahmed I. Zayed, IEEE Singal Processing Letters 5 (1998) 101-103
[78] H. Goldstein, Classical Mechanics, the second edition, Addisson-Wesley, Massachusetts (1980)
[79] Fan Hong-yi and Chen Jun-hua, Commun. Theor. Phys. 38 (2002) 147; 40 (2003) 589; Fan Hongyi and Xu Zhi-hua Phys. Rev. A 50 (1994) 2921; Hong-Yi, F., and J. VanderLinde, Phys. Rev. A 39 (1989) 2987
[80] Hong-yi Fan and Hai-liang Lu, International Journal of Theoretical Physics, in press
[81] A. Wǖnsche, J. Opt. B: Quantum Semiclass. Opt. 1 (1999) R11
[82] T. Alieva and F. Agullo-Lopez, Opt. Comm., 125 (1996) 267
[83] Hong-yi Fan and Hai-liang Lu, Optics Comm., 258 (2006) 51-58
[84] Pang Qian-Jun, Chinese Physics Letters, 22 (2005) 2469
[85] Pang Qian-Jun, Chinese Phys., 14 (2005) 1307
[86] Glasser and M. Lawrence, Montaldi and Emilio, eprint arXiv: math/9307213
[87] Hong-yi Fan and Hai-liang Lu, Optics Letters, 31 (2006) 407
[88] Daubechies, I. Ten Lectures on Wavelets (Philadelphia, PA: Society for Industrial and Applied Mathematics, 1992).
[89] Kaiser, G. A Friendly Guide to Wavelets (Cambridge, MA: Birkhauser, 1994)
[90] Chui, C. K. An Introduction to Wavelets (San Diego, CA: Academic Press, 1992)
[91] Hong-yi Fan and Hai-liang Lu, Physics Letters A, 334 (2005) 132-139
[92] D. Bhaumik, et al, Jour. Phys. A 9 (1976) 1291
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