摘要
经典光学(Fourier光学)的一个重要组成部分是Fresnel衍射(Fresnel变换)及关于衍射的Collins公式。我国学者范洪义等人首先用相干态表象研究了Fresnel衍射的量子对应,利用有序算符内的积分(IWOP)技术他们建立了一个量子力学Fresnel算符,以实现量子光学中的Fresnel变换,它与经典Collins公式相对应。自量子的Fresnel变换提出以来,它被广泛的应用于讨论经典光学与量子光学理论之间的关系,及与其他光学变换的关系。本文用量子光学的观点重新审视若干经典光学变换,采用的是压缩相干态表象和相干纠缠态表象,提出与经典Fresnel变换对应的新的量子光学中的若干幺正算符,如单模和双模广义Fresnel算符,Fresnel-Hadamard互补算符,并给出其物理意义。这些新幺正算符的经典对应给经典光学提供了新变换的可能性。这篇论文研究的出发点是基于如下的考虑:范洪义等人给出的Fresnel算符是通过相干态在相空间的一个代表点运动到另一点sz-rz*而导出的,这里ss*-rr*=1。根据相空间的直观分析,一个相干态对应于图形上面积为h/2的小圆。量子Fresnel变换表明相空间的一个小圆移动到另一个小圆。ss*-rr*=1保证了该变换是辛变换,同时也保证了刘维定理(相体积不变)的满足。考虑到相干态是压缩相干态中的一个特殊态矢,以及压缩相干态在量子光学和量子信息等领域的广泛应用,我们从压缩相干态在相空间的代表点(一个椭圆)的运动及用IWOP技术提出了广义Fresnel算符。鉴于量子纠缠的理论也渗透到量子光学中,我们又基于相干纠缠态表象,提出了一个Fresnel-Hadamard互补算符,它对一个光分束器的两个输出光场和分别起到了Hadamard变换和Fresnel变换的作用(该光分束器的两个输入光场光场分别为a1与a2)。
论文具体安排如下:
第一章,首先介绍本文工作的研究背景,然后介绍经典Fourier光学中的Fresnel衍射公式,以及在经典框架内介绍了衍射的Collins公式的推导过程。
第二章,为了建立起从经典到量子之间的”桥梁”,我们介绍了一些背景知识。首先介绍一些常见表象如坐标、动量、粒子数、及相干态等表象。然后介绍真正起到上述“桥梁”作用的有序算符内的积分(IWOP)技术的提出背景和内涵。
第三章,我们介绍了范等人从量子光学的相干态表象及用IWOP技术提出的与Fresnel变换对应的量子算符,称之为Fresnel算符。从相空间的直观分析,Fresnel算符对应于相干态在相空间的代表点(一个为面积h/2的圆)的运动变换。我们还简单介绍范等人从量子光学的双模相干态表象及用IWOP技术提出的与Fresnel变换对应的量子算符,称之为双模Fresnel算符。
第四章,我们从量子光学的相干压缩态表象及用IWOP技术提出了与Fresnel变换对应的量子算符,称之为广义Fresnel算符。从相空间的直观分析,广义Fresnel算符对应于压缩相干态在相空间的代表点(一个椭圆)的运动。
第五章,我们基于双模相干压缩态表象介绍了与双模Fresnel算符对应的双模广义Fresnel算符。
第六章,基于相干纠缠态表象,我们发现了一个Fresnel-Hadamard互补算符,它对双模光场a1与a2经过一个光分束器的输出光场和分别起到了Hadamard变换和Fresnel变换的作用。以上讨论表明,从量子光学的新表象和有序算符内的积分(IWOP)技术,可以找到新的光学变换。
最后,我们给出一些总结和展望。
One of the important parts in classical optics (Fourier optics) is the Fresnel diffrac-tion and its Collins formula. Scholars of China Fan Hong-yi et.al firstly studied the quantum correspondence of Fresnel diffraction by using the coherent state. By using the Integral technique within ordered product (IWOP) of operators they constructed a quantum Fresnel operator to realize the Fresnel transformation in quantum optics and it corresponds to the classical Collins formula. Since the introduction of the quantum Fresnel transformation, it has been widely applied to the discussion of the relation be-tween the classical optics and the quantum optics and the relations with other optical transformations. In this paper will use the view of quantum optics to resurvey some classical optical transformation by using the squeezed coherent state and coherent en-tangled state representation to put forward some new unitary operators (e.g. the single mode and two mode generalized Fresnel operators, the Fresnel-Hadamard complimen-tary operator) in quantum optics corresponding to the classical Fresnel transformation and give their physical meanings. The classical correspondence of these new unitary operators may provide probability for some new transformations in classical optics. The starting point of our work is based on the below considering:the Fresnel operator given by Fan Hong-yi et.al is derived from a coherent's moment in phase space from point to point sz-rz*, where ss*- rr*= 1. According to the intuitional analysis, a coherent state is graphically corresponding to a small round. The quantum Fresnel transformation indicates a small round moving to another small round. The condition ss*-rr*=1 guarantees the transformation is symplectic and also guar-antees satisfying the Liouvil theorem (the invariability of phase volume). According to that the coherent state is a special case of squeezed coherent state and thinking that the squeezed coherent state has been got widely used in quantum optics and quantum information, the generalized Fresnel operator we introduce in this paper is just based on the moment of squeezed coherent state(graphically corresponding to a ellipse in phase space) and by using the IWOP technique. Whereas the filtering of the theory of quantum entanglement to quantum optics, based on the coherent entangled state, we introduce a Fresnel-Hadamard complementary operator, which can play the roles of Hadamard transformation and Fresnel transformation respectively for the two output fields of and of a beamsplitter (The two input fields of the beamsplitter is a1 and a2).
My Ph.D dissertation is arranged as following:
In chapter one, we introduce some background knowledge of our work, the Fres-nel diffraction formula in classical Fourier optics and the deriving process to obtain the Collins formula in classical frame.
In chapter two, in order to bridge the classics to quantum, we introduce some background knowledge. Firstly, we introduce some basic representation like the co-ordinate, momentum, particle number and the coherent state representations, then we introduce the background and meaning of the Integral technique within ordered product (IWOP) of operators which really has the above bridge function.
In chapter three, we introduce the quantum operator corresponding to the classi-cal Fresnel transformation, i.e. the Fresnel operator constructed by Fan et.al, which is based on the coherent state representation in quantum optics and the IWOP technique. According to the intuitional analysis in phase space, the Fresnel operator corresponds to the moving transformation of the representing point(a round with h/2 area) of co-herent state in phase space. we also introduce the quantum operator corresponding to the classical Fresnel transformation, i.e. the two-mode Fresnel operator constructed by Fan et.al, which is based on the two-mode coherent state representation in quantum optics and the IWOP technique.
In chapter four, based on the squeezed coherent state representation in quantum optics and the IWOP technique, the quantum operator corresponding to the classical Fresnel transformation, i.e. the generalized Fresnel operator is introduced. According to the intuitional analysis in phase space, the generalized Fresnel operator corresponds to the moving transformation of the representing point(an ellipse with h/2 area) of squeezed coherent state in phase space.
In chapter five, based on the two-mode squeezed coherent state representation we introduced the two-mode generalized Fresnel operator corresponding to the two-mode Fresnel operator.
In chapter six, based on the coherent entangled state, we find a Fresnel-Hadamard complementary operator. For the optical fields and considered as two out-put fields after the two mode optical fields a1 and a2 passing through a beamsplitter, the Fresnel-Hadamard complementary operator can play the roles of Hadamard trans-formation and Fresnel transformation respectively. Above discussion indicates that from the new representation in quantum optics and using the Integral technique within ordered product (IWOP) of operators we can find new optical transformations.
Finally, we give some conclusions and expectations.
引文
[1]R. J. Glauber. Large-n limit of the heisenberg-hubbard model:Implications for high-tc superconductors. Phys. Rev.,130:2529,1963.
[2]J. R. Klauder, B.-S. Coherent States. Singapore-World Scientific,2003.
[3]E. Schrodinger. Naturwiss,14:664,1926.
[4]P. A. M. Dirac. The Principles of Quantum Mechanics. Oxford:Clarendon Press,1930.
[5]C. Cohen-Tannoudji C. DeWitt, A. Blandin. Optique et Electronique Quantiques-Quantum Optics and Electronics. New York-Gordon and Breach,1965.
[6]L. Mandel and E. Wolf. Optical Coherence and Quantum Optics. Cambridge University Press,1996.
[7]G. Nienhuis and L. Allen. Paraxial wave optics and harmonic oscillators. Phys. Rev. A,48:656,1993.
[8]D. Dragoman. Phase space correspondence between classical optics and quan-tum mechanics. ROGRESS IN OPTICS,42:433-496,2002.
[9]P.A. Belanger. Beam propagation and the abed ray matrix. Opt. Lett.,16:196, 1991.
[10]S.D. Silvestri. V. Magni, G. Cerullo. Abed matrix analysis of propagation of gaussian beams through kerr media. Opt. Commun.,96:348,1993.
[11]Pierre Pellat-Finet. Fresnel diffraction and the fractional-order fourier transform. Opt. Lett.,19:1388,1994.
[12]K. Vogel and H. Risken. Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. Phys. Rev. A,40: 2847-2849,1989.
[13]J. Cooper M. G. Raymer T. D. Smithey, M. Beck and A. Faridani. Measurement of the wigner distribution and the density matrix of a light mode using optical homodyne tomography:Application to squeezed states and the vacuum. Phys. Rev. Lett.,70:1244,1993.
[14]Y. Meyer See e.g., S. Jaffard and R. D. Ryan. Wavelets, Tools for Science& Technology. Philadelphia:SIAM,2001.
[15]M. A. Pinsky. Introduction to Fourier Analysis and Wavelets. Cole,2002.
[16]H. P. Yuen. Two-photon coherent states of the radiation field. Phys. Rev. A,13: 2226,1976.
[17]R. Loudon and P. L. Knight. Theory of the spin gap in high-temperature super-conductors.J. Mod. Opt.,34:709,1987.
[18]Hong yi Fan and H. R. Zaidi. New approach for calculating the normally ordered form of squeeze operators. Phys. Rev. D,35:1831,1987.
[19]Hong yi Fan. Quantum Semiclass. Opt.,18:1387-1455,2004.
[20]Hong yi Fan and J. R. Klauder. Eigenvectors of two particles'relative position and total momentum. Phys. Rev. A,1994.
[21]A. Wunsche.J. Opt. B:Quantum Semiclass. Opt,1:11,1999.
[22]Hai-liang Lu Hong-yi Fan.and Yue Fan. Newton (?)eibniz integration for ket (?)bra operators in quantum mechanics and derivation of entangled state rep-resentations. Ann. Phys.,321:480-494,2006.
[23]H. R. Zaidi Hong-yi Fan and J. R. Klauder. New approach for calculating the normally ordered form of squeeze operators. Phys. Rev. D,35:1831-1834, 1987.
[24]Hong yi Fan. Representation and Transformation Theory in Quantum Mechan-ics. Shanghai Scientific and Technical Publishers,1997.
[25]Hong yi Fan. Entangled State Representations in Quantum Mechanics and Their Applications. Shanghai Jiao Tong University Press,2001.
[26]Hong yi Fan and J. Vanderlinde. Mapping of classical canonical transformation to quantum unitary operators. Phys. Rev. A,39(2987),1989.
[27]Hong-Yi Fan. Unitary transformation for four harmonically coupled identical oscillators. Phys. Rev. A,1990.
[28]Hong yi Fan and J. VanderLinde. Simple construction of the unitary operator to transform from individual particles to jacobi coordinates. Phys. Rev. A,43:1604, 1991.
[29]Hong yi Fan and Zhi hua Xu. Symplectic transformations in the n-mode coherent state representation using integration within an ordered product of operators. Phys. Rev. A,50:2921,1994.
[30]Hong yi Fan and Hai liang Lu. Eigenmodes of fractional hankel transform de-rived by the entangled state method. Optics letters,28:680,2003.
[31]Hong yi Fan. Fractional hankei transform studied by charge-amplitude state representations and complex fractional fourier transform. Opt. Lett.,28:2177, 2003.
[32]Hong yi Fan. Application of iwop technique to quantum-classical correspon-dence for spherical polar coordinate system. Phys. Lett. A,127:405,1988.
[33]Hong yi Fan and Li yun Hu. New approach for analyzing time evolution of density operator in a dissipative channel by the entangled state representation. Opt. Commun.,281:5571,2008.
[34]Hong yi Fan and Hai-Liang Lu. Wave-function transformations by general su(1, 1) single-mode squeezing and analogy to fresnel transformations in wave optics. Opt. Commun.,258:51-58,2006.
[35]H. Kogelnik and T. Li. Laser beams and resonators. Appl. Opt.,5:1550,1966.
[36]M. Born and E. Wolf. Principles of Optics. Cambridge-World Scientific,1999.
[37]J. W. Goodman. Introduction to Fourier Optics. McGraw-Hill,1972.
[38]Wang S M and Zhao D M. Matrix optics. China Highher Education Press Beijing and Springer-Verlag Berlin Heidelberg,2000.
[39]吕百达.激光光学-激光束的传输变换和光束质量控制.第二版.四川大学出版社,1992.
[40]J. M. Burch. A. G erfad. Introduction to Matrix Methods in optics. Joho Wiley, 1975.
[41]陈惠芬志鸣,沈元华.光学.北京:高等教育出版社,2000.
[42]钟锡华.光波衍射和变换光学.北京:高等教育出版社,1985.
[43]钟锡华赵凯华.光学.北京:北京大学出版社,1984.
[44]C.C. Hull. M.H. Freeman. Optics.世界图书出版公司,2005.
[45]M.Young. Optics and lasers.科学出版社,2007.
[46]W.T. Welford. Optics. Oxford University Press,1976.
[47]P. A. M. Dirac. Recollections of an exciting area. New York:Academic Press, 1977.
[48]P. A. M. Dirac. Directions in Physics. New York:John Wiley,1978.
[49]P. A. M. Dirac. The Physicist's conception of Nature. New York:Jagdish Mehra, 1973.
[50]范洪义.量子力学表象与变换-Driac's符号法进展.上海:上海科学技术出版社,1997.
[51]Hong yi Fan. Entangled states, squeezed states gained via the route of develop-ing dirac's symbolic method and their applications. Inter. J. Mod. Phys. B,18: 1387,2004.
[52]Hong yi Fan. Quantum-mechanical unitary operators gained via integration over a type of ket-bra operators. Europhys. Lett.,17:285,1992.
[53]H. R. Zaidi Hong-yi Fan and J. R. Klauder. New approach for calculating the normally ordered form of squeeze operators. Phys. Rev. D,35:1831,1998.
[54]Hong yi Fan and J. R. Klauder. Canonical coherent-state representation of some squeeze operators. J.Phys.A,21:725,1988.
[55]Hong yi Fan. Operator ordering in quantum optics theory and the development of dirac's symbolic method. J. Opt. B:Quantum Semiclass. Opt.,5(4):8147, 2003.
[56]A. Wunsche. About integration within ordered products in quantum optics.J. Opt. B:Quantum Semiclass. Opt.,1(11),1999.
[57]R. J. Glauber. Coherent and incoherent states of the radiation field. Phys. Rev., 2766,1963.
[58]R. J. Klauder and B. S. Skargerstam. Coherent States. Singapore:World Scien-tific,1985.
[59]Hong yi Fan and Yong Ren. New applications of the coherent state in calculating the class operator of rotation group. J. Phys. A,21:1971,1988.
[60]Hong-Yi Fan. New two-mode fermionic coherent state.J. Phys. A,23:259, 1990.
[61]R. J. Glauber Glau. The quantum theory of optical coherence. Phys. Rev.,130: 2529,1963.
[62]Hong yi Fan and J. R. Klauder.J. Phys. A,21(715):1114,1988.
[63]Hong yi Fan and Hui-Zou. Similarity transformation operators as the images of classical symplectic transformations in coherent state representation. Phys. Lett. A,252:281-287,1999.
[64]N. N. Bogolyubov. Lectures on Quantum Statistics. New York:Gordon and Breach,1987.
[65]Hong yi Fan and J.Vanderlinde. Generalized bogolyubov transformation. J. Phys. A,23:1113,1990.
[66]Hong yi Fan. Squeezed states:Operators for two types of one-and two-mode squeezing transformation. Phys. Rev. A,41:1526,1990.
[67]Hong yi Fan. Entangled states, squeezed states gained via the route of devel-oping dirac's symbolic method and their applications. Inter. J. Mod. Phys.,18: 1387,2004.
[68]Hong yi Fan and Hai ling Cheng. Nonlinear coherent state representation and related generalized wigner function. J. Opt. B:Quan. Semiclass. Opt.,3:388, 2001.
[69]Li-Yun Hu and Hong yi Fan. J. Mod. Opt.,55:242,2008.
[70]Hong yi Fan and Hai-Liang Lu. Opt. Lett.,31:2622,2006.
[71]Hong yi Fan and Li-Yun Hu. Quantum optical abed theorem in two-mode case. Chin. Phys.B,17:1640,2008.
[72]Li-Yun Hu and Hong yi Fan.J. Mod, Opt.,55:242,2008.
[73]Li-Yun Hu and Hong yi Fan.J. Mod. Opt.,55:835,2008.
[74]Hong yi Fan and Li-Yun Hu. Fresnel-transform's quantum correspondence and quantum optical abed law. Chin. Phys. Lett.,24:2238,2007.
[75]Hong yi Fan and Li-Yun Hu. On the scaled fractional fourier transformation operator. Chin. Phys. Lett.,25:513,2008.
[76]Hai-Liang Lu Hong-yi Fan and Yue Fan. Abed rule for gaussian beam propaga-tion in the context of quantum optics derived by the iwop technique. Ann. Phys., 321:2116-2127,2006.
[77]Hong yi Fan. Quantum-mechanical unitary operators gained via integrations over a type of ket-bra operators. Euro. phys. Lett.,17:4,1993.
[78]Hong yi Fan. Newton (?)eibniz integration for ket-bra operators in quantum mechanics (v)—deriving normally ordered bivariate-normal-distribution form of density operators and developing their phase space formalism. Ann. Phys., 323:1502-1528,2008.
[79]Glauber R J. Coherent and incoherent states of the radiation field. Phys. Rev. 131:2766-2788,1963.
[80]S. A. Collins. J. Opt. Soc. Am.,60:1168,1970.
[81]Hong yi Fan and Hai-Liang Lu. Collins diffraction formula studied in quantum. Optics Letters,31:2622,2006.
[82]Hong yi Fan and Li-Yun Hu. Fresnel-transform's quantum correspondence and quantum optical abed law. Chin.Phys.Lett.,24:2238,2007.
[83]Hong yi Fan and Li-Yun Hu. Abed rule involved in the product of general su(1,1) squeezing operators and quantum states fresnel transformation. Opt. Commun., 281:1629,2008.
[84]Hai zi Liu. Simple abcd matrix method for evaluating optical coupling system laser diode to single-mode fiber with a lensed-tip. Optik,116:415,2005.
[85]Hong yi Fan. Newton (?)eibniz integration for ket (?)bra operators in quantum mechanics (iv)—integrations within weyl ordered product of operators and their applications. Ann, of Phys.,500-526:480,2008.
[86]Hong yi Fan and Wunsche A. Design of squeezing. J. Opt. Phys. B: Quant.Semiclass. Opt.,2:464,2000.
[87]Hong yi Fan and Hai-Liang Lu. Optical fresnel transformation and quantum tomography. Opt. Commun.,282:3734-3736,2008.
[88]Hong yi Fan and Hai-Liang Lu.2-mode fresnel operator and entangled fresnel transform. Phys. Lett. A,334:132,2005.
[89]Hong yi Fan. Entangled states, squeezed states gained via the route of develop-ing dirac's symbolic method and their applications. Inter. J. Mod. Phys. B,18: 1387,2004.
[90]Hong-Yi Fan. Entangled state representations for describing 2-dimensional elec-tron system in uniform magnetic field. Inter. J. Mod. Phys. B,18:2771,2004.
[91]Hong yi Fan and J. R. Klauder. Eigenvectors of two particles'relative position and total momentum. Phys. Rev. A,49:704,1994.
[92]Hong-Yi Fan and Ye Xiong. Common eigenstates of two particles'center-of-mass coordinate and mass-weighted relative momentum. Phys. Rev. A,51:3343, 1995.
[93]Hong yi Fan. Application of entangled state representationin quantum telepor-tation with continuous variables. Phys. Lett.A,294:253,2002.
[94]Chuan mei Xie, Hong yi Fan, and Shao long Wan. A generalized collins formula derived by virtue of the displacement-squeezing related squeezed coherent state representation. Chin.Phys.B,19:06XXXX,2010.
[95]R.Loudon and P.L.Knight. Rev.Mod. Opt.,34:709,1987.
[96]H.J.Kimble and D.F.Walls. Squeezed states of the electromagnetic field. J.Opt.Sot.Am.B,4:1,1987.
[97]Min Xiao Hong-yi Fan. A special type of squeezed coherent state. Physics Letters A,220:8,1996.
[98]Chuan mei Xie and Hong yi Fan. A generalized freshel operator in two-mode. Submitted.
[99]Hong yi Fan and Jun hua Chen. Coherent state projection operator representa-tion of symplectic transformations as a loyal representation of symplectic group. Commun. Theor. Phys.,38:147,2002.
[100]Fan Hong-yi and Xu Zhi-hua. Symplectic transformations in the n-mode coher-ent state representation using integration within an ordered product of operators. Phys. Rev. A,50:2921,1994.
[101]Zhong xi Zhang and Hong yi Fan. The higher order squeezing for one-and two-mode combination squeezed states. Quan.Opt.,5:149,1993.
[102]Hong yi Fan and Hai ling Cheng. New approach for calculating wigner func-tions of generalized two-mode squeezed state and squeezed number state via entangled state representation. Commun. Theor. Phys.,36:651,2001.
[103]Hong yi Fan and Hui Wang. Entanglement swapping for displaced two-mode squeezed states. Commun. Theor. Phys.,36:455,2001.
[104]Hong yi Fan and Nai le Liu. Mixed squeezing for a pair of two-mode squeezed states. Chin.Phys. Lett.,18:322,2001.
[105]Hong yi Fan. A new representation of two-mode squeezed states. Optics Com-munication,277,2004.
[106]Hong yi Fan and Zhi hua Xu. Symplectic transformations in the n-mode coherent state representation using integration within an ordered product of operators. Phys. Rev. A,50:2921,1994.
[107]Hong yi Fan and Hui Zou. Similarity transformation operators as the images of classical symplectic transformations in coherent state representation. Phys. Lett. A,252:281,1999.
[108]Hong yi Fan. Symplectic group representation of the two-mode squeezing oper-ator in the coherent state basis. Commun. Theor. Phys.,40:589,2003.
[109]Hong yi Fan and Yue Fan. Epr entangled states and complex fractional fourier transformation. Euro. Phys. J. D,21:233,2002.
[110]Hong yi Fan and Yue Fan. Reducing projection calculation in quantum tele-portation by virtue of the iwop technique and schmidt decomposition of—eta(?) state. Commun, Theor. Phys.,37:35,2002.
[111]Li yun Hu and Hong yi Fan. Adaption of collins formula to fractional transform studied in the entangled state representation of quantum optics. J. Mod. Opt., 55:2429,2008.
[112]W. Vogel and W. P. Schleich. Measuring the Quantum State of Light. New York:Cambridge University,1997.
[113]Markus Aspelmeyer. Quantum tomography:Measured measurement. Nature Physics,5:11,2009.
[114]Hong yi Fan. Fractional hankel transform studied by charge-amplitude state representations and complex fractional fourier transform. Opt. Lett.,28:2177, 2003.
[115]D. Dragoman. Classical versus complex fractional fourier transformation. J. Opt. Soc. Am. A,26:274,2009.
[116]L.Allen. G.Nienhuis. Paraxial wave optics and harmonic oscillators. Phys.Rev.A, 48:656,1993.
[117]E. Wolf. L. Mandl. Optical Coherence and Quantum Optics. Cambridge:Cam-bridge University Press,1995.
[118]S.Bose S.Parker and M.B.Plenio. Entanglement quantification and purification in continuous-varible system. Physical Review, 61:032305,2000.
[119]I. L. Chuang M.A.Nielsen. The Quantum Computation and Quantum Informa- tion. Cambridge:Cambridge University Press,2000.
[120]J. Preskill. See e.g. Quantum Information and Computation. California:Cali-fornia Institute of Technology,1998.
[121]Isaac L. Michale A. Nielsen. Quantum Information and Computation. Cam-bridge University,2000.
[122]Hong yi Fan and Xu-Bing Tang. New squeezing operator derived by the bipartite coherent entangled state representation.J. Opt. B:Quantum Semiclass. Opt.,7: 5765,2005.
[123]Hong yi Fan and Hai liang Lu. New two-mode coherent-entangled state and its application.J. Phys. A,37:10993,2004.
[124]Chuan mei Xie and Hong yi Fan. The fresnel-hardmard complementary trans-formation in quantum optics. J.Mod.Opt., accepted.
[125]R. Loudon. and P.L. Knight. Squeezed light.J.Mod. Opt.,34:709,1987.