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g. stokes studied the polarization properties of a quasi - monochromatic plane wave of light in an arbitrary polarization state by introducing four quantities , known since then as the stokes parameters @xcite . the stokes parameters are a set of four quantities which provide intuitive and practical tools to describe the polarization of light @xcite . the stokes parameters give a direct relation between the light ( photons ) polarization and the polarization of elementary particles @xcite . this fact was widely exploited to study many features of radiation of particles and to the scattering problems @xcite . the stokes parameters are formulated in terms of the observables of the electromagnetic field , namely , the amplitudes and the relative phase difference between the orthogonal components of the field @xcite . in fact , the density matrix @xcite and the coherence matrix @xcite for the case of electromagnetic radiation result to be the same @xcite , and are written in terms of these observables .    the standard procedure to describe the polarization of an electromagnetic wave is to set the propagation direction along the @xmath1-axis , and the two components of the polarization field on the @xmath6 and @xmath7directions . however , when the direction of arrival from the source is unknown a priori , the three - dimensional coherence matrix must be used to obtain a complete polarization characterization @xcite . in ref . @xcite , jauch and rohrlich introduced the stokes parameters in the quantum regime , which are called stokes operators . it is at the quantum domain where we can see that a symmetry group structure is related to the stokes operators . when the direction of propagation of light is known , the symmetry is the @xmath8 group @xcite however , when the direction of propagation is unknown , the symmetry group is @xmath9 @xcite . also , other generalizations of stokes operators have been reported @xcite .    in this work we give a new approach to the theory of light polarization . for simplicity , we study the case of a monochromatic plane electromagnetic wave which propagates along the @xmath1-axis . our study is based on a set of quantum operators , constructed from two independent bosons , being three of them the @xmath0 lie algebra generators , and the other one , the casimir operator of this algebra . this work is organized as follows . in section 2 , we deduce the @xmath0 lie algebra generators by the jordan - schwinger map . in section 3 , by taking the expectation value of the algebra generators in a two - mode coherent state , we obtain their classical limit . in section 4 , we define our stokes parameters ( we refer to them stokes - like parameters ) and show that the light polarization ellipse can be written in terms of them . in section 5 , the stokes - like parameters are written in terms of two parameters and it is shown that they define a one - sheet ( poincar hyperboloid ) of a two - sheet hyperboloid . in what follows we will use @xmath10 , where @xmath11 is the mass of each one - dimensional harmonic oscillator and @xmath12 is the frequency of either the electromagnetic wave or the harmonic oscillators . we define the operators @xmath13 with @xmath14 the operators @xmath15 and @xmath16 the left and right annihilation operators of the two - dimensional harmonic oscillator , with the non vanishing commutators @xmath17=[a_2,a_2^{\dag}]=1 $ ] . the matrices @xmath18 are defined as follows : @xmath19 , @xmath20 , @xmath21 and @xmath22 , where @xmath23 are the usual pauli matrices @xcite . + explicitly , the operators @xmath24 , @xmath25 , @xmath26 and @xmath27 are given by @xmath28 the operator @xmath24 can be rewritten as @xmath29 , being @xmath30 the @xmath31component of the angular momentum of the two - dimensional harmonic oscillator , whose hamiltonian is given by @xmath32 . therefore , the operator @xmath27 is essentially the energy of the two - dimensional harmonic oscillator . it can be shown that @xmath33=0 , \hspace{2ex}\ ] ] also , by a straightforward calculation , we show that the commutation relations of the operators @xmath34 are @xmath35=-i \mathcal{k}_3,\hspace{3ex } \left[\mathcal{k}_2,\mathcal{k}_3\right]=i \mathcal{k}_1 , \hspace{3ex } \left[\mathcal{k}_3,\mathcal{k}_1\right]=i \mathcal{k}_3 . \label{comu}\ ] ] therefore , these operators close the @xmath0 lie algebra . the casimir operator @xmath36 for this algebra results to be @xmath37 . hence , we have obtained the generators of the @xmath0 lie algebra , equation ( [ jsm ] ) , by the so called jordan - schwinger map @xcite . in ref . @xcite the classical states of the two - dimensional harmonic oscillator were calculated . on the other hand , the stokes parameters were obtained by evaluating the @xmath38 lie algebra generators in a two - mode coherent state @xcite . the same idea was used to derive one of the stokes parameters generalizations by evaluating the @xmath39 lie algebra generators in a three - mode coherent states @xcite . in this paper we take advantage of these facts . thus , we evaluate the @xmath0 lie algebra generators @xmath40 in a two - mode coherent state to obtain their classical limit . the two - mode coherent states are well known to be given by @xmath41}\sum_{n_1,n_2=0}^\infty{\alpha_1^{n_1}\alpha_2^{n_2 } \over \sqrt{n_1!n_2!}}|n_1,n_2\rangle,\label{tm}\ ] ] which are eigenstates of the annihilation operators @xmath15 and @xmath16 @xmath42 with this procedure , we obtain @xmath43 where @xmath44 are the classical oscillations @xmath45 $ ] with amplitudes @xmath46 and phases @xmath47 . this leads to @xmath48 where @xmath49 . thus , the classical limit of the lie algebra generators in a time - dependent two - mode coherent state is time dependent . this is because the @xmath0 lie algebra generators do not commute with the hamiltonian of the two - dimensional harmonic oscillator . it is well known that the standard stokes parameters obtained as the classical limit of the @xmath38 generators are time - independent @xcite . this is due to the @xmath38 lie algebra generators commute with the two - dimensional harmonic oscillator hamiltonian . we set an electromagnetic wave with arbitrary polarization , which propagates along the @xmath1-axis , given by @xmath50 where the complex amplitudes are defined by @xmath51}$ ] . it can be written in the form @xmath52 where @xmath53 and @xmath54 from these equations is easy to obtain the nonparametric quadratic equation @xmath55 by setting the definitions @xmath56 , @xmath57 and @xmath58 , it is shown that @xmath59 . thus , when @xmath60 , equation ( [ trayec ] ) represents an ellipse centered at the origin . equations ( [ campo1 ] ) and ( [ campo2 ] ) allows to write equation ( [ trayec ] ) as @xmath61 this is the polarization ellipse described by the plane wave electromagnetic field . using equations ( [ funda1 ] ) and ( [ funda2 ] ) , and the definition of the phase difference @xmath62 , this equation can be rewritten as @xmath63 in this way we have written the polarization ellipse in terms of the classical values of the @xmath0 lie algebra generators . + we define the classical stokes - like parameters @xmath64 as follows @xmath65 thus , the ellipse polarization can be expressed as @xmath66 we note that the polarization ellipse , equation ( [ elipse ] ) , depends on the phase difference @xmath67 , whereas the classical limit of the @xmath0 generators do not . this fact has forced us to define the stokes - like parameters as those of equations ( [ stokes1 ] ) and ( [ stokes2 ] ) , which depends on the phase difference and are time - independent . the procedure to map the standard stokes parameters to the poincar sphere is explained in full detail in sec . 1.4.2 of ref . @xcite . in the present section , we translate the born and wolf ideas to our stokes - like parameters . we shall show that they are mapped to a one - sheet of a two - sheet hyperbolic paraboloid . we perform an hyperbolic rotation to the fields @xmath68 and @xmath69 by an angle @xmath70 , in such a way that the fields @xmath71 and @xmath72 are obtained on the principal axis of the ellipse . thus , @xmath73 equation ( [ campo ] ) can be written as @xmath74 where @xmath75 , and @xmath76 , @xmath77 . + on the other hand , the oscillations on the principal axis of the ellipse are given by @xmath78 being @xmath79 a constant phase . the signs @xmath80 are for right and left polarization on the coordinates system @xmath81 . from the substitution of equations ( [ oscilaciones ] ) and ( [ oscilaxi ] ) into equation ( [ rot1 ] ) , we deduce the equalities @xmath82 similarly , by substituting equations ( [ oscilaciones ] ) and ( [ oscilaeta ] ) into equation ( [ rot2 ] ) , we deduce @xmath83 dividing ( [ ocho ] ) by ( [ seis ] ) and ( [ nueve ] ) by ( [ siete ] ) , we obtain @xmath84 the second equality and some elementary hyperbolic identities lead to @xmath85 the definition @xmath86 and the identity for @xmath87 permit to write this equation as @xmath88    on squaring and adding ( [ seis ] ) and ( [ siete ] ) , we obtain @xmath89 also , on squaring and adding ( [ ocho ] ) and ( [ nueve ] ) : @xmath90 the subtracting of equation ( [ bcuad ] ) from equation ( [ acuad ] ) allows to obtain @xmath91 if we multiply ( [ seis ] ) by ( [ ocho ] ) , and ( [ siete ] ) by ( [ nueve ] ) and adding the respective results , we get @xmath92 from the last two equalities it is implied that @xmath93 if we introduce the definition @xmath94 , this equation result to be @xmath95 now , the definition @xmath96 above , leads to @xmath97    equation ( [ funda4 ] ) can be written in terms of equations ( [ funda1 ] ) , ( [ funda2 ] ) and ( [ oscilaciones ] ) as @xmath98 where for the last equality , we have used equation ( [ stokes1 ] ) . this equation leads to @xmath99 or depending on the sign of the @xmath100 : @xmath101 similarly , from equations ( [ funda3 ] ) , ( [ funda1 ] ) , ( [ funda2 ] ) , and ( [ stokes2 ] ) , we can show that @xmath102 on the other hand , from equations ( [ funda1 ] ) and ( [ funda2 ] ) it is immediate to show that @xmath103 where for the last equality we have used equations ( [ stokes1 ] ) and ( [ stokes2 ] ) . equations ( [ tild1 ] ) , ( [ tild2 ] ) and ( [ forc ] ) lead to @xmath104 however , because of equation ( [ stokes2 ] ) , @xmath105 , thus @xmath106 substituting this equation into equation ( [ tild2 ] ) we have @xmath107    equations ( [ tild1 ] ) , ( [ tild3 ] ) and ( [ tild4 ] ) are the stokes - like parameters written in terms of @xmath108 and @xmath70 . we have plotted the vector function @xmath109 with the different setting of signs : a ) ( + , + , + ) , b ) ( + , ,+ ) , c ) ( ,+,+ ) and d ) ( ,,+ ) . it must be emphasized that for each of these setting , the polarization surface generated for the stokes - like parameters results to be as that shown in fig . 1 , been @xmath110 the distance from the origin of coordinates to the minimum of the hyperboloid sheet . hence , because of the plus sign in equation ( [ tild3 ] ) , the stokes - like parameters generate a one - sheet ( poincar hyperboloid ) of a two - sheet hyperboloid , instead an sphere . fig . 1 shows how the poincar hyperboloid varies as a function of @xmath110 . +    . the greater one is for @xmath111 and the little one is for @xmath112 . ] [ f1 ]    now , we are interested in investigating the regions on the poincar hyperboloid in which occurs the different polarizations : i)linear polarization @xmath113 , ii)circular polarization @xmath114 and iii ) elliptical polarization @xmath115 . + i ) for linear polarization @xmath113 , @xmath116 , @xmath117 @xcite , equations ( [ stokes1 ] ) and ( [ stokes2 ] ) imply that @xmath118 and @xmath119 to a real quantity . also , for @xmath116 , @xmath120 a linear polarization holds . for this case , @xmath118 and @xmath119 to a real quantity . in this way , we have showed that each point on the curve @xmath113 of fig . 2 represents to a linear polarization . + ii ) the circular polarization @xmath114 , occurs when @xmath121 , ( @xmath80 represents right ( left ) circular polarization , respectively ) , @xmath122 and @xmath123 . in order to have a graphical interpretation to this case , we consider a limit process @xmath124 and @xmath125 . under this assumption : a ) the minimum of the poincar hyperboloid approaches to the origin of coordinates , as shown in fig . 1 , and b ) with @xmath121 , equations ( [ funda1 ] ) , ( [ funda2 ] ) , ( [ stokes1 ] ) and ( [ stokes2 ] ) implies that @xmath126 and @xmath127 . in fact , for right circular polarization , @xmath128 , @xmath129 and @xmath127 , and for left circular polarization , @xmath130 , @xmath131 and @xmath127 . hence , each point of the curve @xmath114 on the poincar hyperboloid represents a circular polarization ( see fig . + iii ) it is well known that right elliptical polarization @xmath132 holds for @xmath133 , and left elliptical polarization @xmath134 holds for @xmath135 @xcite . since for @xmath132 , @xmath136 , from equations ( [ stokes1 ] ) and ( [ stokes2 ] ) we deduce that @xmath137 whereas @xmath138 can takes positive and negative values . similarly , for @xmath134 , @xmath139 and @xmath138 can takes both positive and negative values . in fig 2 we have represented these two regions .    . we have set @xmath140 , @xmath141 and @xmath142 . the curve @xmath113 represents the points where linear polarization occurs , and in each point on the curve @xmath114 there exists a circular polarization . each point on the region @xmath132 represents a right elliptical polarization , and the points on the region @xmath134 are those on which left elliptical polarization holds . ]    [ f2 ] we have given an alternative definition of the stokes parameters ( stokes - like parameters ) and showed that the polarization ellipse can be written in terms of these parameters . based on the procedure to map the standard stokes parameters to the poincar sphere @xcite , we achieve to map our stokes - like parameters to a one - sheet ( poincar hyperboloid ) of a two - sheet hyperboloid . we note that in the equations ( [ stokes1 ] ) and ( [ stokes2 ] ) , the definitions of @xmath143 and @xmath138 can be interchanged without any essential change in the results .    finally , we emphasize that the case we have treated in this paper is when the direction of arrival of the electromagnetic wave is known a priori . our results rest on the lie algebra @xmath144 . when the direction of arrival of the electromagnetic wave is unknown a priori , a generalization of the results of the present paper could be related with the generators of the @xmath145 lie algebra . this work was partially supported by : cofaa - ipn , edi - ipn , edd - ipn and cgpi - ipn project numbers 20161727 and 20160108 .    99 stokes g. g. , trans . cambridge philos . soc . , * 9*,(1852 ) 399 ( reprinted in stokes g. g. , mathematical and physical papers , johnson reprint corporation , n. y. and london , 1966 ) . shurcliff w. a. and ballard s. s. , polarized light , van nostrand company , princeton , n.j . , 1964 . mcmaster w. h. , am . j. phys . * 22 * , ( 1954 ) 351 . mcmaster w. h. , rev . * 33 * , ( 1961 ) 8 . collett e. , am . j. phys . * 36 * , ( 1968 ) 713 . wolf e. , nuovo cimento * 12 * , ( 1954 ) 884 . wolf e. , nuovo cimento , * 13 * , ( 1959 ) 1165 . blaskal s. and kim y. s. , arxiv : quant - ph/0501050 . roman p. , nuovo cimento * 13 * , ( 1959 ) 974 . ramachandran g. , murthy m. v. n. and mallesh k. s. , pramana * 15 * , ( 1980 ) 357 . carozzi t. , karlsson r. and bergman j. , phys . e * 65 * , ( 2000 ) 2024 . mota r. d. , xicotncatl m. a. and granados v. d. , j. phys a : math.and gen . * 37 * , ( 2004 ) 2835 . jauch j. m. and rohrlich f. , the theory of photons and electrons , springer - verlag , berlin , 1976 . mota r. d. , xicotncaltl m. a. and granados v. d. , can . j. phys . * 82 * , ( 2004 ) 767 . abouraddy a. f. , sergienko a. v. , saleh b. e. a. and teich m. c. , opt . commun . * 201 * , ( 2002 ) 93 . jaeger g. , teodorescu - frumosu m. , sergienko a. b. , saleh b. e. a. and teich m. c. , phys . rev . a * 67 * , ( 2003 ) 032307 . biedenharn l. c. and louck j. d. , angular momentum in quantum physics , addison - wesley publishing company , massachusetts , 1981 . cohen - tannoudji c. , diu b. and lale f. , quantum mechanics , john wiley @xmath146 sons , paris , 1977 . hecht e. , optics , addison wesley , san francisco , 2002 . born m. and wolf e. , principles of optics , pergamon press , oxford , 1975 .
we introduce an alternative approach to the polarization theory of light . 
 this is based on a set of quantum operators , constructed from two independent bosons , being three of them the @xmath0 lie algebra generators , and the other one , the casimir operator of this algebra . by taking the expectation value of these generators in a two - mode coherent state 
 , their classical limit is obtained . 
 we use these classical quantities to define the new stokes - like parameters . 
 we show that the light polarization ellipse can be written in terms of the stokes - like parameters . 
 also , we write these parameters in terms of other two quantities , and show that they define a one - sheet ( poincar hyperboloid ) of a two - sheet hyperboloid . our study is restricted to the case of a monochromatic plane electromagnetic wave which propagates along the @xmath1 axis . 
 @xmath2 escuela superior de ingeniera mecnica y elctrica , unidad culhuacn , instituto politcnico nacional , av . 
 santa ana no . 
 1000 , col . 
 san francisco culhuacn , delegacin coyoacn , c.p . 
 04430 , mxico d. f. , mexico . 
 + @xmath3 escuela superior de cmputo , instituto politcnico nacional , av . juan de dios batiz , esq . 
 av . 
 miguel othon de mendizbal , col . 
 lindavista , del . 
 gustavo a. madero , c.p . 
 07738 , mxico d. f. , mexico . 
 + @xmath4 escuela superior de fsica y matemticas , instituto politcnico nacional , ed . 
 9 , unidad profesional adolfo lpez mateos , c.p . 
 07738 , mxico d. f. , mexico . 
 +    keywords : @xmath5 lie algebra , light polarization , poincar hyperboloid