Vectorial Pauli algebraic approach in polarization optics. II. Interaction of light with the canonical polarization devices

Optik 121 (2010) 2149–2158

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Vectorial Pauli algebraic approach in polarization optics. II. Interaction of light with the canonical polarization devices Tiberiu Tudor a,b a b

Faculty of Physics, University of Bucharest, P.O. Box MG-11, 0771253 Bucharest, Magurele, Romania Academy of Romanian Scientists, 54 Splaiul Independentei, 050094 Bucharest, Romania

a r t i c l e in fo

abstract

Article history: Received 10 March 2009 Accepted 25 August 2009

A theoretical approach to the interaction between polarized light and polarization devices, based on the vectorial and pure operatorial form of the Pauli algebra, is presented. In the first part of the paper we have established the vectorial Pauli-algebraic forms of the operators corresponding to various polarization devices and states of light polarization. In this second part we give the vectorial Paulialgebraic treatment of the interaction between the canonical polarization devices and the various forms of light polarization. Unlike the standard (Jones and Mueller) approaches, this formalism does not appeal to any matrix representation of the involved operators. This approach establishes a bridge between the Hilbert space of the density operators of the polarization states and the Poincare´ space of their geometric representations and gives a rigorous justification of the handling of the interactions between the polarization states and polarization systems on the Poincare´ sphere and in the Poincare´ ball. In such an approach, unlike the standard ones, the three relevant quantities that characterize the interaction – the gain, the Poincare´ vector of the outgoing light and its degree of polarization – result straightforwardly, in block. A generalized form of Malus’ law, for any dichroic device and partially polarized light is also obtained this way. & 2009 Elsevier GmbH. All rights reserved.

Keywords: Light polarization Quantum optics Pauli algebra

1. Introduction From a mathematical viewpoint, the action of various linear polarization systems (devices/media) on the various states of light polarization (SOPs) is a question of linear algebra. The old language of linear algebra is the matrix language and even now the great majority of the papers in polarization theory use this language. A visible reluctance is manifested in adopting more elevated forms of linear algebra in this field, keeping away from clearer, more intuitive, compact and elegant approaches. It is worth noting that 50 years ago the situation was the same in the linear algebra itself: New techniques have been developed which depend less on the choice or introduction of a coordinate system and not at all upon the use of matrices. Fortunately, in most cases these techniques are simpler than the older formalisms, and they are invariably clearer and more intuitive. These newer techniques have long been known to the working mathematician but a curious inertia has kept them out of books on linear algebra at the introductory level until very recently.[1] In the last 50 years there were some remarkable tentatives to introduce the new ‘‘coordinate–free’’ and ‘‘non-matrix’’ techniques E-mail address: [email protected] 0030-4026/$ - see front matter & 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2009.08.001

in the polarization theory. I have listed them in Section 1 of the first part of the paper [2]. The polarization systems are two-state systems (they have two eigenstates) and, consequently, the topology of their Hilbert state space is isomorphic with that of the Poincare´–Bloch ball, S13. This is, actually, the deep root of Poincare´’s intuition in constructing his sphere to represent the SOPs and their interaction with the polarization systems. This is also the reason for which one can handle the polarization problems in such an intuitive geometric way, namely in R3 [3–7]. There are various means of calculus to assist quantitatively the handling of the SOPs on the Poincare´ sphere: spherical trigonometry [8–12], quaternionic algebra (e.g. [13]), Pauli algebra (e.g. [5–7,14–18]), turns algebra [19–20]. Generally, both the quaternionic and the Pauli algebra are used in their matrix forms. It is Whitney [21], who introduced a Pauli algebraic pure operatorial (non-matrix) approach in analyzing some device (‘‘instrument’’) operators, and in generalizing Jones and Hurwitz’ theorems concerning the composability of polarization optical systems. She noted: ‘‘the well-known Jones and Mueller formalisms and the spherical-trigonometric approach y have virtually supplanted vector-field algebraic methods.’’ More recently, non-matrix descriptions of the states of polarized light (SOPs) and of the operators of the ‘‘canonical’’ [5] polarization devices have been given in the quaternionic language [22,23] and in a Clifford algebraic approach [24].

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Quaternionic algebra is isomorphic, with minor differences, to the Pauli algebra, and both can be embedded in the Clifford algebra Cl3. From these three algebraic ‘‘dialects’’, the Pauli algebra is by far more familiar to the physicists. The aim of this paper is to give a vectorial and pure operatorial Pauli-algebraic treatment of the action of the deterministic canonical polarization devices on the polarized light. A first advantage of this approach over the standard ones is that it leads straightforwardly, in the most direct manner and only in few lines of calculus to the whole group of three quantities which characterize the action of the system on the SOP: the gain g the Poincare´ unit vector so of the polarization state and the degree of polarization po of the emergent light. For reaching them, the standard approaches make a long round about way: they go through a more or less explicit matrix representation of the operators and then come back to their invariants, the trace and the determinant, on the basis of which are expressed – invariably in the standard approaches – the gain and the degree of polarization. Moreover, in the standard approaches, the gain, the output SOP and its degree of polarization are reached on separate lines of calculus. Here, all the relevant quantities, so, po, g appear in block in the expression of the polarization density operator of the output state. It is a unique expression which contains all the information about the interaction which occurred. We shall discuss in Section 5 the roots of this compactness and simplicity.

2. Action of various device operators on the density operator of totally polarized light For the sake of self-consistency of this second part of the paper we shall summarize here the Pauli-algebraic expressions of the device operators corresponding to various canonical polarization devices and of the density operators of various polarization states of light, we have deduced in the first part of the paper [2]. The canonical polarization devices are of two kinds: birefringent and dichroic. All these devices have orthogonal eigenvectors and, consequently, their operators are normal operators. The first class comprises the (linear, circular, generally elliptical) retarders. Their operators are unitary operators and have the Pauli-algebraic form   d d Un ðdÞ ¼ eiao eiðd=2Þn  r ¼ eiao s0 cos  in  r sin ; ð1Þ 2 2 where d is the phase-shift introduced by the retarder, a0 the isotropic phase-shift and n the Poincare´ axis of the device. In the second class enter the (linear, circular, generally elliptical) orthogonal dichroic polarizers. Their operators are Hermitian, and have the Pauli algebraic form  Z Z Hn ðr; ZÞ ¼ er eðZ=2Þn  r ¼ er s0 cosh þn  r sinh ; ð2Þ 2 2 where er ¼ eðZ1 þ Z2 Þ=2 ; eZ ¼ eZ1 Z2

ð3Þ ð4Þ

are the isotropic and the relative amplitude transmittances of the device, respectively, with eZ1 and eZ2 its principal (eigen-) transmittances. We shall take the Pauli (here reducible to the Poincare´) axis n of the dichroic device as corresponding to its major eigenstate (defined as the state of maximum transmittance, irrespective of the fact that the device is a diamplifier or a diattenuator). Particularly, the Pauli algebraic form of the operator of an ideal polarizer is Pn ¼ 12ðs0 þ n  rÞ:

ð5Þ

Concerning the density operators of the polarization states of the light, the polarization density operator of a pure state (totally polarized light) of unit intensity is J ¼ 12ðs0 þs  rÞ;

ð6Þ

where s is the Poincare´ axis of the state. The Pauli algebraic form of the polarization density operator of a mixed state (partially polarized light) has the general form J¼

J ðs0 þps  rÞ: 2

ð7Þ

The three parameters which characterize the beam of light – the intensity J , the degree of polarization, p, and the Poincare´ unit vector of its polarization state, s (or, globally, the Poincare´ vector ps of its state) – appear all, in block, in the expression of the density operator of the state. The action of the device on the polarized light may be analyzed at the level of the light spinor only for the pure states. For mixed states it can be analyzed only at the level of the density operator of the state, where it takes the operatorial form [4–6] Jo ¼ DJi Dy

ð8Þ

irrespective of the algebra in which we handle this action. Here D is the operator of the device and Ji and Jo the polarization density operators of the incident and emergent light, respectively. If we consider the polarization density operator of the incident light normalized to unit intensity: Ji ¼ 12ðs0 þ pi si  rÞ;

ð9Þ

then the density operator of the emergent (output) light has the generic form Jo ¼ 12gðs0 þ po so  rÞ:

ð10Þ

Here g is the ratio of the intensity of the emergent light to that of the incident light, the so-called gain of the transformation [3,4]; it is lower or higher than unity, according to whether the medium is absorbent or amplifier. We have to point out that our approach, unlike the standard ones, uses the primary definition of the gain. The algorithm of this vectorial Pauli algebraic approach we propose for analyzing the interaction of the light with the optical devices/media is the following:

 one gives the characteristics of the incident light, pi, si embodied in its polarization density operator, Eq. (9),

 one gives the characteristics of the device – e.g. a0, d, n, Eq. (1) – embodied in its operator,

 we apply the general law of transformation of the polarization density operator, Eq. (8), and we get the characteristics of the emergent light po, so and g, embodied in its polarization density operator. All the characteristics of the emergent light result this way straightforwardly, compactly, in block. On this way, from a physical viewpoint, we recover some particular results that were obtained in the frame of Jones or Mueller matrix formalisms [4–6] and we obtain some new, more general results, among which the largest generalization of Malus’ law (for any dichroic device acting on any state of partially polarized light). From a mathematical viewpoint, we give a new formalism of approaching the problem of interaction between the polarized light and the polarization devices. The results of this approach are, in comparison with those obtained by other approaches, very symmetric and hence more expressive (see especially Section 4.3). This is because we address to the right parameters revealing the

T. Tudor / Optik 121 (2010) 2149–2158

internal symmetries and structures of both the devices and the states of polarization. 2.1. Action of a unitary operator on the density operator of a pure state In polarization optics this corresponds to the action of a retarder (linear, circular, generally elliptical) on a (linear, circular, elliptical) totally polarized beam of light. We shall take the Pauli expansion of the unitary operator with the sign minus at the exponent, as it is usually taken in the literature of polarization. We shall see, even in this section, which is the reason of this widespread option:   d d Un ðdÞ ¼ eiao eiðd=2Þnr ¼ eiao s0 cos  in  r sin : ð11Þ 2 2 Under the action of this operator, the density operator of a pure state (completely polarized light) of the Poincare´ axis si and of unit intensity: Ji ¼ 12ðs0 þ si  rÞ

ð12Þ

becomes [4,5] Jo ¼ Un ðdÞJi Uyn ðdÞ;

ð13Þ

where we have labeled by the indices i and o the input and output states, respectively. It is worth to note that the phase factor eiao which appears in the general form of the unitary operator plays no role in this action; it is eliminated in Eq. (13). In this first case, we shall present the calculus in detail:     1 d d d d s0 cos  in  r sin ðs0 þsi  rÞ s0 cos þ in  r sin Jo ¼ 2 2 2 2 2   1 d d d d d s0 cos  in  r sin s0 cos þ in  r sin þsi  r cos ¼ 2 2 2 2 2 2  d d þis0 si  n sin  ðsi  nÞ  r sin 2 2  1 d d d d ¼ s0 cos2 þ in  r sin cos þ si  r cos2 2 2 2 2 2 þis0 si  n sin in  r sin

d 2

d 2

cos

cos

þðn  si Þ  r sin

d 2

d 2

d 2

 ðsi  nÞ  r sin

þðn  rÞ2 sin2 cos

d

cos

d 2

 is0 n  si sin

þðn  rÞðsi  nÞsin

2

d 2

cos

Jo ¼

Eq. (17) gives the law of transformation of the Poincare´ axis of a pure state under the action of a unitary operator on this state. It expresses a rotation in R3 , here a rotation of angle d of the unit vector si around the Poincare´ axis n of the operator. We can recognize in Eq. (17) the operator of the R3 rotation [20]: so ¼ Rn ðdÞsi ;

ð18Þ

Rn ðdÞ ¼ cos d þ ð1  cos dÞnðnÞ þsin dðnÞ:

ð19Þ

A matrix equivalent of this result is given in [25]. A positive rotation d around the n axis in Eq. (19) corresponds to the ‘‘right-hand screw’’ rule [18]. Should we have taken the sign plus in the Pauli expansion of the unitary operator, the last term in Eq. (19) would change the sign, i.e. the rotation operator would correspond this time to a ‘‘left-hand screw’’ rule. This is the reason of the usual option of the sign in the Pauli expression of the unitary operator, Eq. (11). It is gratifying to emphasize that the unitary operator Un, Eq. (11), acts, Eq. (13), on the state of optical polarization expressed by its polarization density operator, hence in the Hilbert space of the polarization density operators of the states, whereas the rotation operator Rn, Eq. (19), acts, Eq. (18), on a R3 vector, namely the Poincare´ vector, si, abstractly associated to this state (Fig. 1). The main result of this section is establishing an algebraic bridge between the Hilbert space of the SOPs and the S13 space of their Poincare´ axes: each unitary action, Eqs. (13), (11), in the first space is mapped in a rotation, Eqs. (18), (19), in the second space. This is the mathematical ground when handling the actions of the phase-shifters on the SOPs as rotations on the Poincare´ sphere. In fact, obviously, the phase-shifter acts, in the real physical space, on the electric vector of the light wave. This action is mapped in a unitary action Un ðdÞ in the Hilbert space of the SOPs and further in a rotation Rn ðdÞ on the Poincare´ sphere S13. From a physical viewpoint we recover the well-known result (e.g. [4–6]) that a phase-shifter modifies the SOP of the completely polarized incident light without changing its intensity, nor its degree of polarization.

d 2

d

2  d d  si ðn  nÞ  r sin2 þnðn  si Þ  r sin2 þis0 n  ðsi  nÞsin 2 2 2    1 d d 2d 2d þ 2ðn  si Þ  r sin cos ¼ s0 þ si  r cos  sin 2 2 2 2 2  2d þ2n  si ðn  rÞ sin 2 2

2

d 2

d 2

2151

d

2.2. Action of an orthogonal projector on the density operator of a pure state An orthogonal projector in polarization optics corresponds to an ideal polarizer.

  1 d s0 þ si  r cos d þ ðn  si Þ  r sin d þ 2n  si ðn  rÞ sin2 : 2 2 ð14Þ

We have used here Dirac’s identity: ða  rÞðb  rÞ ¼ a  b þ iða  bÞ  r;

ð15Þ

0

where a and b are three-dimensional vectors. The density operator of the output light, Eq. (14), has, as it is expectable, a form corresponding to a pure state of unit intensity: Jo ¼ 12ðs0 þ so  rÞ

ð16Þ

(i.e. g= 1, po = 1) with so ¼ si cos d þ n  si sin d þ 2ðn  si Þn sin2

d 2

:

ð17Þ

Fig. 1. Rotation of the Poincare´ axis of the density operator of a SOP around the Poincare´ axis of the operator of a retarder. Notations.

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We shall consider now an orthogonal projector expressed in the Pauli algebraic form Pn ¼ 12ðs0 þ n  rÞ

ð20Þ n

and the operator of a pure state expressed in the same form Ji ¼ 12ðs0 þ si  rÞ:

si

ð21Þ α

For an orthogonal projector the components of the unit vector n are real. It is worth stressing that here si and n correspond, generally, to elliptical pure SOPs and to elliptical polarizers, respectively. The density operator of the output SOP is Jo ¼ Pn Ji Pyn

0

ð22Þ

that is 1 ðs0 þ n  rÞðs0 þ si  rÞðs0 þ n  rÞ 8 1 1þ n  si 1 ðs0 þ n  rÞ: ¼ ½s0 þn  r þ ðn  si Þðs0 þ n  rÞ ¼ 4 2 2

Jo ¼

ð23Þ

Fig. 2. Poincare´ projection of the axis of the density operator of a SOP on the axis of the operator of a polarizer. Notations.

Finally we get Jo ¼

1 þ n  si Jn ; 2

ð24Þ

where we label by Jn the normalized pure state corresponding to the ideal polarizer of operator Pn. Hence the SOP of the emergent light (totally polarized, po = 1) is the passed eigenstate of the polarizer, so ¼ n

ð25Þ

and the intensity of the beam is reduced in the ratio (i.e. the gain is) 1þ n  si : ð26Þ 2 If the Poincare´ axes of the incident state, si, and of the polarizer, n, coincide, we get g¼

Jo ¼ 12ðs0 þ si  rÞ ¼ Ji

a

2.3. Action of a Hermitian operator on the density operator of a pure state

ð27Þ

and the gain is gmax = 1, i.e. the input SOP remains, obviously, unmodified by the polarizer. If, on the contrary, the polarizer is orthogonal to the incident state, i.e. their Poincare´ vectors are antipodal, n =  si, the incident light is blocked by the polarizer: Jo ¼ 0 . Labeling by a the angle between n and si, Eq. (24) becomes Jo ¼ cos2

passes through the polarizer is given in a Poincare´ representation by the square of the cosine of half the angle of the Poincare´ projection of si on n (Fig. 2). This is a well-known result in polarization theory [5,8], but the traditional deduction, based on the decomposition of the, generally elliptical, polarized light in linear components and on the spherical trigonometry, is considered, even by its authors, tedious [26]. The possibility of obtaining it in the few lines of calculus of Eq. (23) is due to the facts that our approach is adapted to the (spherical) symmetry of the polarization state space and it uses the most suited and compact mathematic formalism of operating in such a space – the vectorial Pauli algebra.

Jn : ð28Þ 2 We have here a first generalized form of Malus’ law: the fraction of the intensity of the (in general elliptically polarized) incident light passing through a (generally elliptical) polarizer is the square of the cosine of half the angle between the Poincare´ axis of the polarizer and that of the polarization state of the incident light. On the Poincare´ sphere the angle a is the angle of the geodesic arc between the tops of the Poincare´ vector of the SOP of the incident light, si, and the Poincare´ vector of the polarizer, n. The action of the (generally elliptical) polarizer on a (generally elliptically polarized) SOP is represented on the Poincare´ sphere ´ projection of si on by what we could denominate as the Poincare n – a movement of the top of si to the top of n along the geodesic arc (unfortunately the term ‘‘spherical projection’’ is consecrated, in spherical trigonometry, to another signification). After passing through the polarizer, a pure state remains pure, i.e. its Poincare´ vector remains on the Poincare´ sphere S12. The length of its Poincare´ vector is not modified. We cannot represent the modification of the intensity, the gain, by a reduction of the state vector, but we can give to this modification a Poincare´ representation: the decrease of the intensity of the light which

In polarization optics the Hermitian operators are known especially as describing some passive optical elements – the homogeneous diattenuators [27,28] – e.g. the dichroic polarizers [3]. But some active elements (with optically pumped crystals) are conceived [4,29] which act, on the contrary, as diamplifiers, or as squeeze devices (amplifiers on one channel, attenuators on the other). The results we shall obtain in this section are valid for all these devices, for homogeneous diamplifiers, squeeze devices or diattenuators. We have to adopt a generic term for all these devices. From a mathematical viewpoint, the operators corresponding to all of them are Hermitian. Hence, they could be denominated Hermitian devices. From a physical viewpoint, they have in common the anisotropy of the absorption or of the amplification. Even of a disputable adequacy [3], the term dichroism is consecrated for absorboanizotropy. Maybe by an extension of this term to the anisotropy of amplification [30] it would be reasonable to join together the diattenuators, diamplifiers and squeeze devices under the name of dichroic devices. The most general Hermitian operator  Z Z Hn ðr; ZÞ ¼ er eðZ=2Þnr ¼ er s0 cosh þn  r sinh ð29Þ 2 2 acting on a pure state Ji ¼ 12ðs0 þ si  rÞ

ð30Þ

gives another pure state: Jo ¼ Hn ðr; ZÞJi Hyn ðr; ZÞ;

ð31Þ

T. Tudor / Optik 121 (2010) 2149–2158

1 2r  Z Z e s0 cosh þ n  r sinh ðs0 þ si  rÞ 2 2 2  Z Z  s0 cosh þ n  r sinh 2 2  1 ¼ e2r s0 cosh Z þ n  r sinh Z þ si  r þ s0 si  n sinh Z 2 i 2Z ; þ2n  si ðn  rÞ sinh 2 1 2r n s0 ðcosh Z þ si  n sinh ZÞ Jo ¼ e 2h i o 2Z r : þ si þ n sinh Z þ 2ðn  si Þn sinh 2 This expression is, indeed, that of a pure state:

The emergent light is totally polarized in the minor eigenstate of the dichroic device and the gain is minimum: gmin ¼ e2Z2 . Generally, if we label by a the angle between the Poincare´ axis of the (major eigenstate of the) device, n, and the Poincare´ axis of the SOP of the incident light, si, i.e.

Jo ¼

Jo ¼ 12gðs0 þso  rÞ:

n  si ¼ cos a;

ð44Þ

results (34), (35) become ð32Þ

g ¼ e2r ðcosh Z þ cos a sinh ZÞ ¼ e2r

 Z  e þ eZ eZ  eZ þcos a 2 2

 a a 2Z1 a a ¼ e cos2 þe2Z2 sin2 ; ¼ e2r eZ cos2 þ eZ sin2 2 2 2 2

ð45Þ

ð33Þ 2

(po =1) where g is the gain given by the device (it may be lower or higher than unity depending on the values of Z1, Z2 and a): g ¼ e2r ðcosh Z þ si  n sinh ZÞ ¼ e2r ðcosh Z þ cos a sinh ZÞ

ð34Þ

and the Pauli unit (see further) vector of the output light is given by 2

so ¼

2153

si þ n sinh Z þ 2ðn  si Þn sinh ðZ=2Þ : cosh Z þ si  n sinh Z

ð35Þ

A dichroic device moves the Poincare´ vector of the completely polarized incident light on a great circle of the Poincare´ sphere, in a plane determined by the Poincare´ axis of the device, n, and that of the incident light, si. We have to note that, in contrast with the case of the unitary operators, in this case the isotropic factor in Eq. (29), is kept (squared: e2r) in the expression of the output state of the Hermitian device, Eq. (32). It affects only the gain given by the device. The modification of the SOP is determined by the relative transmittance, more precisely, by the coefficient of dichroism Z = Z1  Z2. Let us consider now some particular cases of results (32)–(35), applied to polarization optics. If the SOP of the incident light coincides with the major eigenstate of the dichroic device (we bear in mind generally an elliptical state), the Poincare´ vectors of the device and of the incident SOP are parallel: si ¼ n case in which   2 n 1 þ sinh Z þ 2 sinh ðZ=2Þ ¼ n ¼ si ; so ¼ cosh Z þ sinh Z

ð36Þ

ð37Þ

ð38Þ

Jo ¼ e2Z1 12 ðs0 þ si  rÞ ¼ e2Z1 12ðs0 þ n  rÞ ¼ gmax Jn :

ð39Þ

The emergent light is totally polarized in the major eigenstate of the dichroic device (coincident with the SOP of the incident light) and its intensity is ‘‘amplified’’ (attenuation included) by gmax ¼ e2Z1 . If, on the contrary, the SOP of the incident light is aligned with the minor eigenstate of the dichroic device:

then Eqs. (35), (34) and (32) give   2 n 1  sinh Z þ 2 sinh ðZ=2Þ ¼  n ¼ si ; so ¼  cosh Z  sinh Z

ð46Þ

Expression (45) of the gain constitutes another generalization of the Malus law (of course still limited to the action on pure states). It is worth noting that the final position of the Poincare´ vector of the state depends only of the coefficient of relative transmittance, Z. The isotropic transmittance of the device does not appear in Eq. (46). For the validity of all the conclusions drawn in this subsection, it is essential for so in Eq. (35) be a unit vector. We have postponed proving this assertion. Expression (46) is more suitable for this handling. The modulus squared of the numerator in Eq. (46) is 2

2

1þ sinh Z þcos2 a cosh Z  2 cos2 a cosh Z þ cos2 a þ2 cos a sinh Z þ 2 cos2 a cosh Z  2 cos2 a þ 2 cos a sinh Z cosh Z  2 cos a sinh Z ¼ ðcosh Z þcos a sinh ZÞ2 ;

ð47Þ

i.e. it is equal with the square of the denominator in Eq. (46). Hence so is a unit vector. It is gratifying to note that in the limit Z1-0 and Z2- N, Z = Z1  Z2-N, when a diattenuator passes in an ideal polarizer (pure projector), from Eqs. (45) and (46) we get g-cos2

so -n

a 2

;

tanh Z þcos a 1 þ cos a -n ¼ n; 1 þ tanh Z cos a 1 þ cos a

ð48Þ

ð49Þ

concordant with the corresponding results of Section 2.2.

g ¼ gmax ¼ e2r ðcosh Z þsinh ZÞ ¼ e2Z1 ;

si ¼  n

si þn sinh Z þ2n cos a sinh ðZ=2Þ cosh Z þ cos a sinh Z si þ n sinh Z þ n cos að cosh Z  1Þ ¼ : cosh Z þcos a sinh Z

so ¼

ð40Þ

3. Action of various device operators on the density operator of natural (unpolarized) light 3.1. Action of a unitary operator on the density operator of the unpolarized light The normalized density operator of the incident unpolarized light being Ji ¼ 12s0

ð50Þ

from Eq. (13), with Eq. (11) we get ð41Þ

gmin ¼ e2r ðcosh Z  sinh ZÞ ¼ e2Z2 ;

ð42Þ

Jo ¼ e2Z2 12 ðs0 þ si  rÞ ¼ e2Z2 12ðs0  n  rÞ:

ð43Þ

Jo ¼ eiao eiðd=2Þn  r 12 s0 eiao eiðd=2Þn  r ¼ 12s0 ; Jo ¼ Ji :

ð51Þ

A phase-shifter does modify neither the state, nor the intensity of the unpolarized incident light.

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3.2. Action of an orthogonal projector on the density operator of unpolarized light

Jo ¼ 12ðs0 þpo so  rÞ

From Eq. (22), with Eqs. (20) and (50) one gets

po ¼ pi :

¼ 18 ½s0 þ 2n  r þ ðn  rÞ2  ¼ 14ðs0 þn  rÞ; ð52Þ

The emergent light is completely polarized in the (major) eigenstate of the polarizer and its intensity is half the intensity of the incident light (the gain is 1/2).

ð54Þ

and with a Poincare´ unit vector: so ¼ n:

The Hermitian operator partially polarizes the incident unpolarized light in its major eigenstate. The gain is equal with the arithmetic mean value of the two principal intensity transmittances of the device: e2Z1 þ e2Z2 : ð57Þ g¼ 2 For an ideal polarizer (Z1 =0, Z2- N,Z-N), the results of the Section 3.2 are recovered: po ¼ 1;

so ¼ n;

g ¼ 12:

ð58Þ

4. Action of various device operators on the density operator of partially polarized light 4.1. Action of a unitary operator on the density operator of a mixed state In polarization optics, mixed states correspond to partially polarized light. We shall consider again a unitary operator, expressed in a Pauli algebraic form, Eq. (11), acting, this time, on a mixed (partially polarized) state [2]: Ji ¼ 12ðs0 þ pi si  rÞ;

ð59Þ

where pi is the degree of polarization of the incident light. The polarization state density operator of the emergent light is     1 d d d d Jo ¼ s0 cos  in  r sin ðs0 þ pi si  rÞ s0 cos þ in  r sin ; 2 2 2 2 2 Jo ¼

:

ð63Þ

4.2. Action of an orthogonal projector on the density operator of a mixed state

ð53Þ

it follows that it represents partially polarized light with a degree of polarization: po ¼ tanh Z

d 2

ð56Þ

Jo ¼ er eðZ=2Þnr 12 s0 er eðZ=2Þnr ¼ 12e2r ðs0 cosh Z þ n  r sinh ZÞ;

Jo ¼ 12gðs0 þ po so  rÞ;

so ¼ si cos d þ n  si sin d þ 2ðn  si Þn sin2

ð55Þ

From Eq. (31) with Eqs. (29), (50), (3) and (4) we get

1 e2Z1 þ e2Z2 ðs0 þ n  r tanh ZÞ: 2 2 If we compare this result with the standard equation,

ð62Þ

The Poincare´ unit vector (of the totally polarized component) of the output state is

The gain is unitary (g= 1), the intensity of the partially polarized incident light is not modified by a phase-shifter. This is a well-known result: any (generally elliptic) retarder changes the state of polarization of the incident light without affecting its intensity and degree of polarization [4]. Under the action of a retarder, a mixed state remains mixed with the same degree of polarization. Loosely speaking, we say that a unitary operator moves the Poincare´ vector on the Sp2i Poincare´ sphere (pi being the radius of the inner sphere corresponding to the degree of polarization pi in the Poincare´ ball). In fact, this means that to the unitary, SU(2), operator acting on the Jones vector of the incident light corresponds a rotation O(3) operator acting on the Poincare´ vector of the incident SOP. We recognize in Eq. (63) the O(3) operator Rn ðdÞ, Eq. (19), determining the rotation of the Poincare´ ball, which corresponds (in R3 ) to the action of the SU(2) operator Un ðdÞ in the Hilbert space of the density operator of the states, Eq. (13).

3.3. Action of a Hermitian operator on the density operator of the unpolarized light

Jo ¼

ð61Þ

with the same degree of polarization as that of the incident light:

Jo ¼ 12 ðs0 þ n  rÞ12 s0 12 ðs0 þ n  rÞ ¼ 18ðs0 þ n  rÞ2

Jo ¼ 12Jn :

The density operator of the output light has, as it is expectable, the form corresponding to a mixed (partially polarized) state:

 

1 d s0 þ pi si  r cos d þ ðn  si Þ  r sin d þ 2n  si ðn  rÞsin2 : 2 2

ð60Þ

The action of an orthogonal projector, Eq. (20), upon a mixed (partially polarized) state, Eq. (59), is given by 1 ðs0 þ n  rÞðs0 þ pi si  rÞðs0 þn  rÞ 8 1 ¼ ½s0 þ n  r þpi ðn  si Þðs0 þn  rÞ 4 1 1þ pi cos a ðs0 þn  rÞ; ¼ 2 2

Jo ¼

1 þpi cos a Jn : 2 Hence the characteristics of the output SOP are

Jo ¼

ð64Þ

so ¼ n;

ð65Þ

po ¼ 1;

ð66Þ

g ¼ 12ð1 þ pi cos aÞ:

ð67Þ

The output light is, evidently, totally polarized in the major eigenstate of the polarizer and the (lower than unity) gain depends firstly on the Poincare´ angle between the SOP of the incident light and the eigenstate of the polarizer, and secondly on the degree of polarization of the incident light. Results (64)–(67) can also be obtained by decomposing the incident light into its unpolarized and totally polarized components, and by using then the results of Sections 2.2 and 3.2, in other words by using the ‘‘polarized – unpolarized dichotomy’’ [3]. In what concerns the gain, such an approach (see Appendix A) leads straightforwardly to an equivalent form of Eq. (64):   a 1  pi Jo ¼ pi cos2 þ Jn ð68Þ 2 2 and gives more insight into the physical aspects of the action of the polarizer on the partially polarized incident light.

T. Tudor / Optik 121 (2010) 2149–2158

Unlike in the global approach, leading to Eq. (64), in this last approach we get a more intuitive grasp concerning the gain (here the loss of intensity given by the polarizer): it comes from the losses of cos2(a/2) and of 1/2 for totally polarized (pi =1) and unpolarized (pi = 0) incident light, respectively. Eq. (68) is a new step in the generalization of Malus’ law, namely for ideal polarizers and partially polarized light. Let us consider now some particular cases of the general results (64)–(67). If the Poincare´ axis of the totally polarized component of the incident state, si, and the Poincare´ axis, n, of the polarizer are parallel, from Eq. (64) one obtains Jo ¼

1 þpi 1 1 þ pi 1þ pi ðs0 þ si  rÞ ¼ Ji ¼ Jn : 2 2 2 2

1  pi 1 1  pi ðs0 þ n  rÞ ¼ Jn : 2 2 2

1 2r n e s0 ðcosh Z þ pi si  n sinh ZÞ h2

ð70Þ

The emergent light is, obviously, in the pure state imposed by the polarizer, and its intensity is reduced in the fraction(1  pi)/2: the totally polarized component (of fraction pi in the incident light) is blocked, the unpolarized component (of fraction 1–pi) is reduced to the fraction (1  pi)/2, which is all the percent of the incident light which passes through the polarizer; the gain is (1 pi)/2, in agreement with Eq. (68). Varying the angle between the Poincare´ axis, n, of the polarizer and that of the (totally polarized component of the) incident light, si, for a fixed value of the degree of polarization, pi, of the incident light, the (lower than unity) gain varies between the values:

Zi 2

o r :

ð74Þ

where g is the gain given by the dichroic device: g ¼ e2r ðcosh Z þ pi si  n sinh ZÞ ¼ e2r ðcosh Z þ pi cosa sinh ZÞ   1 þ pi cos a 1  pi cos a þ eZ ; ¼ e2r eZ 2 2 g ¼ e2Z1

1þ pi cos a 1  pi cos a þ e2Z2 2 2

ð75Þ

and the degree of polarization and the Poincare´ unit vector of the output light are given by 2

po so ¼

pi si þ n sinh Z þ 2pi ðn  si Þn sinh ðZ=2Þ : cosh Z þ pi si  n sinh Z

p o so ¼

pi si þn sinh Z þpi n cos aðcosh Z  1Þ : cosh Z þ pi cos a sinh Z

si ¼ n

ð71Þ

case in which

gmin ¼

1  pi for n ¼  si : 2

ð72Þ

Jo ¼ e2r

ð77Þ

i  1h Z s0 ðcosh Z þpi sinh ZÞ þ pi þsinh Z þ 2pi sinh2 n  r 2  2 1 sinh Z þ pi cosh Z nr ¼ e2r ðcosh Z þ pi sinh ZÞ s0 þ 2 cosh Z þpi sinh Z   tanh Z þ pi 2r 1 ðcosh Z þ pi sinh ZÞ s0 þ nr : ð78Þ ¼e 2 1 þpi tanh Z

In this particular case it is straightforward to separate po and so. In Eq. (78), n is a unit vector, so that so ¼ n ¼ si :

Let us consider now the action of a Hermitian operator, Eq. (29) on a mixed state, Eq. (59). With Eq. (31) we get  1 Z Z Jo ¼ e2r s0 cosh þ n  r sinh ðs0 þ pi si  rÞ 2 2 2 Z Z  s0 cosh þ n  r sinh ; 2 2

ð76Þ

We have labeled here by a the angle between the Poincare´ axis of the polarizer, n, and that of the incident light, si. Expression (75) of the gain constitutes the largest generalization of Malus’ law, valid for partially polarized light passed through any dichroic device. In the general case, the separation of the Poincare´ vector so and of the degree of polarization po of the output light is not straightforward. We have to point out that, in fact, the vector poso gives a complete characterization of the SOP of the, generally partially polarized, light. Generally we are not interested separately in po and so. We prefer to postpone this separation and to analyze now some particular cases of results (75), (76) for getting first some physical insight concerning these results. If the SOP of the incident light is ‘‘aligned’’ with the major axis of the dichroic device, i.e. the Poincare´ vectors of the device and of the SOP are parallel:

1þ pi for n ¼ si ; 2

4.3. Action of a Hermitian operator on the density operator of a mixed state

ð73Þ

This expression is that of a mixed state: Jo ¼ 12g½s0 þ po so  r;

gmax ¼

The SOP of any partially polarized incident light is projected by the polarizer on the Poincare´ sphere S12 (in the point n) but the gain, Eq. (67) cannot be longer represented by the square of the cosines of half the angle between n and si, as in the case of totally polarized incident light. Malus’ law, expressed now by Eq. (68), loses the direct Poincare´ – geometrical interpretation it had in the simpler case of the totally polarized incident light.

2

þ pi si þn sinh Z þ2pi ðn  si Þn sinh

ð69Þ

The SOP of the light emerging from the polarizer is identical to the SOP of the totally polarized component of the incident light and the gain is (1+pi)/2. This value of the gain – at a first sight somewhat curious – can be clarified on the basis of Eq. (68). For a = 0 ðn  si Þ the totally polarized fraction of the incident light passed by the polarizer remains pi (the corresponding intensity is not reduced), while the fraction of the unpolarized light is reduced from 1  pi to (1 pi)/2. This means that, by passing through the polarizer, the total intensity of the incident light is reduced by the coefficient (1+pi)/2. If the Poincare´ axis n of the polarizer is antiparallel to the Poincare´ axis, si, of the totally polarized component (the major axis of this component is crossed with the passing axis of the polarizer) the polarization density operator of the emergent light becomes Jo ¼

Jo ¼

2155

ð79Þ

The emergent light is partially polarized in the major eigenstate of the device (coincident with the SOP of the incident light) and its intensity is reduced or amplified by   1 þ pi 1  pi þe2Z2 : gmax ¼ e2r ðcosh Z þ pi sinh ZÞ ¼ e2Z1 2 2

ð80Þ

The degree of polarization of the output SOP depends both on the degree of polarization of the input SOP, pi,

2156

T. Tudor / Optik 121 (2010) 2149–2158

and on the device, Z: po ¼

coefficient

of

relative

transmittance

of

sinh Z þpi cosh Z tanh Z þ pi ¼ : cosh Z þ pi sinh Z 1 þ pi tanh Z

the

ð81Þ

which can be put in the physically expressive form " #1=2 1  p2i po ¼ 1  : ½cosh Z þ pi n  si sinh Z2

ð90Þ

If, on the contrary, the SOP of the incident light is ‘‘aligned’’ with the minor axis of the device, then

From (76) with (90) the Poincare´ unit vector of the output state has the following expression:

si ¼  n

so ¼

ð82Þ

and Eqs. (73), (75) and (76) give   1 sinh Z  pi cosh Z nr Jo ¼ e2r ðcosh Z  pi sinh ZÞ s0 þ 2 cosh Z  pi sinh Z   1 tanh Z  pi ¼ e2r ðcosh Z  pi sinh ZÞ s0 þ nr : 2 1  pi tanh Z

so ¼ ð83Þ

In this case the gain is gmin ¼ e2r ðcosh Z  pi sinh ZÞ ¼ e2Z1

1  pi 1 þpi þ e2Z2 : 2 2

ð85Þ

By consequence, in this case, the emergent light is partially polarized in the major or the minor eigenstate of the device, n and n, respectively, depending on the sign of the coefficient of n . r in Eq. (83). It is gratifying to note that formulae (81) and (85) are strongly similar with some formulae in the theory of special relativity [31]. This similarity is not a casual one. Its roots stand in the isomorphism between the group of transformations SL(2, C), used all along this paper (in a pure operatorial representation) and the Lorentz group O(3, 1) which describe the transformations in the special relativity. Well-known, this isomorphism was largely exploited in the last decade in the quasirelativistic formulation of the theory of polarization and, more generally, of the ‘‘two-state’’ (‘‘two-beam’’) systems [27,32–36]. Let us turn now back to the general result, Eq. (76), for separating here the degree of polarization po and the Poincare´ unit vector of the emergent light. Straightforwardly, the degree of polarization is jpi si þ n sinh Z þ pi n cos aðcosh Z  1Þj : jcosh Z þ pi cos a sinh Zj

ð86Þ

The square of the numerator can be processed as follows: ½pi si þn sinh Z þpi ðn  si Þðcosh Z  1Þn2 2

¼ p2i þsinh Z þ p2i ðn  si Þ2 ðcosh Z  2 cosh Z þ 1Þ 2

þ 2p2i ðn

 si Þ ðcosh Z  1Þ þ2pi n  si sinh Z þ2pi n  si sinh Zðcosh Z  1Þ 2

2

¼ p2i þsinh Z þ p2i ðn  si Þ2 cosh Z  p2i ðn  si Þ2 þ2pi n  si sinh Z cosh Z 2

2

¼ p2i þsinh Z þ 2pi n  si sinh Z cosh Z þ p2i ðn  si Þ2 sinh Z;

ð87Þ

whereas the square of the denominator is ðcosh Z þ pi n  si sinh ZÞ2 2

2

¼ cosh Z þ p2i ðn  si Þ2 sinh Z þ2pi n  si sinh Z cosh Z:

pi si þ n sinh Z þ pi ðn  si Þnðcosh Z  1Þ ½p2i  1 þ ðcosh Z þ pi n  si sinh ZÞ2 1=2

:

;

ð91Þ

It is straightforward to obtain from Eqs. (90) and (91) the particular results (81) and (85). We present the calculus in Appendix B.

5. Conclusions

jsinh Z  pi cosh Zj jtanhZ  pi j ¼ : po ¼ cosh Z  pi sinh Z 1  pi tanh Z

2

2

ð84Þ

Concerning the degree of polarization, because it is a positive definite quantity, we have to take

po ¼ jpo so j ¼

pi si þ n sinh Z þ pi ðn  si Þnðcosh Z  1Þ 2

½p2i þsinh Z þ p2i ðn  si Þ2 sinh Z þ 2pi n  si sinh Z cosh Z1=2

ð88Þ

Thus, " #1=2 2 2 p2 þ sinh Z þp2i ðn  si Þ2 sinh Z þ2pi n  si sinhZ cosh Z ; po ¼ i 2 2 cosh Z þp2i ðn  si Þ2 sinh Z þ2pi n  si sinh Z cosh Z ð89Þ

The Pauli algebra has been used for a long time in polarization optics, mainly in its scalar and usually in its matrix form, for describing the states of light polarization and the canonical optical devices. In this paper we have used the Pauli algebra in its very compact, vectorial form for analyzing the action of the canonical devices on the partially polarized light. A first, visible, advantage of the vectorial Pauli algebraic approach in comparison with the traditional methods arises from its compactness. Unlike in the standard approaches, the characteristics of the emergent light – its Poincare´ unit vector so, the polarization degree po and the intensity – result straightforwardly, in block, in the expression of the polarization density operator of the output state. It is one of the computational advantages of this approach with respect to the traditional Jones or Mueller matrix formalism, in which the expressions of the gain and polarization degree are reached by a cumbersome calculus involving the traces and determinants of various matrices [4,5]. A second, deeper, advantage of the vectorial Pauli algebraic approach is that it is straightforwardly and intimately connected with the Poincare´ sphere representation, which plays a central role in the theory and practice of optical polarization. The vectorial Pauli algebraic approach brings into focus the Pauli axes of the operators, which have a direct correspondent in the Poincare´ ball geometric representation: the Poincare´ vectors of the polarization states and of the devices. Consequently, the vectorial Pauli algebra is the shortest bridge between the actions of the device operators in the Hilbert space of the states and the corresponding movements of the Poincare´ vectors associated to these states on the Poincare´ sphere or in the Poincare´ ball. We have illustrated this fact in Sections 2.1 and 4.1, by establishing the rotation operator Rn in the R3 Poincare´ space corresponding to the unitary operator Un of a phase shifter, which acts in the Hilbert space of the SOPs. This connection is equally straightforward in the second case we have analyzed, that of the Hermitian operators corresponding to the dichroic devices. Another advantage of the vectorial Pauli algebraic method, which has become visible in the analysis of the action of the dichroic devices on partially polarized light (Section 4.3) is that the results have a high degree of symmetry – e.g. Eqs. (75). This symmetry, as well as the compactness of the results, is due to the fact that our approach is parameterized in a manner well adapted to the symmetries of both the polarization states space and of the devices. On one hand, our approach is performed in the Hilbert space of the (density operators of the) polarization states and in the Poincare´ S13 space, isomorphic with the first. Well known, the

T. Tudor / Optik 121 (2010) 2149–2158

description of a physical system in the abstract space of the states of the system (e.g. the space of configurations) reflects its essential properties and symmetries better than its description in the real physical space. This is the case of the Hilbert space of the SOPs too. On the other hand, concerning the devices, our approach addresses to their eigenstates – the Poincare´ axis of the device operator joints its eigenstates – and to their eigenvalues – e.g. Eqs. (1), (2), (5) – which both reflect the symmetry of the device. Particularly, the generalized Malus’ law (for any dichroic device and any state of polarization) we have established this way, Eq. (75), has also a beautiful symmetric form. Finally, we have to point out that the field of applicability of the formalism we have given in this paper is much larger than that of the light polarization. It can be applied, in the same form, to any ‘‘two-state’’ (‘‘two-beam’’) system (interferometric systems, multilayer systems, geometrical optic systems, spin 1/2 systems, a.s.o.). But, above all, the vectorial Pauli algebraic approach to the interaction polarization systems – polarization states is a new step in introducing the ‘‘coordinate-free’’ paradigm ([2] and Refs. [13– 29] herein; [22–24]) in the polarization theory. The above presented analysis is a proof that the main classical results of this theory can be more efficiently established and that new insightful results can be obtained in this frame – e.g. the Section 2.1, the whole Section 4.3, particularly the most general form of Malus’ law, for partially polarized light and elliptical partial polarizers, Eq. (75).

which can be compressed to the form obtained by the direct calculus, Eq. (64): 1 1 þ pi cos a ð1 þpi n  si Þðs0 þ n  rÞ ¼ Jn : ðA:7Þ 4 2 Quite generally, an advantage of such an approach is the possibility of looking separately, from the very beginning, at the modification undergone by the totally polarized and unpolarized components of the incident light under the action of various devices. As we shall see immediately, the detailed result, Eq. (A.6), can give us more grasp into the intuitive physical aspects of the interaction of the partially polarized incident light with the polarizer. Eq. (A.6) may be put in a more expressive form   a 1 1 Jo ¼ pi cos2 þ ð1  pi Þ ðs0 þ n  rÞ 2 2 2   1  pi 2a þ Jn : ðA:8Þ ¼ pi cos 2 2 Jo ¼

In this form, its signification is quite clear:

 After passing through an ideal polarizer (represented by an



Appendix A We will give here only one of the multitude of possible examples of how the main results of the paper can be developed in a more detailed analysis of the interaction of polarized light with various polarization devices, for getting more physical insight into this interaction. Results (64)–(67) can be obtained by decomposing the incident light into its unpolarized and totally polarized components, and by using then the results of Sections 2.2 and 3.2, in other words by using the ‘‘polarized–unpolarized dichotomy’’ [3]. If we label by J0tpi and J0upi the normalized polarization density operators of the totally polarized component (of Poincare´ vector si) and of the unpolarized component of the incident light, respectively: J0tpi ¼ 12ðs0 þsi  rÞ;

ðA:1Þ

1 s0 ; 2

ðA:2Þ

J0upi ¼

ðA:3Þ

Under the action of an ideal polarizer of Poincare´ axis n, the normalized density operator of the totally polarized component of the incident light is modified according to Eq. (23), i.e.: 1 a1 ðs0 þn  rÞ: Jtpo ¼ ð1þ n  si Þðs0 þn  rÞ ¼ cos2 4 22

g ¼ pi cos2

a 2

þ

1  pi : 2

ðA:9Þ

Unlike in the global approach, leading to Eq. (64), in this approach, based on the ‘‘polarized–unpolarized dichotomy’’, we get a more intuitive grasp concerning the loss of intensity given by the polarizer: it comes from the losses of cos2(a/2) and of 1/2 for totally polarized (pi = 1) and unpolarized (pi =0) incident light, respectively.

In the cases si 7n, the general result in Eq. (89) leads to the previous particular forms, Eqs. (81) and (85). Indeed, from Eq. (89) one obtains 2

p2o ¼

ðA:4Þ

ðA:5Þ

(its intensity is reduced by half). Adding (incoherently) the totally polarized component, Eq. (A.4), and the unpolarized component, Eq. (A.5), of the light emerging from the polarizer, one obtains Jo ¼ 14 pi ð1 þ n  si Þðs0 þ n  rÞ þ ð1  pi Þ14ðs0 þ n  rÞ ¼ 14½pi ð1 þ n  si Þ þ ð1  pi Þðs0 þn  rÞ;

The parenthesis in Eq. (A.8) is the ‘‘gain’’ (of course, subunitary):

¼

(it is no longer normalized to unit intensity), while, according to Eq. (52), the unpolarized component of the output is Jupo ¼ 12ðs0 þn  rÞ



orthogonal projector) the partially polarized incident light becomes totally polarized in the major eigenstate of the (generally elliptical) polarizer. The fraction pi of the incident light corresponding to its totally polarized component is reduced in the ratio cos2(a/2), where a is the angle between the Poincare´ vector, si, of the incident light SOP and the Poincare´ vector, n, of the polarizer. The fraction 1  pi of the incident light corresponding to its unpolarized component is reduced to half of its value.

Appendix B

we can write Ji ¼ pi J0tpi þð1  pi ÞJ0upi :

2157

ðA:6Þ

2

p2i þ sinh Z þ p2i sinh Z 7 2pi sinh Z cosh Z 2

2

cosh Z þ p2i sinh Z 72pi sinh Z cosh Z ðsinh Z 7 pi cosh ZÞ2 ðcosh Z 7pi sinh ZÞ2

:

ðB:1Þ

By looking at the denominator, one can see that the value between parentheses is always positive, because cosh Z is always greater than sinh Z and pi is lower than unity. However, the value between parentheses at the numerator can, in certain cases, be lower than zero. This can be the case only if the sign between the two terms is minus. Taking into consideration that the degree of polarization is positively valued, we have to take, generally, po ¼

jsinh Z 7 pi cosh Zj : cosh Z 7 pi sinh Z

ðB:2Þ

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T. Tudor / Optik 121 (2010) 2149–2158

In the particular cases si = 7n, Eq. (91) becomes so ¼ ¼

7 pi n þ n sinh 7 pi nðcosh 2 2 2 ½pi þ sinh þp2i sinh 72pi sinh

Z

Z

Z  1Þ Z cosh Z1=2

Z

ðsinh Z 7pi cosh ZÞn 2

2

½sinh Z þ p2i cosh Z 72pi sinh Z cosh Z1=2 ðsinh Z 7pi cosh ZÞn ðsinh Z 7pi cosh ZÞn : ¼ ¼ 2 1=2 jsinh Z 7pi cosh Zj ½ðsinh Z 7pi cosh ZÞ 

ðB:3Þ

If si =n, Eq. (B.3) becomes so ¼

ðsinh Z þpi cosh ZÞn ¼n jsinh Z þ pi cosh Zj

ðB:4Þ

because the sum at the nominator is positive. If si =  n, Eq. (B.3) becomes so ¼

ðsinh Z  pi cosh ZÞn ; jsinh Z  pi cosh Zj

which is equal sinh Z o pi cosh Z.

to

n

ðB:5Þ if

sinh Z 4pi cosh Z

and

to

–n

if

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