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Alexeyev et al.

Vol. 25, No. 3 / March 2008 / J. Opt. Soc. Am. A

643

Angular momentum flux of counterpropagating paraxial beams Constantine N. Alexeyev,1,* Maxim A. Yavorsky,1 and Vladlen G. Shvedov2 1

Taurida National V.I. Vernadsky University, Vernadsky Prospekt, 4, Simferopol, 95007, Crimea, Ukraine 2 Nonlinear Physics Center, Research School of Physical Science and Engineering, The Australian National University, Canberra, ACT, 0200, Australia, and Taurida National V.I. Vernadsky University, Vernadsky Prospekt, 4, Simferopol, 95007, Crimea, Ukraine *Corresponding author: [email protected] Received September 4, 2007; accepted January 7, 2008; posted January 15, 2008 (Doc. ID 87189); published February 11, 2008

We study the angular momentum (AM) of the arbitrary superposition of counterpropagating paraxial beams that have the same magnitude of the wavenumber. We derive compact analytical expressions for the total AM in a transverse cross section (linear AM density) and the total AM flux through the cross section. We demonstrate that whereas for the time-averaged linear AM density its separation into the spin and orbital parts is not, generally, observed, the total time-averaged AM flux is separated into well-identifiable spin and orbital constituents. Moreover, we show that such a flux is also naturally separated into the fluxes of forward- and backward-propagating beams. © 2008 Optical Society of America OCIS codes: 260.0260, 260.2110.

1. INTRODUCTION The interest in a long-standing problem of the angular momentum (AM) of electromagnetic fields [1–3] has been revived since the work of Allen et al. on the AM of Laguerre–Gaussian (LG) beams [4]. Along with the orbital AM (OAM), they have also studied the spin AM (SAM) of such beams—the constituent not present in scalar fields. One of the main results of this seminal work was demonstrating the separation of the total AM of paraxial LG beams into the spin and the orbital parts. In particular, it was established that the ratio of the cycleaveraged linear AM density 具Mzlin典 to the averaged linear energy density 具Wlin典 equals 共l + ␴兲 / ␻, where l is the orbital number of the beam, ␴ describes polarization (spin), and ␻ is frequency. Such a separation has also been established in a general case of paraxial propagating fields [5] (see also [3]). In the most concise mathematical form this circumstance is formulated as [6] 具Mzlin典

1 具␺兩共lˆz + ␴ˆ z兲兩␺典

Et =

冉 冊冕 ␣

␤

k

dtE共t兲exp共il␸兲exp共iz冑k2 − t2兲Jl共tr兲,

0

where Et is the transverse component of electric field (the expressions for longitudinal electric field and magnetic field can be found from Maxwell’s equations [8]), Jl is the Bessel function of order l, k is the wavenumber, E共t兲 is the mode function, and the complex constants ␣ , ␤ satisfy 兩␣兩2 + 兩␤兩2 = 1. As has been exhaustively demonstrated by Barnett [8], one should search for the reason for such a failure in the type of physical characteristics one applies to study the AM of the field. Instead of the AM stored by the field, which is given by the AM density, one should rather study the AM transmitted by the same field. The last quantity is given by the AM flux density and conventionally defined as Mik = ␧imnxm

冋

1 2

册

␦nk共␧0E2 + ␮0H2兲 − ␧0EnEk − ␮0HnHk ,

共1兲

共2兲

␺+ Ex − iEy ⬅ 1 / 冑2 , is the operator of the z ␺− Ex + iEy component of the OAM, and ␴ˆ z is the Pauli matrix that describes here the SAM. Cylindrical–polar coordinates 共r , ␸ , z兲 are implied. Here and throughout the paper the superscript lin stands for “linear density.” The scalar ␺+ , S beproduct is defined here as 具␸ 兩 ␺典 ⬅ 兰兰dxdy共␸+* , ␸−*兲 ␺− S ing the total transverse cross section of the beam. However, attempts to go beyond the paraxial approximation immediately revealed that no such lucid separation takes place for a quite common class of nonparaxial beams [7]:

where ␧imn is the Levi–Civitta symbol. As has been shown in [8], being applied to the study of the AM of nonparaxial beams, the notion of the AM flux now allows one to regain the “lost” separation of the AM flux into the spin and orbital parts. This remarkable result shows that, although for paraxial beams the AM density is proportional to the AM flux density, in a general case, it is more correct to study the AM transfer using appropriate analytical constructions. It might have seemed, however, that with respect to noncompatibility of the results obtained by use of the AM density concept and the AM flux density, one is restricted to nonparaxial situations, whereas within the paraxial propagation the two descriptions provide well-correlating

具Wlin典 where 兩␺典 =

共 兲

=

共

␻

具␺兩␺典

,

兲

共 兲

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© 2008 Optical Society of America

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J. Opt. Soc. Am. A / Vol. 25, No. 3 / March 2008

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results. In the present paper we give an example illustrating a discrepancy between these two approaches even within the framework of the paraxial propagation of light beams. We demonstrate that for a superposition of paraxially counterpropagating beams the description of their AM within the concept of the AM density and the AM flux density gives two completely different pictures. While for the AM density no separation of the total AM into the SAM and OAM is generally observed, the AM flux density of counterpropagating paraxial beams is well separated into the spin and orbital parts. Moreover, the total AM flux of such beams is also separated into the fluxes of forward- and backward-propagating beams. We also derive an analytical expression analogous to Eq. (1) for the ratio of the AM flux to radiation power that explicitly conveys the idea of the above-mentioned separation.

Ht ⬇

k

␻␮0

where nz is the unit vector in the z direction. As is obvious, this connection also differs from the standard one that takes place at B = 0, i.e., Ht ⬇ 共 k / ␻␮0 兲nz ⫻ Et. The expression for Hz has the standard form

共3兲

where k is the wavenumber, A and B are some complex functions, and the z axis is assumed to be the direction of propagation. Paraxiality of the beams implies that A and B are the slowly varying functions of coordinates, which enables one to disregard their derivatives in comparison with the derivatives of the exponentials. In what follows we will use one of the main ideas of [5,6], which is expressing all the components of the fields E, H in terms of the transverse electric field Et. This can be done using the Maxwell equations for the monochromatic field: rot E = − ikH,

共4兲

div E = 0.

共5兲

It should be noted, that at B = 0 we have the standard connection between the longitudinal electric field Ez and ជ · E , where ⵜ ជ = 共 ⳵ / ⳵x , ⳵ / ⳵y 兲. However, Et, i.e., Ez ⬇ 共 i / k 兲ⵜ t t t if E is given by Eq. (3) its derivative with respect to z is no longer proportional to Ez, which violates one of the main assumptions made in deriving Eq. (1). That is why Eq. (1) may be inapplicable for the description of counterpropagating beams. To establish the correct expression for Ez one has to integrate Eq. (5) over z, which readily gives i

ជ · A eikz − ⵜ ជ · B e−ikz , Ez ⬇ ⵜ t t t t k k

i

Hz ⬇ −

␻␮0

ជ ⫻E . ⵜ t t

共8兲

Now we are in a position to derive the expression for the AM of the superposition of counterpropagating paraxial beams.

Consider a superposition of two paraxial beams with the same magnitude of wavenumber that counterpropagate in the void. The electric field E of such a superposition can be represented in the form

i

共7兲

A. Angular Momentum Density One of the relevant quantities that characterize the AM of electromagnetic field is the AM density, which is defined as [9]

2. ANGULAR MOMENTUM OF COUNTERPROPAGATING BEAMS

E共x,y,z兲 = A共x,y,z兲eikz + B共x,y,z兲e−ikz ,

共nz ⫻ Ateikz − nz ⫻ Bte−ikz兲,

1 M=

where the subscript t means the transverse component of the corresponding vector. Here, as usual, we disregard the derivatives of A, B with respect to z. In the same approximation, one obtains from Eq. (4) the following expression for the transverse magnetic field:

r ⫻ 关E ⫻ H兴,

共9兲

c being the speed of light. For monochromatic fields the time-averaged AM density 具M典 can be calculated using a well-known rule: 具M典 =

1

Re r ⫻ 关E ⫻ H*兴.

2c2

共10兲

For propagating beams the AM can be conventionally characterized by a standard quantity: 具M2lin典 =

冕冕

具Mz典dxdy.

共11兲

S

This quantity can be readily shown as 具Mzlin典 =

1 2c2

冕冕

Re

rt ⫻ 关Et ⫻ Hz* + Ez ⫻ Ht*兴dxdy.

S

共12兲 Here the subscript z stands for the longitudinal component of a vector. Using Eqs. (3) and (6)–(8) one can express all the fields in Eq. (12) in terms of At, Bt and their derivatives. Upon such a substitution Eq. (12) assumes the form 具Mzlin典 =

共6兲

c2

i 4c ␻␮0 2

ជ · − 共ⵜ t

冕冕

ជ ⫻ E +*兴 rt ⫻ 兵Et+ ⫻ 关ⵜ t t

S

+ Et−兲Et *

+ c.c.其dxdy,

共13兲

where Et± ⬅ At exp共ikz兲 ± Bt exp共−ikz兲. This expression can be brought into a compact form using the definition of the scalar product implied in Eq. (1). After a little algebra one can obtain the following form for the linear AM density:

Alexeyev et al.

Vol. 25, No. 3 / March 2008 / J. Opt. Soc. Am. A

1

具Mzlin典 =

4c ␻␮0 2

冋

1 具A兩Jˆz兩A典 + 具B兩Jˆz兩B典 2 2 1

册

ˆ 兩A典 + c.c. , + 2e2ikz具B兩M where Jˆz = lˆz + ␴ˆ z and

ˆ = M

冤冉

冉

e−2i␸ lˆz + r

0

e2i␸ lˆz − r

⳵ ⳵r

冊

共14兲

⳵ ⳵r

0

冊

冥

1 2

冕冕

具W 典 lin

=

共具A兩A典 + 具B兩B典兲.

共16兲

具A兩A典 + 具B兩B典

. 共17兲

At 兩B典 = 0 this result goes over into the known Eq. (1). B. Angular Momentum Flux The other characteristic of the electromagnetic field’s AM is the density of the AM flux, which is given by Eq. (2). For symmetry reasons the only component of Mik that gives a nonzero overall contribution to the flux through a transverse cross section is Mzz. The cycle average of the flux of Mzz through the total cross section is given by tot 具Mzz 典=

1 2

冕冕

Re

关y共␧0ExEz* + ␮0HxHz*兲 − x共␧0EyEz*

S

+ ␮0HyHz*兲兴dxdy.

共18兲

Once again, using Eqs. (3) and (6)–(8) and integrating by parts certain members that have the structure ˆ are some linear differential op兰兰EiLˆikEk* dxdy, where L ik S

erators and i , k are either x or y, one can bring Eq. (18) into the form tot 具Mzz 典=

␧0 2k

共20兲

共21兲

Finally, for the desired ratio one has tot 典 具Mzz

具N典

=

1 具A兩Jˆz兩A典 − 具B兩Jˆz兩B典

␻

具A兩A典 − 具B兩B典

.

共22兲

Together with Eq. (17) this relation represents the main result of the present paper.

3. DISCUSSION

ˆ 兩A典 + c.c.兲 1 具A兩Jˆz兩A典 + 具B兩Jˆz兩B典 + 共e2ikz具B兩M

␻

E ⫻ H*dxdy,

共15兲

For the ratio “AM density/energy density” one obtains the final result 具Mzlin典

Re

1 具N典 = ␧0c共具A兩A典 − 具B兩B典兲. 2

In the same manner this leads to

2

2

which readily yields

.

共␧0E2 + ␮0H2兲dxdy.

␧0

冕冕

1

S

S

具Wlin典 =

To obtain a physically “recognizable” result one has to divide this quantity by the total power transmitted through the cross section, which is determined by the z component of the Poynting vector P. For the cycleaveraged flux of P through a transverse cross section (radiation power N) one has 具N典 =

To obtain Eq. (14) we have integrated some terms in Eq. (13) by parts, which is a usual technique in dealing with the total AM [5–7]. Physically lucid results are obtained if one divides the linear AM density by the linear energy density, which is defined as Wlin =

645

共具A兩Jˆz兩A典 − 具B兩Jˆz兩B典兲.

共19兲

As is well established, the AM density and the AM flux density form a set of complementary mathematical constructions used for the description of the AM that are mutually connected through the continuity equation [9]:

⳵ Mi ⳵t

+

⳵ Mik ⳵ xk

= 0.

共23兲

Each of them provides unique information about the AM of the field, and it is generally impossible to get a complete description of the AM using only one of these two physical quantities. They describe different aspects of the field AM: Whereas the AM density describes the AM stored by the field, the AM flux density is connected with the ability of the field to transmit the AM. It is, therefore, not surprising that in applying these quite different quantities to the description of the AM one obtains completely different pictures. In particular, in our case this is expressed in two major distinctions in the structure of the formulas (17) and (22) for the linear AM density and the total AM flux through the cross section, respectively. The first distinction is connected with separation of the AM into spin and orbital constituents. As is evident from Eqs. (14) and (17), if one uses the concept of the AM density for the description of the AM of counterpropagating beams, one does not obtain a complete separation of the linear AM density into distinct contributions that can be attributed either to spin or to orbital parts of the AM. Esˆ describes the sentially, the term comprising the matrix M spin–orbit coupling if one adopts the point of view of Allen et al. [10]. However, for some special cases that separation does take place. Such is the case, for example, of counterpropagating beams of the same circular polarization (defined in the same frame of reference). Indeed, the operator ˆ couples the vectors with the opposite polarization, so it M gives zero contribution for any beams 兩A典, 兩B典 of the same circular polarization. In particular, this takes place if the

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beam 兩B典 is the beam 兩A典 that is normally reflected from a flat mirror. In the case of reflection the separation is also present for some more complicated polarizations of 兩A典, for example, when 兩A典 is the TE or TM mode. In contrast, the flux of the AM through the same cross section is always well separated into the spin and orbital parts, as is clearly seen from Eqs. (19) and (22). In this context the average 具A 兩 lˆz 兩 A典 should be treated as the flux of the OAM, whereas the term 具A 兩 ␴ˆ z 兩 A典 is the flux of the SAM. The second distinction concerns an additional separation into the fluxes of backward- and forward-propagating waves. Of course, it would have seemed intuitively illogical to obtain separation of the AM density into the contributions of forward- and backward-propagating beams. As a matter of fact, in some above specified cases this separation takes place, but even in those cases they are additive and enter Eq. (14) symmetrically. On the contrary, one intuitively expects of the flux the separation into the flux of forward-propagating and backward-propagating waves. Moreover, these fluxes should be of opposite sign. Exactly these features are present in Eqs. (19) and (22). This result has a standard and clear physical explanation. An analogous separation into the contributions of forward- and backward-propagating waves also takes place for the Poynting vector’s flux through a cross section [see Eq. (21)]. One can also find analogous examples in quantum mechanics [11]: It is well-known that if the wave function ␺ has the form A exp共ipx兲 + B exp共−ipx兲, the probability density current is proportional to 兩A兩2 − 兩B兩2 (or 具A 兩 A典 − 具B 兩 B典, if one uses the standard definition of the scalar product). In contrast, the probability density 兩␺兩2 would be given by 兩A兩2 + 兩B兩2 + 关exp共2ipx兲AB* + c.c.兴, which can also be written in an equivalent form as 具A 兩 A典 + 具B 兩 B典 + 关exp共2ipx兲具B 兩 A典 + c.c.兴, which evidently resembles Eq. (14). As a rule, to obtain a consistent description of the superposition of two counterpropagating fields one has to apply the concept of the appropriate flux density rather than the concept of corresponding density of the physical quantity. The uniqueness of the situation with AM is that in applying the right concept, one simultaneously obtains not only its separation into contributions of opposite fluxes, but the separation of those fluxes into the spin and orbital parts as well. In conclusion, consider the question of why in the paraxial limit the total AM flux turns out not to be proportional to the linear AM density. It might have seemed that this should have been the case, since in [8] the proportionality of the AM flux density and AM density for any paraxial beam was shown. However, that proof was based on the fact that for paraxial beams the following approximate relations hold true: E x ⬇ c ␮ 0H y,

E y ⬇ − c ␮ 0H x .

sity and the AM flux density. This, in its turn, gives rise to a question of whether the superposition of two paraxial counterpropagating waves forms a paraxial beam. The standard criterion of paraxiality operates with the angles that the beam’s rays make with the direction of propagation. In our situation it seems reasonable to study the Poynting vector’s direction to decide whether the field Eq. (3) is indeed a paraxial one. As can be readily shown, the cycle-averaged longitudinal component of P is 具Pz典 ⬇

k 2␻␮0

共兩At兩2 − 兩Bt兩2兲.

共25兲

The transverse component of P has a much more complicated form, but the order of its modulus can be assessed as 具Pt典 ⬀ 1 / 2␻␮0l 兩 At兩2, where l is the characteristic scale of transverse variation of A and B. As is evident, 具Pt典 / 具Pz典 ⬀ ␭ / l Ⰶ 1, ␭ being the wavelength; that is, the energy flows almost along the z axis. However, the angle of P with the z axis will be close to either 0 or ␲, depending on the sign of the projection Pz. Of course, such a field cannot be considered a paraxial one. The deviation from paraxiality is even more drastic at At = Bt (the case of a standing wave), in which case the energy flows in the transverse direction. Nevertheless, the results, Eqs. (17) and (22), are valid even in this limiting case, where the propagation of energy in the beam of Eq. (3) is paraxial in no cross section. Indeed, while deriving Eqs. (17) and (22) we have as a matter of fact used a weaker restriction on the field (3): paraxiality of the partial beams A exp共ikz兲 and B exp共−ikz兲, which is equivalent to the slowness of the functions A and B as compared with the exponentials exp共±ikz兲.

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5. 6. 7.

共24兲

8.

In our case, however, as follows from Eq. (7), neither of these relations is true: The connection between Ht and Et is much more complicated. This is an immediate mathematical reason for the difference between the AM den-

9. 10. 11.

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