May 15, 2005 / Vol. 30, No. 10 / OPTICS LETTERS
1207
Wigner representation and geometric transformations of optical orbital angular momentum spatial modes Gabriel F. Calvo Grup de Física Teòrica and Institut de Fisica d’Altes Energies, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain Received December 3, 2004 An exact Wigner representation of optical spatial modes carrying orbital angular momentum is found in closed form by exploiting the underlying SU(2) Lie-group algebra of their associated Poincaré sphere. Orthogonality relations and observables of these states are obtained within the phase space picture. Development of geometric phases on mode transformations is also elucidated. © 2005 Optical Society of America OCIS codes: 030.4070, 080.2720, 260.0260, 350.1370, 350.5500.
One of the most appealing and profound aspects of wave optics and wave mechanics is their underlying structural similarity. This similarity becomes manifest between paraxial and Schrödinger equations and has allowed researchers to describe spatial beam propagation by applying an operator formalism equivalent to that for eigenstates of the quantum harmonic oscillator.1 Similarly, Padgett and Courtial proposed a Poincaré sphere to represent paraxial first-order-mode states carrying orbital angular momentum (OAM) analogous to that of polarization and atomic-spin coherent states.2 The inherent SU(2) symmetry of the sphere was subsequently shown.3 In this picture the poles of the sphere correspond to Laguerre–Gaussian modes4 LGpl with radial-node number p = 0, and winding index l = ± 1 (the plus and minus represent the north and south pole, respectively). Points on the sphere with azimuthal and axial angles and are associated with stable mode superpositions (up to a global phase):
, = cos共/2兲LG01 + exp共i兲sin共/2兲LG0−1 .
共1兲
For example, in the equatorial plane of the sphere, combinations with = / 2 and = 0 ( = / 2, = ) give rise to Hermite–Gaussian modes HGnx,ny with indices nx = 1, ny = 0 (nx = 0, ny = 1). Although the first-order orbital Poincaré sphere constitutes an elegant framework in which to represent families of states bearing OAM and their transformation, as points and paths connecting these points on the sphere, higher-order modes cannot be described by Eq. (1). That is, states on higher-order Poincaré spheres involve more complex superpositions of Laguerre–Gaussian modes. A generalization would provide, among many possibilities, direct means by which to visualize the development of geometric phases in optical beams5,6 and to exploit the multidimensional Hilbert structure of higher-ordermode spaces, which has been recognized as a potential resource for performing multibit quantum information processing and quantum computation.7 The aim of this Letter is to carry out the abovementioned generalization to all higher-order orbital Poincaré spheres. To this end, we use an SU(2) Lie0146-9592/05/101207-3/$15.00
group operator algebra8 to construct the spatial modes carrying OAM that correspond to coherent states on such spheres. Remarkably, these states are represented in a compact form by the Wigner function (WF), allowing us to uncover the hidden symmetries. The present approach starts by proposing sets of OAM coherent states 兩 , 典l,p as points 共 , 兲 on orbital Poincaré spheres Ol,p labeled l and p (or equivalently l and sphere order N = 2p + 兩l兩 艌 0). This continuum of states on Ol,p can be generated from Laguerre–Gaussian vector states 兩l , p典 (fixed on the poles of the sphere) through unitary operations: ˆ · u 兲兩l,p典 ⬅ U ˆ 共, 兲兩l,p典. 兩, 典l,p = exp共− iL
共2兲
ˆ Here L is an OAM operator and u = 共−sin , cos , 0兲 is a unit vector in the equatorial plane of the sphere. The action of the unitary (metaˆ on states 兩l , p典 can be interpreted plectic) operator U as a counterclockwise rotation of about u takes the axis containing the poles in the direction with unit vector ur = 共cos sin , sin sin , cos 兲. Under this ˆ generates multidimensional superposirotation, U tions of LGpl modes [including the particular case of Eq. (1)]. We assume, without lost of generality, that all Ol,p have l 艌 0. Spheres with l ⬍ 0 exhibit a state configuration identical to those with l ⬎ 0, after inversion with respect to their centers. We can obtain the 共 , 兲 distribution of states (2) on Ol,p by resorting to the WF representation. In the optical phase space, let q = 共x , y兲 and p = 共px , py兲 denote the transverse position and momentum variables, reˆ be the associated canonical Herspectively, and qˆ , p mitian operators 共pˆ → −i⑄q兲. The only nonvanishing commutation relations among these operators are 关xˆ , pˆx兴 = 关yˆ , pˆy兴 = i⑄. Here the reduced wavelength ⑄ = / 共2兲 is the optical analog of ប in quantum mechanics. For convenience, we arrange the phase space variables and the canonical operators in column vectors = 共x , y , px , py兲 and ˆ = 共xˆ , yˆ , pˆx , pˆy兲. In terms of ˆ , ˆ are the components of the OAM operator L © 2005 Optical Society of America
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OPTICS LETTERS / Vol. 30, No. 10 / May 15, 2005
ˆ = L x
ˆ = L y
ˆ = L z
xˆ2 − yˆ2 +
2w2 xˆyˆ w2
+
共pˆ2x − pˆ2y 兲w2 8⑄2
pˆxpˆyw2 4⑄
xˆpˆy − yˆpˆx 2⑄
共3a兲
,
共3b兲
,
共3c兲
.
Equations (3) satisfy the usual angular momentum ˆ ˆ 兴 = i⑀ L ˆ ,L commutators 关L a b abc c, where ⑀abc is the totally antisymmetric Levi–Civita tensor and w is the mode width at the beam waist. The Wigner representation of an optical field amplitude 共q兲 = 具q 兩 典 in the transverse plane is9,10 ⬁ 2 W共兲 = 兰−⬁ d exp共−ip · / ⑄兲共q + ”2兲*共q − / 2兲共2⑄兲2. We may now invoke two remarkable properties: (i) on account of the Stone–von Neumann theorem, unitary ˆ ) induce operators that are quadratic in ˆ (such as U ˆ : ˆ ⬘ → S ˆ ˆ , in the linear canonical transformations, S ˆ the WF optical phase space; (ii) under the action of S experiences a simple point transformation S : W共兲 → W⬘共兲 = W共S−1兲.11 In our case the linear canonical transformation generated by quadratic operators (3) ˆ −1ˆ U ˆ and reads as ˆ ˆ = U results from the relation S
S=
冤
c
0
− z 0s s
z 0s c
0
c
z 0s c
z 0s s
c
0
0
c
s s z0 −
s c z0
− −
s c z0 s s z0
冥
, 共4兲
where c = cos共 / 2兲, s = sin共 / 2兲, c = cos , s = sin , and z0 = w2 / 共2⑄兲 is the Rayleigh range. Notice that S has the form of a symplectic ray-transfer matrix of a generally anisotropic first-order system, and its action is independent of the chosen states = = 0. The key point is thus to observe that knowledge of the WF of one given state on Ol,p allows one to determine the WF of all states on that same sphere. Laguerre–Gaussian states constitute the convenient choice here. Using their Wigner representation,12 together with property (ii) and Eq. (4) yields the normalized WF: Wl,p共 ; , 兲 =
共− 1兲
N
2⑄ 2
exp共− Q0兲LN−l/2共Q0
− 4Q · ur兲LN+l/2共Q0 + 4Q · ur兲,
共5兲
where Q0 = 2关x2 + y2 + 共p2x + p2y 兲z02兴 / w2, Lm共兲 are the mth-order Laguerre polynomials, and the quadratic ˆ , polynomials Q共兲 ⬅ 共Qx , Qy , Qz兲 follow from Lˆx , L y ˆ ˆ and L in Eqs. (3) by replacing → . When = 0 共 z
= 兲 one recovers from Eq. (6) the WF of Laguerre– Gaussian modes LGpl 共LGp−l兲. If = / 2 and = 0 ( = / 2 and = ), one obtains the WF of Hermite– Gaussian states HGnx,N−nx共HGN−ny,ny兲. Equation (5) is a strictly positive and angleindependent Gaussian distribution only when l = p = 0; its associated Poincaré sphere becomes degenerate [all points 共 , 兲 on the sphere represent the same state]. Moreover, although Wl,p共 ; , 兲 does not explicitly contain propagation variable z, its spatial evolution along z can be fully described by applying a Galilean boost q → q − zp. The orthogonality relations (scalar product) satisfied by states 兩 , 典l,p and 兩⬘ , ⬘典l⬘,p⬘ are given by the overlap integral of their associated WFs:
兩l⬘,p⬘具⬘, ⬘兩, 典l,p兩2 = 共2⑄兲2
冕
⬁
−⬁
=
冤
d4 Wl⬘,p⬘共 ; ⬘, ⬘兲Wl,p共 ; , 兲
共N−l兲/2
兺
共− 1兲k
k=0
⫻ N␦l,l⬘␦p,p⬘ ,
冢 冣冢 冣冉 冊 冥 N+l
N−l
2
2
k
k
1−
2
k
共6兲
with = 关1 + cos cos ⬘ + sin sin ⬘cos共 − ⬘兲兴 / 2 (notice that 0 艋 艋 1). Equation (6) implies the following: (i) any two states belonging to different spheres are mutually orthogonal; (ii) within a given nondegenerate sphere, if p = 0, only states corresponding to antipodal points are mutually orthogonal. However, if p ⬎ 0, additional points exist on the sphere (apart from the antipodal) where their associated states are also orthogonal [e.g., if p = 1, Eq. (6) vanishes when = 共l + 1兲 / 共l + 2兲]. The marginal distributions of Eq. (5) give rise to the intensity transverse profiles in position Il,p共q ; , 兲 = 兩具q 兩 , 典l,p兩2 and momentum ˜Il,p共p ; , 兲 = 兩具p 兩 , 典 兩2. They are related by ˜I 共p ; , 兲 l,p
l,p
= z02Il,p共−z0p ; , 兲. Figure 1(a) depicts Il,p共q ; , 兲 for a third-order-mode Ol,p sphere 共N = l = 3兲 along representative directions of the unit vector ur. Let us now consider the normalized moments ruvw of the Wigner distribution: z0u+v
冕
⬁
d4 xryspux pvyW共兲, 共7兲 wr+s+u+v −⬁ where integers r , s , u , v 艌 0. Interestingly enough, the ˆ 典, of the z component of the expectation value, 具L z OAM carried by states 兩 , 典l,p may be expressed in terms of the second-order moments (in units of ប per ˆ 典 = 2ប关 photon) as 具L z 1001 − 0110兴, making the concept of angular momentum extensible to partially coherent beams.13–16 Employing Eqs. (5) and (7), the ten second-order moments (for which r + s + u + v = 2) possess the structure of rotationally symmetric beams: ursuv =
May 15, 2005 / Vol. 30, No. 10 / OPTICS LETTERS
1209
evaluate ⌽B exactly by use of the phase space formalism. First, the Berry connection may be written, with the help of Eqs. (2) and (3), as il,p具 , 兩ⵜ,兩 , 典l,p ˆ · u 兩l , p典 − u 具l , p兩关L ˆ · u + 共1 − 2 cos 兲Lˆ 兴兩l , p典/ = u 具l , p兩L
r
z
sin, where u = 共cos cos , sin cos , −sin 兲. Next, via the Weyl operator-ordering rule,10 the expectation values can be expressed as the phase space inteˆ · u 兩l , p典 = 兰⬁ d4共Q · u 兲W 共 ; 0 , 0兲=0 and grals 具l , p兩L l,p −⬁ ˆ · u + 共1 − 2 cos 兲Lˆ 兴兩l , p典 = 兰⬁ d4关共Q · u 兲 + 共1 − 2 具l , p兩关L r
z
−⬁
r
⫻cos 兲Qz兴Wl,p共 ; 0 , 0兲=l共1 − cos 兲 / 2. Making use of Stokes theorem, we can conveniently transform line integral (10) into a surface integral, and we finally obtain
Fig. 1. (a) Orbital Poincaré sphere of third-order modes. (b) Geometric phase development under mode transformations and its equivalent mode-converter setup. 1
rsuv = 16 兩r + s − u − v兩兩r − s + u − v兩共N + 1兲 1
1
+ 4 共rv − su兲l cos + 4 共rs + uv兲l sin sin 1
+ 8 共兩r − u兩 − 兩s − v兩兲l cos sin .
共8兲
ˆ 典 results in Combining Eq. (8) with 具L z ˆ 典 = lប cos , 具L z
共9兲
which has a simple geometric interpretation. It is the projection of unit vector ur corresponding to 兩 , 典l,p along the vertical axis of Ol,p. Hence arbitrary states on any sphere bear fractional OAM. The limiting cases, represented by LGpl and HGnx,ny beams, have the well-known lប and zero values, respectively.7 Closely connected to OAM is the geometric phase arising under mode transformations. It has been shown theoretically5 and experimentally6 that when transverse patterns of first-order LGpl and HGnxnymodes undergo a cycle of unitary transformations they acquire, in addition to a dynamic phase [for paraxial propagation this corresponds to the Gouy phase17 ⌽G = −2 arctan共z / z0兲], a geometric phase dictated by the sequence of transformations. On Ol,p these transformations are associated with rotations with respect to angles and : rotations can be performed by pairs of parallel cylindrical lenses (mode converters18), whereas rotations are exerted with coaxial Dove prisms [see Fig. 1(b)]. Let us extend the above-mentioned concept of geometric phase to all transformations that preserve mode order N on any sphere Ol,p. In terms of states (2), one can define the geometric phase ⌽B as ⌽B = i
冖
C
l,p具, 兩ⵜ,兩, 典l,p
· d艎.
共10兲
Equation (10) is the line integral of the so-called Berry connection,19 and it clearly depends on the holonomy of the closed path C followed on the Poincaré sphere [see the top inset in Fig. 1(b)]. It is possible to
共11兲 ⌽B = − 共l/2兲⍀, where ⍀ is the solid angle enclosed by path C. Thus the geometric phase acquired by states (2) during a cyclic evolution is equal to the solid angle ⍀ swept by vector nr along the closed circuit C on Ol,p. To conclude, based on Eqs. (6) and (11) we may state the following corollary: one can always find a basis containing at least 关2N + 3 + 共−1兲N兴 / 4 orthogonal states (2), each associated with Poincaré spheres of equal order N but different l, such that, under the same cycle of order-preserving transformations, allthese states acquire an identical Gouy phase ⌽G = −共N + 1兲arctan共z / z0兲 but differ in ⌽B = −l⍀ / 2. The author (
[email protected]) thanks A. Acín and J. I. Cirac for helpful discussions and the Spanish Ministry of Education and Science for a Juan de la Cierva grant. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
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