## Classical and quantum properties of cylindrically polarized states of light |

Optics Express, Vol. 19, Issue 10, pp. 9714-9736 (2011)

http://dx.doi.org/10.1364/OE.19.009714

Acrobat PDF (1093 KB)

### Abstract

We investigate theoretical properties of beams of light with non-uniform polarization patterns. Specifically, we determine all possible configurations of cylindrically polarized modes (CPMs) of the electromagnetic field, calculate their total angular momentum and highlight the subtleties of their structure. Furthermore, a hybrid spatio-polarization description for such modes is introduced and developed. In particular, two independent Poincaré spheres have been introduced to represent simultaneously the polarization and spatial degree of freedom of CPMs. Possible mode-to-mode transformations accomplishable with the help of Bconventional polarization and spatial phase retarders are shown within this representation. Moreover, the importance of these CPMs in the quantum optics domain due to their classical features is highlighted.

© 2011 OSA

## 1. Introduction

*cylindrical*polarization of the investigated beams. We introduce all possible configurations of these modes and calculate the Schmidt rank, as this is a convincing way to determine the potential inseparability of these modes. We complete the theoretical investigations by deducing the total angular momentum of these CPMs in Sec. 3. In Sec. 4, we introduce our Poincaré sphere representation for the hybrid polarization/spatial degrees of freedom characterizing CPMs. In Sec. 5, we discuss the manipulations of these states of light on these hybrid Poincaré spheres with the help of conventional phase retarders. Furthermore, in Sec. 6, we extend our formalism to the quantum domain and illustrate the importance of these modes in quantum optics experiments. Finally, in the Appendix, we furnish the reader with the mathematical tools which are essential to fully cover the topics of the main text.

## 2. Classical description of light beams with cylindrical polarization

### 2.1. Introduction to rotation principles

*λ*= 2

*π*/

*k*propagating along the

*z*-axis. The corresponding electric field can be written as where

*χ*=

*k*(

*ct*–

*z*) –

*π*/2 [14] and

**r**=

*x*

**x̂**+

*y*

**ŷ**+

*z*

**ẑ**. Furthermore,

**u**(

**r**) is a paraxial mode function of the form

**x̂**and

**ŷ**indicate the polarization in the

*x*and

*y*directions and

*f*

_{1}(

**r**) and

*f*

_{2}(

**r**) are two independent solutions of the paraxial wave equation.:

*x*=

*r*cos

*θ*and

*y*=

*r*sin

*θ*, we can rewrite

*x*=

*x*(

*r*,

*θ*) and

*y*=

*y*(

*r*,

*θ*), having assumed

*r*,

*θ*as independent variables. Then, by definition, and These two equations can be rewritten in compact vector notation as: Now, since the argument of the functions

**u**

^{±}(

*r*,

*θ*) =

**u**

^{±}(

**x**) on the left side of Eq. (14) is exactly

**x**(

*r*,

*θ*±

*φ*), we can use the result above to rewrite Eq. (14) as: or, equivalently, At this point, one should remember that any 2-dimensional vector field

**V**(

*x*,

*y*) transforms under a

*global*counterclockwise active rotation by an angle

*φ*into the new field

**W**(

*x*,

*y*) whose functional expression is determined according to the following rule: Thus Eq. (20a) coincides

*exactly*with the definition of a vector field which is invariant with respect to a

*global*rotation by an angle

*φ*. This means that co-rotating beams have a polarization pattern

*invariant*with respect to global rotations. This result is intuitively obvious for radial and azimuthally polarized beams. Conversely, the corresponding Eq. (20b) for the counter-rotating beams has a less straightforward geometric interpretation. For this case, one can show that Eq. (20b) can be recast in the form of a global rotation as: This equation simply shows that a global rotation by an angle

*φ*upon the counter-rotating modes is equivalent to a

*local*rotation by an angle 2

*φ*on the same mode.

### 2.2. Deduction of cylindrically polarized modes

*x*=

*r*cos

*θ*and

*y*=

*r*sin

*θ*, Eq. (24a) and Eq. (24b) can be rewritten as: where

*α*(

*r*) and

*β*(

*r*) can be determined by keeping in mind that

*f*

_{1}and

*f*

_{2}need to satisfy the paraxial Eq. (3). In particular, we want to express the functions

*f*

_{1}(

**r**) and

*f*

_{2}(

**r**) in terms of the first-order Hermite-Gauss [15

15. M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. **24**(7), 430–432 (1999). [CrossRef]

*z*= 0, as where

*w*

_{0}is the beam waist. Thus, from Eq. (25a) and Eq. (25b) and the fact that

*x*∝

*ψ*

_{10}and

*y*∝

*ψ*

_{01}, it follows that: where

*A*,

*B*are arbitrary numerical constants. In general, with

*ψ*=

_{nm}*ψ*(

_{nm}*x*,

*y*,

*z*) (

*n*,

*m*∈ ℕ

_{0}) we denote the Hermite-Gauss solutions of the paraxial wave equation of the order

*N*=

*n*+

*m*[15

15. M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. **24**(7), 430–432 (1999). [CrossRef]

*A*= 1,

*B*= 0} or {

*A*= 0,

*B*= 1} and inserting this in Eq. (28), one obtains after normalization: The same steps that lead us to Eq. (29a) and Eq. (29b) could be repeated for the counter-rotating modes. However, it is more instructive to note that, from a visual inspection, one can immediately deduce that

*ψ*

_{10},

*ψ*

_{01}} mode bases and the polarization vectors {

**x̂**,

**ŷ**}: However, the two vectors {

**u**(

**x**) and

**v**(

**x**) are two arbitrary vector functions. It should be stressed that the basis vectors

6. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. **91**(23), 233901 (2003). [CrossRef] [PubMed]

12. C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. **106**(6), 060502 (2011). [CrossRef] [PubMed]

### 2.3. Schmidt rank of radially and azimuthally polarized modes

*non-separable*, in the sense that it is not possible to write any of these vector functions as the product of a uniformly polarized vector field times a scalar function. Such a “degree of non-separability” may be quantified by the Schmidt-rank of the function, remembering that all separable functions have a Schmidt rank of 1. Conversely, we are now going to prove that

**ê**

_{1}and

**ê**

_{2}be two orthogonal unit vectors defined as

**ê**

_{1}=

**x̂**and

**ê**

_{2}=

**ŷ**, with and let

*v*

_{1}and

*v*

_{2}be two orthogonal functions defined as where, because of the orthogonality of the Hermite-Gauss modes, one has: Then we can immediately write down, for example for the

*λ*

_{1}=

*λ*

_{2}= 1/2. The right side of Eq. (35) represents the Schmidt decomposition of the vector function

*v*

_{1}and

*v*

_{2}defined as This completes the proof.

*A*|

^{2}+ |

*B*|

^{2}= 1 to guarantee normalization: (

**û**

*,*

_{AB}**û**

*) = 1. From Eq. (36) it follows that*

_{AB}*AB**) = 0. Conversely,

**f**

_{2}=

*C*

**f**

_{1}, namely when −

*B*=

*CA*,

*A*=

*CB*, where

*C*is a proportionality constant. The substitution of the latter equation in the previous one furnishes −

*B*=

*C*

^{2}

*B*which implies

*C*= ±

*i*, namely

*A*= ±

*iB*. In this case it is easy to see that Eq. (36) reduces to which means that the only separable cylindrically symmetric co-rotating modes are the circularly polarized Laguerre-Gauss modes with one unit (|ℓ| = 1) of orbital angular momentum:

## 3. Angular momentum of cylindrically polarized optical beams

17. M. V. Berry, “Optical currents,” J. Opt. A, Pure Appl. Opt. **11**(9), 094001 (2009). [CrossRef]

*linear momentum*carried by the paraxial mode

**u**of the electromagnetic field as: where ∇

_{⊥}=

**x̂**∂

*+*

_{x}**ŷ**∂

*and*

_{y}*c*

^{2}

*͛*

_{0}/

*ω*has been omitted. The symbol “×” denotes the ordinary cross product in ℝ

^{3}. In Eq. (41), the first term

**p**

_{orb}(

**r**) is called the orbital Poynting current and is independent of the polarization state of the beam. Conversely the second term, the so-called spin Poynting current

**p**

_{sp}(

**r**), contains the local “density of spin” of the beam via the term Im (

**u*** ×

**u**) [17

17. M. V. Berry, “Optical currents,” J. Opt. A, Pure Appl. Opt. **11**(9), 094001 (2009). [CrossRef]

*k*indicates the wave vector

*k*=

*ω*/

*c*. The power of a light beam as actually measured by a photo-detector in the laboratory equals the flow of the Poynting vector of the beam across the detector surface. Since the electromagnetic linear momentum density

**p**is equal to 1/

*c*

^{2}times the Poynting vector, it follows that a quantity experimentally observable is

**P**·

**ẑ**, where is the linear momentum of the field per

*unit length*. Without loss of generality, we have assumed that the detector surface coincides with the

*xy*-plane normal to the beam propagation axis

**ẑ**. The double integration in Eq. (43) is extended to the whole

*xy*-plane. This means that for any pair of functions

*f*

_{1}and

*f*

_{2}, normalizable in the

*xy*-plane, which drop to zero at infinity, the “spin” surface terms

**P**

_{sp}= 0. It is worth noting that although

*f*

_{1}and

*f*

_{2}are both functions of

**r**, as opposed to

**x**we can practically evaluate Eq. (43) at

*z*= 0 because the quantity

**P**is independent of

*z*[18

18. A. Aiello, N. Lindlein, Ch. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin hall effect of light,” Phys. Rev. Lett. **103**(10), 100401 (2009). [CrossRef] [PubMed]

*total angular momentum density*is where the

*orbital angular momentum density*is and the spin angular momentum density is We proceed as in the linear momentum case and write the angular momentum of the field per

*unit length*as were, similarly to

**P**, also

**L**and

**S**are independent of

*z*[18

18. A. Aiello, N. Lindlein, Ch. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin hall effect of light,” Phys. Rev. Lett. **103**(10), 100401 (2009). [CrossRef] [PubMed]

*f*

_{1}(

**r**) and

*f*

_{2}(

**r**) that vanish sufficiently fast for |

*xy*| → ∞, integration by parts leads to

*S*= 0 =

_{x}*S*, and to From this relation it follows straightforwardly that Equation (49) clearly reproduces the well-know expression of the helicity

_{y}*S*=

_{z}*i*(

*αβ** −

*α**

*β*)/

*k*of a uniformly-polarized beam when one chooses

*f*

_{1}=

*αψ*(

**r**) and

*f*

_{2}=

*βψ*(

**r**), where

*ψ*(

**r**) is a normalizable function in the

*xy*-plane and

*α*,

*β*a pair of complex number such that |

*α*|

^{2}+|

*β*|

^{2}= 1.

**J**for the set of our four cylindrically polarized modes without performing any actual calculations. In fact, since

**J**is independent of

*z*, we can evaluate it at

*z*= 0 where both

*ψ*

_{10}(

*x*,

*y*,

*z*= 0) and

*ψ*

_{01}(

*x*,

*y*,

*z*= 0) are

*real-valued*functions. Then from Eqs. (45–46) it follows at once that

**S**=

**0**and that However, it is easy to see from Eq. (26a) and Eq. (26b) that by definition |

*ψ*

_{10}(

*x*,

*y*,0)|

^{2}and |

*ψ*

_{01}(

*x*,

*y*,0)|

^{2}are even functions with respect to both

*x*and

*y*coordinates. This automatically implies, because of symmetry, that

*L*= 0 =

_{x}*L*.

_{y}## 4. Hybrid Poincaré sphere representation

*spin*angular momentum eigenstates, respectively. Similarly, the spatial-mode Poincaré sphere [SMPS, Fig. 4(a)] [15

15. M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. **24**(7), 430–432 (1999). [CrossRef]

*orbital*angular momentum eigenstates [21

21. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A **45**(11), 8185–8189 (1992). [CrossRef] [PubMed]

*S⃗*={

*S*

_{1}/

*S*

_{0},

*S*

_{2}/

*S*

_{0},

*S*

_{3}/

*S*

_{0}}, where

*S*,

_{i}*i*∈ {1,2,3} are known as the Stokes parameters [20] and

*S*

_{0}denotes the total intensity of the beam.

*hybrid*Poincaré spheres (HPSs) that embody all characteristics of these complex modes of the electromagnetic field. Up to now, only

*uniform*polarization patterns could have been displayed as points on the PPS. However, with this new representation,

*nonuniform*polarization patterns are represented for the first time. This is especially helpful – let it be in theoretical considerations or in experiments – if one investigates effects deriving from these spatially dependent, complex polarization patterns. Furthermore, manipulations of states within this representation in accordance with the conventional tools such as the PPS and SMPS can be carried out. This is investigated in Sec. 5.

**x̂**,

**ŷ**} and {

*ψ*

_{10},

*ψ*

_{01}}, may be combined in several different manners to produce a four-dimensional mixed polarization-spatial mode space. A

*complete*representation of such a space would require the simultaneous display of eight real numbers which is impossible in a three dimensional space. Consequently, our aim is to find an appropriate way for describing this space by using conventional tools, such as the PPS or the SMPS respectively. We begin by noticing that if we are permitted to discard some information, then a lower-dimension visual representation may be possible. The characteristics of this picture would then be fixed by the kind of degree of freedom (DOF) we choose to represent. Driven by certain recent experimental results [12

12. C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. **106**(6), 060502 (2011). [CrossRef] [PubMed]

*hybrid*representation, which would be suitable to describe physical situations where co- and counter-rotating modes do not mix with each other. This absence of mixing occurs, for example, when the light propagates through cylindrically symmetric optical systems as fibres, non-astigmatic lenses, et cetera. In this case, the disregarded information would amount to the relative phase and amplitude between the sets of co- and counterrotating modes. The representation deriving from these constraints is illustrated in Fig. 4. The North and South Pole of sphere (c) represent radially and azimuthally polarized states whose total angular momentum (orbital plus spin) is equal to zero. Analogously, sphere (d) represent counter-radial and counter-azimuthal states, still with total zero angular momentum. Each of these two pairs of states represent a novel kind of

*hybrid*DOF of the electromagnetic field, since such states describe neither purely polarized nor purely spatial modes of the field.

*ψ*

_{10},

*ψ*

_{01}} and polarization {

**x̂**,

**ŷ**} bases: To span this four dimensional space, we choose the basis {

*superselection rule*[16] which forbids interference between states separated according to the law in Eq. (14). Therefore, we use this law to split the four dimensional space {

*ψ*

_{10}

**x̂**,

*ψ*

_{10}

**ŷ**,

*ψ*

_{01}

**x̂**,

*ψ*

_{01}

**ŷ**} into the Cartesian sum of two subspaces spanned by the “+” and “−” sets of modes: This shows, that depending upon their behavior under rotation, cylindrically polarized optical beams may be subdivided in two independent sets which can be represented by points on the surface of two distinct hybrid Poincaré spheres, as illustrated in Fig. 4. It is worth to stress once again that a beam represented by an arbitrary superposition of either the standard basis vectors {

*ψ*

_{10}

**x̂**,

*ψ*

_{10}

**ŷ**,

*ψ*

_{01}

**x̂**,

*ψ*

_{01}

**ŷ**} or the cylindrical basis vectors {

*hybrid*Stokes parameters, as opposed to the either polarization or spatial Stokes parameters. These hybrid Stokes parameters convey information about both polarization and spatial-mode degrees of freedom simultaneously. They are naturally defined as: where the symbol * denotes the complex conjugation.

**}**, where These amplitudes can be expressed in terms of the spherical coordinates on the two independent hybrid Poincaré spheres (HPSs) as The first sphere of the HPSs represents the “+” modes [Fig. 5(a)] and the second the “−” [Fig. 5(b)] modes. It is possible to show that these Stokes parameters can be actually measured by means of conventional optical elements. To perform such a measurement on either of the HPSs one needs to project an arbitrary cylindrically polarized state of light onto the Stokes basis modes of interest. This can be achieved by a combination of spatial optical elements, such as asymmetric Mach Zehnder interferometers [22

22. H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A **68**(1), 012323 (2003). [CrossRef]

23. M. T. L. Hsu, W. P. Bowen, and P. K. Lam, “Spatial-state stokes-operator squeezing and entanglement for optical beams,” Phys. Rev. A **79**(4), 043825 (2009). [CrossRef]

24. N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and Ch. Silberhorn, “Polarization squeezing and continuous-variable polarization entanglement,” Phys. Rev. A **65**(5), 052306 (2002). [CrossRef]

*θ*

^{±}∈ [0,

*π*],

*ϕ*

^{±}= 0) describes a locally linear polarization state as illustrated in both Fig. 5(a) and Fig. 5(b). Except for these points and the two ones on the

*S*

_{3}axis (

## 5. Manipulating states on the HPSs

21. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A **45**(11), 8185–8189 (1992). [CrossRef] [PubMed]

**24**(7), 430–432 (1999). [CrossRef]

*ψ*

_{10},

*ψ*

_{01}} exactly in the same way as a HWP acts on the pair {

**x̂**,

**ŷ**} in the polarization space [21

21. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A **45**(11), 8185–8189 (1992). [CrossRef] [PubMed]

*θ*∈ [0,

*π*] and

*ϕ*refers to the angles parametrizing a sphere, as shown in Fig. 8. These constraints are due to the fact that phase retarders imprint a spatially uniform phase shift onto the cross section of the beam, which is not, per definition, cylindrically invariant. These constraints are explicitly deduced in the Appendix.

## 6. Quantum properties of cylindrically polarized states of light

*classical*properties of cylindrically polarized states of light so far, we cover their quantum aspects in the following. In this section, we give a few examples of application of our hybrid polarization/spatial formalism to some quantum states of light.

*quantum operator*for paraxial beams of light in the same units of Eq. (1) as: where

**Ê**

^{−}= (

**Ê**

^{+})

^{†}, and with

**ê**

_{1}=

**x̂**and

**ê**

_{2}=

**ŷ**and

*n*,

*m*∈ {0,1,2, …}. The paraxial annihilation operators

*â*satisfy the canonical commutation relations

_{λnm}**u**

*(*

_{AB}**r**) be the generic cylindrical mode (the superscript “+” will be omitted from now on) that now we write as where Eq. (29a) and Eq. (29b) were used in the second line and again |

*A*|

^{2}+|

*B*|

^{2}= 1 is assumed. Since (

**u**

*,*

_{R}**u**

*) = 0, then two different modes*

_{A}**u**

*(*

_{AB}**r**) and

**u**

_{A}_{′}

_{B}_{′}(

**r**) satisfy the relation Now the question is: What is the correct form of the field operator

*â*that annihilates a photon in the mode

_{AB}**u**

*(*

_{AB}**r**)? We seek an answer to this question by requiring that: With the help of Eq. (60) we guess the following form for the sought annihilation operator

*â*: where we have used the polarization indexes

_{AB}*x*for

*λ*= 1 and

*y*for

*λ*= 2. From Eqs. (59,64) it follows that: which, together with Eq. (61), shows that

*â*

_{A}_{′}

_{B}_{′}and

**u**

*(*

_{AB}**r**) and

**u**

_{A}_{′}

_{B}_{′}(

**r**) are orthogonal.

### 6.1. Coherent states

*D̂*(

_{i}*β*),

*i*∈ {1,2, … ,∞} be the

*coherent-state displacement operator*defined as where the single label

*i*embodies

*λ*,

*n*,

*m*, so that Σ

*= Σ*

_{i}*Σ*

_{λ}*Σ*

_{n}*, and we choose the first 4 values of the index*

_{m}*i*in such a way that If we rewrite the field mode functions as then Eq. (58) takes the simpler form Since any coherent state |

*β*〉

*=*

_{i}*D̂*(

_{i}*β*)|0〉 satisfies the relation it immediately follows that if we define |

*α*

_{1},

*α*

_{2},

*α*

_{3},

*α*

_{4}〉 ≡

*D̂*

_{4}(

*α*

_{4})

*D̂*

_{3}(α

_{3})

*D̂*

_{2}(

*α*

_{2})

*D̂*

_{1}(

*α*

_{1})|0〉, then which trivially implies that

### 6.2. Fock states

*Â*(

*ξ*) as: that satisfies the canonical commutation relations Let |1〉 be the single-photon state defined as |1〉 =

*Â*

^{†}(

*ξ*)|0〉. Then Now, if we choose then we obtain in agreement with Eq. (63). Thus, we have proved that Eq. (64) furnishes the correct expression for

*â*.

_{AB}### 6.3. Squeezed states

*classical*inseparability automatically lead to

*quantum*entanglement when the azimuthally polarized beam is prepared in a nonclassical state [12

12. C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. **106**(6), 060502 (2011). [CrossRef] [PubMed]

*squeezing operator*for the azimuthally polarized field is where

*ζ*=

*se*is the complex squeezing parameter. By substituting Eq. (64) evaluated for

^{iϑ}*A*= 0,

*B*= 1 into Eq. (79), it is not difficult to obtain where we have defined the one- and two-mode squeezing operator Thus, the squeezed-vacuum state is clearly entangled, since represents the well-known entangled two-mode squeezed vacuum [26

26. W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. **90**(4), 043601 (2003). [CrossRef] [PubMed]

## 7. Conclusion

**106**(6), 060502 (2011). [CrossRef] [PubMed]

**106**(6), 060502 (2011). [CrossRef] [PubMed]

## 8. Appendix

*classical*modes of the electro-magnetic field and it has nothing to do with

*quantum-mechanical interpretation*. However, for the sake of completeness, our hybrid classical description has been extended to the quantum domain in Sec. 6. Thus, for example, our set of cylindrically polarized modes may be represented as: In passing, we note that these states have the formal structure of the quantum Bell-states. More specifically, if one establishes the following formal equivalence between the two qubits (

*A*,

*B*) Hilbert space and our two degrees of freedom (polarization and spatial mode) space: |0

*〉 ∼*

_{A}**x̂**, |1

*〉 ∼*

_{A}**ŷ**, |0

*〉 ∼*

_{B}*ψ*

_{10}, |1

*〉 ∼*

_{B}*ψ*

_{01}, then it is not difficult to show that our basis is mathematically equivalent to the quantum Bell basis [27]: where

28. C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A **82**(3), 033833 (2010). [CrossRef]

*operators Ĝ*

^{±}(

*φ*) are determined by their action upon the generic basis state |

*ψ*(

_{i}**x**)〉|

*e*〉. According to Eqs. (19) one has: where the rotation operator R̂(

_{j}*φ*) is expressed in terms of the matrix elements

*R*(

_{kj}*φ*) of the rotation matrix

*R*(

*φ*) as: From Eq. (26a) and Eq. (26b) it follows that

*ψ*

_{10}and

*ψ*

_{01}transform as the components of the position vector

**x**, namely: These formulas may be put straightforwardly in the operator form which reads: Note the different order of the indices in the matrix elements appearing in Eq. (91) and Eq. (93). Finally, gathering all these formulas together we obtain:

*A*indicates the transpose of the generic matrix

^{T}*A*, and we used the fact that for orthogonal matrices one has

*R*(

^{T}*φ*) =

*R*(−

*φ*). Here, in the direct product

*R̂*(∓

*φ*) ⊗

*R̂*(−

*φ*) the first operator acts upon the spatial degrees of freedom solely, while the the second one acts on the polarization degrees of freedom. Since in all previous calculations the basis state |

*ψ*(

_{i}**x**)〉|

*e*〉 was arbitrarily chosen, the relation (94) above is perfectly general and may be rewritten as:

_{j}*M̂*and

*P̂*be such that

*T̂*=

*M̂*⊗

*P̂*, the two spatial and polarization operators corresponding to the

*physical*transformation 𝒯 = × 𝒫. Then, by definition such operation transforms |

*u*

^{±}(

**x**)〉 into

*T̂*|

*u*

^{±}(

**x**)〉. It is clear that this new state maintains the same symmetry under rotation of the original state |

*u*

^{±}(

**x**)〉 if and only if: By multiplying both sides of Eq. (89) by

*T̂*, one straightforwardly obtains: Thus, by using Eq. (97) into Eq. (93) one easily obtain: or, equivalently: This equation can be easily recast in commutator form by rewriting it as follows: Of course, any operator

*T̂*that commutes with

*Ĝ*

^{±}(

*φ*), namely [

*Ĝ*

^{±}(

*φ*),

*T̂*] = 0, automatically fulfills Eq. (100). However, this condition is sufficient but not

*necessary*because Eq. (100) represents a much weaker constraint, as it only requires that the kernel (or, at least, part of it) of the operator [

*Ĝ*

^{±}(

*φ*),

*T̂*] coincides with the subspace spanned by either |

*u*

^{+}(

**x**)〉 or |

*u*

^{−}(

**x**)〉. In other words, all what is required to

*T̂*by Eq. (100), is that the commutator [

*Ĝ*

^{±}(

*φ*),

*T̂*] has a null eigenvalue in correspondence of the eigenvector |

*u*

^{±}(

**x**)〉.

*g*

^{±}(

**x**)〉. Obviously, this commutator expression is satisfied whenever the following two conditions are separately satisfied: where the arbitrariness of the angle

*φ*has been exploited. But, again, this condition is sufficient but not at all necessary. In fact, it is not difficult to see that if one choose either or with

*m*

_{1},

*m*

_{2},

*p*

_{1},

*p*

_{2}∈ ℂ, then Eq. (100) is always satisfied but Eqs. (103) are satisfied only by the first choice (104) that represent orthogonal matrices whenever

*α*with respect to the horizontal axis is represented by the Jones matrix which is of the form (105).

## References and links

1. | A. E. Siegman, |

2. | R. Martinez-Herrero, P. M. Mejías, and G. Piquero, |

3. | C. J. R. Sheppard, “Polarization of almost-plane waves,” J. Opt. Soc. Am. A |

4. | C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. |

5. | S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Comm. |

6. | R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. |

7. | N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. |

8. | M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B |

9. | M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A |

10. | P. Banzer, U. Peschel, S. Quabis, and G. Leuchs, “On the experimental investigation of the electric and magnetic response of a single nano-structure,” Opt. Express |

11. | M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous variable entanglement and squeezing of orbital angular momentum states,” Phys. Rev. Lett. |

12. | C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. |

13. | J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Remote preparation of single-photon “hybrid” entangled and vector-polarization states,” Phys. Rev. Lett. |

14. | R. Loudon, |

15. | M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. |

16. | A. Peres, |

17. | M. V. Berry, “Optical currents,” J. Opt. A, Pure Appl. Opt. |

18. | A. Aiello, N. Lindlein, Ch. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin hall effect of light,” Phys. Rev. Lett. |

19. | J. M. Jauch and F. Rohrlich, |

20. | R. M. A. Azzam and N. Bashra, |

21. | L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A |

22. | H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A |

23. | M. T. L. Hsu, W. P. Bowen, and P. K. Lam, “Spatial-state stokes-operator squeezing and entanglement for optical beams,” Phys. Rev. A |

24. | N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and Ch. Silberhorn, “Polarization squeezing and continuous-variable polarization entanglement,” Phys. Rev. A |

25. | J. N. Damask, |

26. | W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. |

27. | M. A. Nielsen and I. L. Chuang, |

28. | C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A |

**OCIS Codes**

(260.0260) Physical optics : Physical optics

(260.5430) Physical optics : Polarization

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: December 20, 2010

Revised Manuscript: February 18, 2011

Manuscript Accepted: February 22, 2011

Published: May 4, 2011

**Citation**

Annemarie Holleczek, Andrea Aiello, Christian Gabriel, Christoph Marquardt, and Gerd Leuchs, "Classical and quantum properties of cylindrically polarized states of light," Opt. Express **19**, 9714-9736 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-10-9714

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### References

- A. E. Siegman, Lasers (University Science Books, 1986).
- R. Martinez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009). [CrossRef]
- C. J. R. Sheppard, “Polarization of almost-plane waves,” J. Opt. Soc. Am. A 17, 335 (2000). [CrossRef]
- C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” N. J. Phys. 9, 78 (2007). [CrossRef]
- S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Comm. 179, 1 – 7 (2000). [CrossRef]
- R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]
- N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6, 273–276 (2001). [CrossRef] [PubMed]
- M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489–492 (2007). [CrossRef]
- M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2006). [CrossRef]
- P. Banzer, U. Peschel, S. Quabis, and G. Leuchs, “On the experimental investigation of the electric and magnetic response of a single nano-structure,” Opt. Express 18(10), 10905–10923 (2010). [CrossRef] [PubMed]
- M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous variable entanglement and squeezing of orbital angular momentum states,” Phys. Rev. Lett. 102(16), 163602 (2009). [CrossRef] [PubMed]
- C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106(6), 060502 (2011). [CrossRef] [PubMed]
- J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Remote preparation of single-photon “hybrid” entangled and vector-polarization states,” Phys. Rev. Lett. 105(3), 030407 (2010). [CrossRef] [PubMed]
- R. Loudon, The Quantum Theory of Light (Oxford University Press, 2000).
- M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24(7), 430–432 (1999). [CrossRef]
- A. Peres, Quantum Theory: Concept and Methods (Kluwer Academic, Boston, 1995).
- M. V. Berry, “Optical currents,” J. Opt. A, Pure Appl. Opt. 11(9), 094001 (2009). [CrossRef]
- A. Aiello, N. Lindlein, Ch. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009). [CrossRef] [PubMed]
- J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons: The relativistic Field Theory of Charged Particles with Spin one-half (Addison-Wesley Publishing Company, 1955).
- R. M. A. Azzam and N. Bashra, Ellipsometry and Polarized Light (North Holland Personal Library, 1988).
- L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]
- H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68(1), 012323 (2003). [CrossRef]
- M. T. L. Hsu, W. P. Bowen, and P. K. Lam, “Spatial-state stokes-operator squeezing and entanglement for optical beams,” Phys. Rev. A 79(4), 043825 (2009). [CrossRef]
- N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and Ch. Silberhorn, “Polarization squeezing and continuous-variable polarization entanglement,” Phys. Rev. A 65(5), 052306 (2002). [CrossRef]
- J. N. Damask, Polarization Optics in Telecommunication (Springer, 2005).
- W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. 90(4), 043601 (2003). [CrossRef] [PubMed]
- M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge University Press, 2000).
- C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82(3), 033833 (2010). [CrossRef]

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