**2. The orbital angular momentum of light**

Light beams carry energy and momentum, the latter both in its linear and angular components. Thus, when an electromagnetic field interacts with matter, an exchange of energy and momentum occurs, manifested in interesting mechanical effects or in changing of the beam properties. Here we will focus on light angular momentum, composed by a spinorial and orbital component. The manipulation of spinorial angular momentum (SAM), commonly associated to the polarization of light, is largely diffused in all the different field of physics. On the other hand, the orbital angular momentum (OAM) of light did not go through the same diffusion for many years, since after seminal works of Poynting (1909) and Beth (1936), only in 1992 Allen et al. (1992) demonstrated that particular solutions of the Helmoltz equation in the paraxial regime exhibit an azimuthal phase structure typical of beams possessing OAM. Such beams, as for example the Laguerre-Gauss beams, can be experimentally manipulated and thus offer a valuable resource in quantum information, where the possibility to exploit the infinite-dimensionality of OAM opens interesting perspectives. Here we present theoretically the main elements that characterize, both in classical and quantum regimes, the orbital angular

the exponential component is the plane wave contribution and *A*(�*r*) ∈ **C** is the complex amplitude whose variations are slow along the beam propagation and thus the following

Manipulation of Photonic Orbital Angular Momentum for Quantum Information Processing 79

*∂*2*A*(�*r*)

*<sup>∂</sup>y*<sup>2</sup> <sup>−</sup> <sup>2</sup>*ik <sup>∂</sup>A*(�*r*)

Replacing in eq.(2) the expression of *U*(�*r*) and considering the paraxial approximation in

This equation admits different families of solutions depending on the coordinates system adopted. For example adopting Cartesian coordinates, the solutions are the Gaussian beams, widely used in optics to describe laser beams propagation. However if the amplitude *A*(�*r*) is modulated by another function slowly varying on direction *z*ˆ, other valid solutions of eq.(4) can be obtained. Among them, a common one is represented by the Hermite-Gauss (HG) modes, where the Gaussian envelope is multiplied by Hermite polynomials. Interestingly, if instead of a beam described by *U*(*x*, *y*, *z*) the Helmoltz equation is solved in cylindrical coordinates that is, the function *U*(*r*, *ϕ*, *z*) is adopted, we obtain another important family of solutions, known as Laguerre-Gauss (LG) modes. As it will be shown in the next section, LG modes are characterized by a dependence of the phase term to the azimuthal angle and hence carry a well-determinate value of orbital angular momentum. Finally, it is worth notice that other classes of solution can be found by adopting polar coordinates (Hypergeometric beams HG Karimi et al. (2007)), or elliptic coordinates (Ince-Gaussian beams), or modifying properly

When the paraxial equation is solved in cylindrical coordinates (*x* = *rcosϕ*, *y* = *rsinϕ*, *z* = *z*) the solution is represented by a function family known as Laguerre-Gauss (LG) modes,

> 2*r*<sup>2</sup> *w*2(*z*)

 *e* <sup>−</sup> *<sup>r</sup>*<sup>2</sup> *<sup>w</sup>*2(*z*) *e*

<sup>−</sup>*<sup>i</sup> kr*2*<sup>z</sup>* 2(*z*2+*z*2 *<sup>R</sup>*) *e* <sup>−</sup>*imϕe*

*<sup>m</sup> Lm p*

beam size along *z*ˆ starting from the beam waist *w*0, and *ς*(*z*)=(2*p* + *m* + 1)*arctan <sup>z</sup>*

Gouy phase term. Here (*p*, *m*) are the radial and azimuthal index, respectively, and determine the order of the mode *N* = 2*p* + |*m*|. In particular (*p*, *m*) are integer numbers with *p* ≥ 0 while *m* can assume both positive or negative values. The lowest order of these modes *p* = *m* = 0 is

Let us focus on the phase term *e*−*imϕ*. This term leads to a phase front that *varies* with the azimuthal angle along the propagation direction: points on the phase front diametrically opposite are indeed dephased by a factor *π*. This means that the phase front follows an

where the gaussian envelope is multiplied by Laguerre polynomials *L<sup>m</sup>*

where *C* is a normalization constant, *zR* is the Rayleigh range, *w*(*z*) = *w*<sup>0</sup>

 *r* √2 *w*(*z*)

*∂*2*A*(�*r*)

*<sup>∂</sup>*<sup>2</sup> , *<sup>∂</sup>*2*A*(�*r*)

*<sup>∂</sup>y*<sup>2</sup> (3)

*<sup>∂</sup><sup>z</sup>* <sup>=</sup> <sup>0</sup> (4)

*<sup>p</sup>* (*τ*). The amplitude

 1 + *<sup>z</sup>*<sup>2</sup> *z*2 *R* is the

*<sup>i</sup>ς*(*z*) (5)

*zR* is the

*∂*2*A*(�*r*) *<sup>∂</sup>z*<sup>2</sup> <<

*∂*2*A*(�*r*) *<sup>∂</sup>x*<sup>2</sup> <sup>+</sup>

eq.(3), we obtain the *paraxial Helmoltz equation*:

the envelopes (Bessel and Mathieu beams).

**3.2 Laguerre-Gauss modes**

*uLG*

distribution of these modes reads:

the Gaussian beam (*TEM*<sup>00</sup> mode).

*<sup>p</sup>*,*m*(*r*, *<sup>ϕ</sup>*, *<sup>z</sup>*) = *<sup>C</sup>*

 1 + *<sup>z</sup>*<sup>2</sup> *z*2 *R*

relation holds:

momentum of light, deriving the solutions of Helmoltz equation and presenting the main ones, namely the Hermite-Gauss and Laguerre-Gauss beams.

#### **3. Classical orbital angular momentum of light**

In classical physics the total angular momentum *J* of an electromagnetic field reads:

$$
\vec{J} = \epsilon\_0 \int d\vec{r} \vec{r} \times (\vec{E} \times \vec{B}),
$$

where *E* and *B* are the electric and magnetic fields, respectively. The angular momentum is a conserved quantity for a free field, due to the invariance under arbitrary rotations of free Marxwell equations Enk & Nienhuis (1994). Moreover, as the electric field can be separated in longitudinal and transverse component *E* = *<sup>E</sup>*|| <sup>+</sup> *<sup>E</sup>*⊥, it is possible to single out the contribution from the transverse part to the angular momentum related to the radiation field *Jrad* whose components reads:

$$
\vec{J}\_{rad} = \vec{L} + \vec{S} \tag{1}
$$

where *S* is the *spin angular momentum* and *L* represent the *orbital angular momentum* contribution. The spin angular momentum does not depend on the specific coordinate system and hence it is often called as the intrinsic angular momentum. Moreover, since for a massless particle, as a photon, it is not possible to define a rest frame and hence a spin vector, it is possible only to define the projection of the spin along the propagation direction *z*ˆ, whose values can be *σ* = ±*s* with *s* both integer or half-integer values Enk & Nienhuis (1994). For the photon, having *s* = 1, the helicity *σ* can takes only values *σ* = ±1 (in ¯*h* units). As a form of angular momentum, spin angular momentum is physically associated to the rotation of the particle around its own axis, and for this reason physically is associated to the polarization of single photons. In particular, right-circular polarization is associated to *σ* = −*h*¯ and left-circular polarization to *σ* = +*h*¯. On the other hand as shown in eq.(1) light beams carry also orbital angular momentum, considered as the extrinsic component of angular momentum since it depends on the chosen coordinates system. It has been demonstrated that OAM is associated to the azimuthal phase structure of light beams, as will be described in section 3.2 (Allen et al. (1992)). Differently from SAM, OAM causes a rotation of the particle around the beam axis. The experimental evidence of the angular momentum and its components can be achieved only by studying the interaction between light and matter, where it is possible to observe a transfer of angular momentum. In particular, differences between SAM and OAM can be appreciated in the paraxial regime.

#### **3.1 The paraxial approximation**

In classical optics a beam that propagates along the *z*ˆ direction is described by the Helmoltz equation:

$$
\nabla^2 \mathcal{U}(\vec{r}) + k^2 \mathcal{U}(\vec{r}) = 0 \tag{2}
$$

where *k* is the wavevector *k* = *ωc*−1. For beams that are characterized by small divergence angle respect to the propagation direction, eq.(2) can be investigated within the *paraxial approximation*. Such approximation provides a proper description for laser beams, whose beam variation along the transverse plane (*x*ˆ*y*ˆ) are slow compared to the ones along the direction of propagation. In this case the beam can be represented as *U*(*r*) = *A*(*r*)*e*−*ikz*, where the exponential component is the plane wave contribution and *A*(�*r*) ∈ **C** is the complex amplitude whose variations are slow along the beam propagation and thus the following relation holds:

$$\frac{\partial^2 A(\vec{r})}{\partial z^2} < < \frac{\partial^2 A(\vec{r})}{\partial^2} \,, \frac{\partial^2 A(\vec{r})}{\partial y^2} \tag{3}$$

Replacing in eq.(2) the expression of *U*(�*r*) and considering the paraxial approximation in eq.(3), we obtain the *paraxial Helmoltz equation*:

$$\frac{\partial^2 A(\vec{r})}{\partial x^2} + \frac{\partial^2 A(\vec{r})}{\partial y^2} - 2ik \frac{\partial A(\vec{r})}{\partial z} = 0 \tag{4}$$

This equation admits different families of solutions depending on the coordinates system adopted. For example adopting Cartesian coordinates, the solutions are the Gaussian beams, widely used in optics to describe laser beams propagation. However if the amplitude *A*(�*r*) is modulated by another function slowly varying on direction *z*ˆ, other valid solutions of eq.(4) can be obtained. Among them, a common one is represented by the Hermite-Gauss (HG) modes, where the Gaussian envelope is multiplied by Hermite polynomials. Interestingly, if instead of a beam described by *U*(*x*, *y*, *z*) the Helmoltz equation is solved in cylindrical coordinates that is, the function *U*(*r*, *ϕ*, *z*) is adopted, we obtain another important family of solutions, known as Laguerre-Gauss (LG) modes. As it will be shown in the next section, LG modes are characterized by a dependence of the phase term to the azimuthal angle and hence carry a well-determinate value of orbital angular momentum. Finally, it is worth notice that other classes of solution can be found by adopting polar coordinates (Hypergeometric beams HG Karimi et al. (2007)), or elliptic coordinates (Ince-Gaussian beams), or modifying properly the envelopes (Bessel and Mathieu beams).

#### **3.2 Laguerre-Gauss modes**

4 Will-be-set-by-IN-TECH

momentum of light, deriving the solutions of Helmoltz equation and presenting the main

*<sup>d</sup><sup>r</sup><sup>r</sup>* <sup>×</sup> (

a conserved quantity for a free field, due to the invariance under arbitrary rotations of free Marxwell equations Enk & Nienhuis (1994). Moreover, as the electric field can be separated

contribution from the transverse part to the angular momentum related to the radiation field

contribution. The spin angular momentum does not depend on the specific coordinate system and hence it is often called as the intrinsic angular momentum. Moreover, since for a massless particle, as a photon, it is not possible to define a rest frame and hence a spin vector, it is possible only to define the projection of the spin along the propagation direction *z*ˆ, whose values can be *σ* = ±*s* with *s* both integer or half-integer values Enk & Nienhuis (1994). For the photon, having *s* = 1, the helicity *σ* can takes only values *σ* = ±1 (in ¯*h* units). As a form of angular momentum, spin angular momentum is physically associated to the rotation of the particle around its own axis, and for this reason physically is associated to the polarization of single photons. In particular, right-circular polarization is associated to *σ* = −*h*¯ and left-circular polarization to *σ* = +*h*¯. On the other hand as shown in eq.(1) light beams carry also orbital angular momentum, considered as the extrinsic component of angular momentum since it depends on the chosen coordinates system. It has been demonstrated that OAM is associated to the azimuthal phase structure of light beams, as will be described in section 3.2 (Allen et al. (1992)). Differently from SAM, OAM causes a rotation of the particle around the beam axis. The experimental evidence of the angular momentum and its components can be achieved only by studying the interaction between light and matter, where it is possible to observe a transfer of angular momentum. In particular, differences between SAM and OAM

In classical optics a beam that propagates along the *z*ˆ direction is described by the Helmoltz

where *k* is the wavevector *k* = *ωc*−1. For beams that are characterized by small divergence angle respect to the propagation direction, eq.(2) can be investigated within the *paraxial approximation*. Such approximation provides a proper description for laser beams, whose beam variation along the transverse plane (*x*ˆ*y*ˆ) are slow compared to the ones along the direction of propagation. In this case the beam can be represented as *U*(*r*) = *A*(*r*)*e*−*ikz*, where

<sup>∇</sup>2*U*(*<sup>r</sup>*) + *<sup>k</sup>*2*U*(*<sup>r</sup>*) = <sup>0</sup> (2)

*Jrad* =

*E* =

*L* +

*<sup>E</sup>* <sup>×</sup> *B*)

*B* are the electric and magnetic fields, respectively. The angular momentum is

*<sup>E</sup>*|| <sup>+</sup>

*<sup>E</sup>*⊥, it is possible to single out the

*S* (1)

*L* represent the *orbital angular momentum*

In classical physics the total angular momentum *J* of an electromagnetic field reads:

*J* = *�*<sup>0</sup> 

ones, namely the Hermite-Gauss and Laguerre-Gauss beams.

**3. Classical orbital angular momentum of light**

in longitudinal and transverse component

can be appreciated in the paraxial regime.

**3.1 The paraxial approximation**

equation:

*S* is the *spin angular momentum* and

where

where

*E* and

*Jrad* whose components reads:

When the paraxial equation is solved in cylindrical coordinates (*x* = *rcosϕ*, *y* = *rsinϕ*, *z* = *z*) the solution is represented by a function family known as Laguerre-Gauss (LG) modes, where the gaussian envelope is multiplied by Laguerre polynomials *L<sup>m</sup> <sup>p</sup>* (*τ*). The amplitude distribution of these modes reads:

$$u\_{p,m}^{\rm LG}(r,\varphi,z) = \frac{\mathbb{C}}{\sqrt{1 + \frac{z^2}{z\_R^2}}} \left(\frac{r\sqrt{2}}{w(z)}\right)^m L\_p^m\left(\frac{2r^2}{w^2(z)}\right) e^{-\frac{r^2}{w^2(z)}} e^{-i\frac{\hbar^2 z}{2(z^2 + z\_R^2)}} e^{-im\varphi} e^{i\zeta(z)}\tag{5}$$

where *C* is a normalization constant, *zR* is the Rayleigh range, *w*(*z*) = *w*<sup>0</sup> 1 + *<sup>z</sup>*<sup>2</sup> *z*2 *R* is the beam size along *z*ˆ starting from the beam waist *w*0, and *ς*(*z*)=(2*p* + *m* + 1)*arctan <sup>z</sup> zR* is the Gouy phase term. Here (*p*, *m*) are the radial and azimuthal index, respectively, and determine the order of the mode *N* = 2*p* + |*m*|. In particular (*p*, *m*) are integer numbers with *p* ≥ 0 while *m* can assume both positive or negative values. The lowest order of these modes *p* = *m* = 0 is the Gaussian beam (*TEM*<sup>00</sup> mode).

Let us focus on the phase term *e*−*imϕ*. This term leads to a phase front that *varies* with the azimuthal angle along the propagation direction: points on the phase front diametrically opposite are indeed dephased by a factor *π*. This means that the phase front follows an

**-**

Manipulation of Photonic Orbital Angular Momentum for Quantum Information Processing 81

Fig. 1. Examples of LG modes. Top: phase profiles for different LG modes and Bottom:

*rdrdϕ*(�*r*<sup>×</sup> <sup>&</sup>lt; �

*rdrdϕ* < �

Equation (9) shows that each photon that constitutes the beam carries an amount of angular momentum equal to *mh*¯ Allen et al. (1992; 1999). It is possible to demonstrate that this value refers to the orbital component of angular momentum and is related to the azimuthal phase factor. Indeed if we consider a circularly polarized beam, i.e. a beam with no vanishing

*A*� = (*αx*ˆ + *βy*ˆ)*u*(*x*, *y*, *z*)*e*

and going through the same procedure carried before, including the transformation to cylindrical coordinates, the ratio between angular momentum and energy density leads to:

We observe from eq.(10) that each photon contributes to the total angular momentum *Jz* with two components. The *σ*/*ω* one belongs to the spinorial contribution, and is due to the polarization of the beam, since *σ* = ±1 depending on the right or left circular polarization. Analogously, we observe a further term purely related to the phase dependence *e*−*im<sup>ϕ</sup>* that does not belongs to the polarization. Such contribution is thus related to the orbital angular momentum, showing that, depending on the twisting velocity of the phase along

The transverse field distribution of laser cavities is typically well described in terms of Hermite-Gauss (HG) modes. These one form a complete orthogonal set that solve the paraxial equation in Cartesian coordinates and their amplitude is given, apart for a phase term, by the product of a gaussian function and Hermite polynomials of indexes (*n*˜, *m*˜) Walborn et al.

*<sup>E</sup>* <sup>×</sup> � *B* >)*<sup>z</sup>*

*<sup>E</sup>* <sup>×</sup> � *B* >*<sup>z</sup>*

*rdrdϕ*(�*r*<sup>×</sup> <sup>&</sup>lt; �

*rdrdϕ* < �

*c*

**-**

*Jz <sup>W</sup>* <sup>=</sup>

contribution of spin angular momentum, its vector potential is:

*c*

the propagation direction, each photon of the beam carries *mh*¯ OAM.

*Jz <sup>W</sup>* <sup>=</sup>

intensity profiles of the same modes.

the transverse plane:

**3.3 Hermite-Gauss modes**

**-**

**-**

*<sup>E</sup>* <sup>×</sup> � *B* >)*<sup>z</sup>*

*<sup>E</sup>* <sup>×</sup> � *B* >*<sup>z</sup>* **-**

<sup>=</sup> *<sup>m</sup>*

*ikz*

<sup>=</sup> *<sup>m</sup> ω* + *σ*

*<sup>ω</sup>* (9)

*<sup>ω</sup>* (10)

helicoidal pattern along *z*ˆ so that in the center of the transverse plane the contact between all different phases causes a *phase singularity*. The torque shown along the directional axis thus leads to an optical vortex, that is LG beams are characterized by a singularity in the center where the field intensity vanishes. For each optical vortex it is possible to associate a number, known as *topological charge*, that refers to the number of twist over a distance of the wavelength *λ*. Such number *q* is always an integer, and can assume both positive or negative values, depending on the direction of the twist. The larger is the number of torsions, greater is the twisting velocity along the beam direction. It is exactly this twisting that leads to a non-vanishing contribution of orbital angular momentum

Fig.(1) reports some examples of LG modes intensities and phases for different values of indices *p*, *m*. For *p* = 0 the intensity in the transverse plane is distributed among an annular pattern with a zero-intensity spot in the center, so that LG modes are known also as "doughnut modes". In the next section we will show that Laguerre-Gauss modes constitute a complete orthogonal basis of orbital angular momentum operator *Lz* eigenvectors, however here we present a notable result of Allen et al. (1992) where for the first time was demonstrated that beam characterized by a helical behavior of the phase front, as for LG modes, carry a well-defined amount of OAM per photon fixed by the value of the azimuthal mode index *m*

Let us first consider a vector potential written in Cartesian coordinates as:

$$
\vec{A}(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \mathfrak{u}(\mathbf{x}, \mathbf{y}, \mathbf{z}) \mathfrak{e}^{\text{l}\mathbf{z}} \tag{6}
$$

Adopting the Lorentz gauge, the relation between magnetic � *B* and electric � *E* fields and potential vector are known, so that is possible to evaluate the mean value of the Poynting vector � *P* that determines the energy carried by the beam:

$$p = \epsilon\_0 < \vec{E} \times \vec{B} > = i\omega \frac{\epsilon\_0}{2} (u\nabla u^\* - u^\*\nabla u) + wk\epsilon\_0|u|^2$$

This result can be extended to the case of a vector potential expressed in cylindrical coordinates, that is where the beam is characterized by the azimuthal angular dependence *e*−*imϕ*. Now we are interested in the estimation of the angular momentum carried by the beam along the *z*ˆ direction. The angular momentum reads:

$$
\vec{j} = \vec{r} \times \vec{p}
$$

where the Poynting vector provides three component �*p* = (*pr*, *pϕ*, *pz*). Thus the component of angular momentum along the propagation direction of the beam *jz* can be calculated as *jz* = *rp<sup>ϕ</sup>* where:

$$p\_{\varphi} = \varepsilon\_0 \frac{wm|u|^2}{r} \tag{7}$$

Just by replacing this result in the *jz* expression, we get:

$$j\_z = \epsilon\_0 w m |u|^2 \tag{8}$$

The value of angular momentum carried by each photon can be obtained considering the ratio between angular momentum density and the energy one, and then integrating over all

Fig. 1. Examples of LG modes. Top: phase profiles for different LG modes and Bottom: intensity profiles of the same modes.

the transverse plane:

6 Will-be-set-by-IN-TECH

helicoidal pattern along *z*ˆ so that in the center of the transverse plane the contact between all different phases causes a *phase singularity*. The torque shown along the directional axis thus leads to an optical vortex, that is LG beams are characterized by a singularity in the center where the field intensity vanishes. For each optical vortex it is possible to associate a number, known as *topological charge*, that refers to the number of twist over a distance of the wavelength *λ*. Such number *q* is always an integer, and can assume both positive or negative values, depending on the direction of the twist. The larger is the number of torsions, greater is the twisting velocity along the beam direction. It is exactly this twisting that leads to a

Fig.(1) reports some examples of LG modes intensities and phases for different values of indices *p*, *m*. For *p* = 0 the intensity in the transverse plane is distributed among an annular pattern with a zero-intensity spot in the center, so that LG modes are known also as "doughnut modes". In the next section we will show that Laguerre-Gauss modes constitute a complete orthogonal basis of orbital angular momentum operator *Lz* eigenvectors, however here we present a notable result of Allen et al. (1992) where for the first time was demonstrated that beam characterized by a helical behavior of the phase front, as for LG modes, carry a well-defined amount of OAM per photon fixed by the value of the azimuthal mode index *m*

*A*� (*x*, *y*, *z*) = *u*(*x*, *y*, *z*)*e*

potential vector are known, so that is possible to evaluate the mean value of the Poynting

This result can be extended to the case of a vector potential expressed in cylindrical coordinates, that is where the beam is characterized by the azimuthal angular dependence *e*−*imϕ*. Now we are interested in the estimation of the angular momentum carried by the

�*<sup>j</sup>* <sup>=</sup>�*<sup>r</sup>* <sup>×</sup>�*<sup>p</sup>*

where the Poynting vector provides three component �*p* = (*pr*, *pϕ*, *pz*). Thus the component of angular momentum along the propagation direction of the beam *jz* can be calculated as

*jz* = *�*0*wm*|*u*|

The value of angular momentum carried by each photon can be obtained considering the ratio between angular momentum density and the energy one, and then integrating over all

*wm*|*u*| 2

*p<sup>ϕ</sup>* = *�*<sup>0</sup>

<sup>2</sup> (*u*∇*u*<sup>∗</sup> <sup>−</sup> *<sup>u</sup>*∗∇*u*) + *wk�*0|*u*<sup>|</sup>

*�*0

*ikz* (6)

*E* fields and

*B* and electric �

2

*<sup>r</sup>* (7)

<sup>2</sup> (8)

non-vanishing contribution of orbital angular momentum

Let us first consider a vector potential written in Cartesian coordinates as:

*B* >= *iω*

Adopting the Lorentz gauge, the relation between magnetic �

*P* that determines the energy carried by the beam:

*<sup>E</sup>* <sup>×</sup> �

beam along the *z*ˆ direction. The angular momentum reads:

Just by replacing this result in the *jz* expression, we get:

*p* = *�*<sup>0</sup> < �

vector �

*jz* = *rp<sup>ϕ</sup>* where:

$$\frac{J\_z}{W} = \frac{\int \int r dr d\varphi (\vec{r} \times <\vec{E} \times \vec{B} >)\_z}{c \int \int r dr d\varphi < \vec{E} \times \vec{B} >\_z} = \frac{m}{\omega} \tag{9}$$

Equation (9) shows that each photon that constitutes the beam carries an amount of angular momentum equal to *mh*¯ Allen et al. (1992; 1999). It is possible to demonstrate that this value refers to the orbital component of angular momentum and is related to the azimuthal phase factor. Indeed if we consider a circularly polarized beam, i.e. a beam with no vanishing contribution of spin angular momentum, its vector potential is:

$$\vec{A} = (\mathfrak{a}\mathfrak{X} + \mathfrak{B}\mathfrak{Y})u(\mathfrak{x}, \mathfrak{y}, \mathfrak{z})e^{ikz}$$

and going through the same procedure carried before, including the transformation to cylindrical coordinates, the ratio between angular momentum and energy density leads to:

$$\frac{J\_z}{W} = \frac{\int \int r dr d\varphi (\vec{r} \times <\vec{E} \times \vec{B} >)\_z}{c \int \int r dr d\varphi < \vec{E} \times \vec{B} >\_z} = \frac{m}{\omega} + \frac{\sigma}{\omega} \tag{10}$$

We observe from eq.(10) that each photon contributes to the total angular momentum *Jz* with two components. The *σ*/*ω* one belongs to the spinorial contribution, and is due to the polarization of the beam, since *σ* = ±1 depending on the right or left circular polarization. Analogously, we observe a further term purely related to the phase dependence *e*−*im<sup>ϕ</sup>* that does not belongs to the polarization. Such contribution is thus related to the orbital angular momentum, showing that, depending on the twisting velocity of the phase along the propagation direction, each photon of the beam carries *mh*¯ OAM.

#### **3.3 Hermite-Gauss modes**

The transverse field distribution of laser cavities is typically well described in terms of Hermite-Gauss (HG) modes. These one form a complete orthogonal set that solve the paraxial equation in Cartesian coordinates and their amplitude is given, apart for a phase term, by the product of a gaussian function and Hermite polynomials of indexes (*n*˜, *m*˜) Walborn et al.

Different indexes that characterize HG and LG are related one set to the other as:

defined as:

of the mode order *N* and the azimuthal index *m*:

Coefficients given in eq.(13) are of crucial importance as they define the matrix element needed for the conversion between HG and LG modes (or viceversa): O'Neil & Courtial (2000). Moreover, the correspondence between HG and LG modes is fixed by the mode order *N*,

Manipulation of Photonic Orbital Angular Momentum for Quantum Information Processing 83

*N* = *n*˜ + *m*˜ = 2*p* + |*m*| Inverting relations given in eq.(16) it is possible to estimate the indexes of HG modes in terms

> *<sup>n</sup>*˜ <sup>=</sup> *<sup>N</sup>* <sup>+</sup> *<sup>m</sup>* 2 *<sup>m</sup>*˜ <sup>=</sup> *<sup>N</sup>* <sup>−</sup> *<sup>m</sup>* 2

All these relations can be exploited to generate Laguerre-Gauss modes starting from Hermite-Gauss modes emitted by lasers. Experimentally such mode converters based on

In 1992 Allen et al. demonstrated that Laguerre-Gauss modes carry a well defined amount of orbital angular momentum through classical mechanics approach, as shown in section 3.2. On the other hand, it is possible to demonstrate this property also following a quantum mechanical approach, showing that LG modes are eigenvectors of the orbital angular

*<sup>L</sup>*<sup>ˆ</sup> <sup>=</sup> <sup>−</sup>*ih*¯(*<sup>r</sup>* × ∇)

 *y ∂ <sup>∂</sup><sup>z</sup>* <sup>−</sup> *<sup>z</sup>*

 *z ∂ <sup>∂</sup><sup>x</sup>* <sup>−</sup> *<sup>x</sup>*

 *x ∂ <sup>∂</sup><sup>y</sup>* <sup>−</sup> *<sup>y</sup>*

*∂ ∂y* 

*∂ ∂z* 

*∂ ∂x* 

Thus the cartesian components of the orbital angular momentum operator reads:

*Lx* = −*ih*¯

*Ly* = −*ih*¯

*Lz* = −*ih*¯

eq.(12) are implemented through cylindrical lenses, as explained in section 4.3.

**3.4 The orbital angular momentum in quantum mechanics**

The classical orbital angular momentum

where the following commutation rules apply:

momentum operator *Lz* (Bransden & Joachain (2003); Sakurai (1994)).

using the definition of operator momentum *p*ˆ = −*ih*¯ ∇, so that

*m* = *n*˜ − *m*˜ (14) *p* = *min*(*n*˜, *m*˜) (15)

*L* =*r* ×*p* finds its counterpart in quantum mechanics

[*Li*, *Lj*] = *ih*¯ *�ijkLk* (18)

(16)

(17)

Fig. 2. Intensity profiles for different HG modes. As can be observed, the zero order of HG coincides with the one of LG modes that is, the gaussian mode.

(2005):

$$\ln u\_{\hbar,\hbar\mathbb{I}}^{\mathrm{HG}} = \sqrt{\frac{2}{\Im^{(\mathfrak{H}+\mathfrak{M})\,\pi\hbar\mathbb{I}\,\mathfrak{M}!}}} \frac{1}{w(z)} H\_{\hbar} \left( \frac{\sqrt{2}\mathfrak{x}}{w(z)} \right) H\_{\hbar\mathbb{I}} \left( \frac{\sqrt{2}\mathfrak{y}}{w(z)} \right) e^{-\frac{\mathfrak{x}^{2}+\mathfrak{y}^{2}}{\mathfrak{u}^{2}(z)}} e^{-i\mathfrak{k}\frac{(\mathfrak{x}^{2}+\mathfrak{y}^{2})z}{2(\mathfrak{x}^{2}+z\_{R}^{2})}} e^{-i(\mathfrak{H}+\mathfrak{M}+1)\xi(z)} \tag{11}$$

where *ς*(*z*) = *actan <sup>z</sup> zr* and the last exponential term is the Gouy phase. This solution of the paraxial Helmoltz equation represents modes structurally stables, so that their intensity remain the same along the propagation Allen et al. (1999). The mode indexes *n*˜, *m*˜ define the shape of the beam profile along the *x*ˆ and *y*ˆ direction, respectively (Saleh & Teich (1991)). Some examples of intensity distribution of such modes are reported in Fig.(2). We observe that the intensity *In*˜,*m*˜ <sup>=</sup> <sup>|</sup>*uHG n*˜,*m*˜ | <sup>2</sup> has *n*˜ nodes on the horizontal direction, and *m*˜ along the vertical, while for *n*˜ = *m*˜ = 0 the shape is the one of a Gaussian beam (*TEM*<sup>00</sup> mode). The order of the mode is defined as *N* = *n*˜ + *m*˜ . As can be observed from eq.(11), in HG modes each component of the transverse plane travel in phase, as for a plane wave. Indeed this means that surfaces that include all the points with same phase are planes separated by a distance equal to *λ*. For this reason there is a nil component of linear momentum along the axial direction, which means that such modes do not carry a well defined amount of OAM, as deducible by the lack of the phase term *eimϕ*. Hermite-Gauss modes form a complete basis set so that any arbitrary field distribution can be written as a superposition of HG modes with indexes *n*˜, *m*˜ . However also Laguerre-Gauss modes form a complete basis set, hence a HG mode can be rewritten as a linear superposition of LG modes and *viceversa* through the relations:

$$u\_{m,p}^{LG}(\mathbf{x},y,z) = \sum\_{k=0}^{N} i^k b(m,p,k) u\_{N-k,k}^{HG}(\mathbf{x},y,z) \tag{12}$$

where

$$b(m, p, k) = \frac{(N - k!k!)^{\frac{1}{2}}}{2^N m! p!} \frac{1}{k!} \frac{d^k}{dt^k} [(1 - t)^m (1 + t)^p]\_t = 0 \tag{13}$$

8 Will-be-set-by-IN-TECH

**-**

**-**

**-**

Fig. 2. Intensity profiles for different HG modes. As can be observed, the zero order of HG

 <sup>√</sup>2*<sup>y</sup> w*(*z*)

the paraxial Helmoltz equation represents modes structurally stables, so that their intensity remain the same along the propagation Allen et al. (1999). The mode indexes *n*˜, *m*˜ define the shape of the beam profile along the *x*ˆ and *y*ˆ direction, respectively (Saleh & Teich (1991)). Some examples of intensity distribution of such modes are reported in Fig.(2). We observe that the

for *n*˜ = *m*˜ = 0 the shape is the one of a Gaussian beam (*TEM*<sup>00</sup> mode). The order of the mode is defined as *N* = *n*˜ + *m*˜ . As can be observed from eq.(11), in HG modes each component of the transverse plane travel in phase, as for a plane wave. Indeed this means that surfaces that include all the points with same phase are planes separated by a distance equal to *λ*. For this reason there is a nil component of linear momentum along the axial direction, which means that such modes do not carry a well defined amount of OAM, as deducible by the lack of the phase term *eimϕ*. Hermite-Gauss modes form a complete basis set so that any arbitrary field distribution can be written as a superposition of HG modes with indexes *n*˜, *m*˜ . However also Laguerre-Gauss modes form a complete basis set, hence a HG mode can be rewritten as a

 *e* <sup>−</sup> *<sup>x</sup>*2+*y*<sup>2</sup> *<sup>w</sup>*2(*z*) *e*

*zr* and the last exponential term is the Gouy phase. This solution of

<sup>2</sup> has *n*˜ nodes on the horizontal direction, and *m*˜ along the vertical, while

**-**

**-**

coincides with the one of LG modes that is, the gaussian mode.

 <sup>√</sup>2*<sup>x</sup> w*(*z*)

linear superposition of LG modes and *viceversa* through the relations:

*N* ∑ *k*=0 *i*

> 1 <sup>2</sup> 1 *k*! *dk*

2*Nm*!*p*!

*kb*(*m*, *p*, *k*)*uHG*

*<sup>m</sup>*,*p*(*x*, *y*, *z*) =

*<sup>b</sup>*(*m*, *<sup>p</sup>*, *<sup>k</sup>*) = (*<sup>N</sup>* <sup>−</sup> *<sup>k</sup>*!*k*!)

*uLG*

 *Hm*˜

1 *w*(*z*) *Hn*˜

(2005):

*uHG <sup>n</sup>*˜,*m*˜ =

where

2

where *ς*(*z*) = *actan <sup>z</sup>*

intensity *In*˜,*m*˜ <sup>=</sup> <sup>|</sup>*uHG*

2(*n*˜+*m*˜)*πn*˜!*m*˜ !

*n*˜,*m*˜ |

**-**

**-**

> <sup>−</sup>*ik* (*x*2+*y*2)*<sup>z</sup>* 2(*z*2+*z*2 *<sup>R</sup>*) *e*

*<sup>N</sup>*−*k*,*k*(*x*, *<sup>y</sup>*, *<sup>z</sup>*) (12)

*dt<sup>k</sup>* [(<sup>1</sup> <sup>−</sup> *<sup>t</sup>*)*m*(<sup>1</sup> <sup>+</sup> *<sup>t</sup>*)*p*]*<sup>t</sup>* <sup>=</sup> <sup>0</sup> (13)

<sup>−</sup>*i*(*n*˜+*m*˜ <sup>+</sup>1)*ς*(*z*) (11)

Different indexes that characterize HG and LG are related one set to the other as:

$$
\hbar \mathfrak{m} = \mathfrak{n} - \mathfrak{m} \tag{14}
$$

$$p = \min(\mathfrak{n}, \mathfrak{m}) \tag{15}$$

Coefficients given in eq.(13) are of crucial importance as they define the matrix element needed for the conversion between HG and LG modes (or viceversa): O'Neil & Courtial (2000). Moreover, the correspondence between HG and LG modes is fixed by the mode order *N*, defined as:

$$N = \mathfrak{n} + \mathfrak{m} = 2p + |m|.$$

Inverting relations given in eq.(16) it is possible to estimate the indexes of HG modes in terms of the mode order *N* and the azimuthal index *m*:

$$
\begin{aligned}
\tilde{n} &= \frac{N+m}{2} \\
\tilde{m} &= \frac{N-m}{2}
\end{aligned}
$$

All these relations can be exploited to generate Laguerre-Gauss modes starting from Hermite-Gauss modes emitted by lasers. Experimentally such mode converters based on eq.(12) are implemented through cylindrical lenses, as explained in section 4.3.

#### **3.4 The orbital angular momentum in quantum mechanics**

In 1992 Allen et al. demonstrated that Laguerre-Gauss modes carry a well defined amount of orbital angular momentum through classical mechanics approach, as shown in section 3.2. On the other hand, it is possible to demonstrate this property also following a quantum mechanical approach, showing that LG modes are eigenvectors of the orbital angular momentum operator *Lz* (Bransden & Joachain (2003); Sakurai (1994)).

The classical orbital angular momentum *L* =*r* ×*p* finds its counterpart in quantum mechanics using the definition of operator momentum *p*ˆ = −*ih*¯ ∇, so that

$$\hat{L} = -i\hbar (r \times \nabla)$$

Thus the cartesian components of the orbital angular momentum operator reads:

$$\begin{aligned} L\_x &= -i\hbar \left( y \frac{\partial}{\partial z} - z \frac{\partial}{\partial y} \right) \\ L\_y &= -i\hbar \left( z \frac{\partial}{\partial x} - x \frac{\partial}{\partial z} \right) \\ L\_z &= -i\hbar \left( x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x} \right) \end{aligned} \tag{16}$$

(17)

where the following commutation rules apply:

$$\mathbb{E}\left[L\_{i\prime}L\_{j}\right] = i\hbar \varepsilon\_{ijk}L\_{k} \tag{18}$$

different hologram is then needed. Each hologram is then characterized by the transmittance function, related to the pattern that has to be impressed on the film. Generally, if we consider as reference optical vortex the beam *E*(*ϕ*, *z*) = *E*0*eimϕe*−*ikz*, and the oblique-propagation plane

Manipulation of Photonic Orbital Angular Momentum for Quantum Information Processing 85

<sup>2</sup> = 1 + *I*<sup>0</sup> + 2

2*π*

. When a generic wave *E*(*ρ*, *ϕ*) = *E*(*ρ*)*eim<sup>ϕ</sup>* impinges on the

*<sup>i</sup>*(*m*+1)*ϕe*

2*π* <sup>Λ</sup> *<sup>x</sup>* + *e*

Taking the Fourier transform of the interference pattern, it is possible to obtain the generic

where Λ is the period of the hologram grating, and *α* the amplitude of phase modulation. In order to illustrate how a hologram actually works, let us consider a standard single-fork

(*ϕ*(*x*, *<sup>y</sup>*) + <sup>2</sup>*<sup>π</sup>*

*<sup>I</sup>*0*cos*(*m<sup>ϕ</sup>* <sup>+</sup> *kx*) (20)

<sup>Λ</sup> *rcosϕ*)} (21)

<sup>Λ</sup> *<sup>x</sup>*)} (22)

*<sup>i</sup>*(*m*−1)*ϕe*

−2*π* <sup>Λ</sup> *<sup>x</sup>*] (23)

(*kx x* − *kzz*), the interference pattern at *z* = 0 reads:

*T*(*r*, *ϕ*) = *exp*{*iαcos*(*mϕ* +

1 2

<sup>2</sup> *<sup>e</sup>*

*im<sup>ϕ</sup>* +

Looking at the last term in eq.(23), it emerges how after the hologram the main part of the field is undiffracted and keeps the same value of the initial OAM. On the two first-order diffracted

*E*(*ρ*) <sup>4</sup> [*<sup>e</sup>*

**-**

**-**

**-**

Fig. 3. **a)** Schematic representation of a hologram and its action on an optical field.**b)**Action of a single-fork hologram on different input states. In the yellow box have been enlightened the

*I* = |*E*(*ϕ*, *z*) + *U*(*x*, *z*)|

hologram (Δ*m* = 1), characterized by the transmittivity function:

*<sup>T</sup>* <sup>=</sup> *cos*4{

hologram, the output field is determined by <sup>√</sup>*TE*(*ρ*, *<sup>ϕ</sup>*), that gives rise to:

*im<sup>ϕ</sup>* <sup>=</sup> *<sup>E</sup>*(*ρ*)

transmittance function for a CGH in polar coordinates:

*y* 

<sup>Λ</sup> *<sup>x</sup>*)]*<sup>e</sup>*

modes that can be coupled to single mode fiber for analysis.

wave *U*(*x*, *z*) = *e<sup>i</sup>*

where *<sup>ϕ</sup>*(*x*, *<sup>y</sup>*) = *arctg <sup>x</sup>*

<sup>2</sup> [<sup>1</sup> <sup>+</sup> *cos*(*ϕ*(*x*, *<sup>y</sup>*) + <sup>2</sup>*<sup>π</sup>*

 


 -

*E*(*ρ*)

being *�ijk* the Levi-Civita tensor. It is convenient to study the orbital angular momentum in polar coordinates (*r*, *θ*, *ϕ*) so that the orbital angular momentum operator *Lz* can be written as:

$$L\_z = -i\hbar \frac{\partial}{\partial \rho}$$

The eigenvectors equation *Lz*Ψ(*ϕ*) = *E*Ψ(*ϕ*) with *E* = *mh*¯ leads to:

$$\Psi\_m(\varphi) = \frac{1}{\sqrt{2\pi}} e^{im\varphi} \tag{19}$$

Moreover the imposition on the eigenvectors to be single valued leads to a constraint on the values of *m*, that is restricted to positive or negative integers or zero. It emerges from eq.(19) that in paraxial approximation LG modes represent indeed the eigenvectors of the orbital angular momentum operator *Lz* with eigenvalue *mh*¯. Generally speaking, any mode with a phase dependence as in eq.(19) can be considered eigenvector of *Lz* and thus carry a well defined amount of OAM. Laguerre-Gauss modes form a complete Hilbert basis so that can properly represent the quantum photon states in paraxial regime, where spin and orbital contribution can be considered separately. Indeed photon represented by a single LG mode are in a quantum state related to a well-defined value of OAM. It is worth notice that as the eigenvalues of *Lz* can assume all integer values both positive or negative, the OAM is defined in a infinite-dimensional Hilbert space, thus opening exciting perspectives in the quantum information field.

#### **4. Manipulation of orbital angular momentum**

The experimental investigation of orbital angular momentum started in the '90s, thus allowing the idea of OAM as a relatively "'young"' degree of freedom of light. While many devices have been developed for the efficient manipulation of polarization states through birefringent media, as waveplates and polarizing beam splitter, the optical tools for generating and controlling the OAM photon states are rather limited. In last years many efforts have been made in order to implement a device able to generate and manipulate with high efficiency LG modes. In this section we will present the most common and reliable optical devices that exploiting the properties of beams with azimuthal phase structure *eimϕ*, have allowed the implementation of several quantum optics experiments.

#### **4.1 Computer generated holograms**

Computer generated holograms (CGH) are typical tools of diffractive optics, and are probably one of the most common device adopted for the generation and analysis of OAM states. CGHs can be described as diffractive gratings impressed on a film depending on the specific computer-calculated interference pattern formed by a reference plane wave and the desired beam that has to be generated, for example an optical vortex, propagating over a small angle respect to the reference beam. Indeed if a plane wave impinges on the hologram, we would expect to generate on the first diffracted mode an optical vortex equal to the former one. For example, if the reference beam is a LG with OAM *m*, its interference pattern with a plane wave to be impressed on the hologram is made by a diffraction grating with *m* lines forming a multi-pronged fork. Depending on the optical vortex that has to be generated/analyzed, a 10 Will-be-set-by-IN-TECH

being *�ijk* the Levi-Civita tensor. It is convenient to study the orbital angular momentum in polar coordinates (*r*, *θ*, *ϕ*) so that the orbital angular momentum operator *Lz* can be written

*Lz* <sup>=</sup> <sup>−</sup>*ih*¯ *<sup>∂</sup>*

<sup>Ψ</sup>*m*(*ϕ*) = <sup>1</sup>

The eigenvectors equation *Lz*Ψ(*ϕ*) = *E*Ψ(*ϕ*) with *E* = *mh*¯ leads to:

*∂ϕ*

*im<sup>ϕ</sup>* (19)

<sup>√</sup>2*<sup>π</sup> e*

Moreover the imposition on the eigenvectors to be single valued leads to a constraint on the values of *m*, that is restricted to positive or negative integers or zero. It emerges from eq.(19) that in paraxial approximation LG modes represent indeed the eigenvectors of the orbital angular momentum operator *Lz* with eigenvalue *mh*¯. Generally speaking, any mode with a phase dependence as in eq.(19) can be considered eigenvector of *Lz* and thus carry a well defined amount of OAM. Laguerre-Gauss modes form a complete Hilbert basis so that can properly represent the quantum photon states in paraxial regime, where spin and orbital contribution can be considered separately. Indeed photon represented by a single LG mode are in a quantum state related to a well-defined value of OAM. It is worth notice that as the eigenvalues of *Lz* can assume all integer values both positive or negative, the OAM is defined in a infinite-dimensional Hilbert space, thus opening exciting perspectives in the quantum

The experimental investigation of orbital angular momentum started in the '90s, thus allowing the idea of OAM as a relatively "'young"' degree of freedom of light. While many devices have been developed for the efficient manipulation of polarization states through birefringent media, as waveplates and polarizing beam splitter, the optical tools for generating and controlling the OAM photon states are rather limited. In last years many efforts have been made in order to implement a device able to generate and manipulate with high efficiency LG modes. In this section we will present the most common and reliable optical devices that exploiting the properties of beams with azimuthal phase structure *eimϕ*, have allowed the

Computer generated holograms (CGH) are typical tools of diffractive optics, and are probably one of the most common device adopted for the generation and analysis of OAM states. CGHs can be described as diffractive gratings impressed on a film depending on the specific computer-calculated interference pattern formed by a reference plane wave and the desired beam that has to be generated, for example an optical vortex, propagating over a small angle respect to the reference beam. Indeed if a plane wave impinges on the hologram, we would expect to generate on the first diffracted mode an optical vortex equal to the former one. For example, if the reference beam is a LG with OAM *m*, its interference pattern with a plane wave to be impressed on the hologram is made by a diffraction grating with *m* lines forming a multi-pronged fork. Depending on the optical vortex that has to be generated/analyzed, a

as:

information field.

**4. Manipulation of orbital angular momentum**

implementation of several quantum optics experiments.

**4.1 Computer generated holograms**

different hologram is then needed. Each hologram is then characterized by the transmittance function, related to the pattern that has to be impressed on the film. Generally, if we consider as reference optical vortex the beam *E*(*ϕ*, *z*) = *E*0*eimϕe*−*ikz*, and the oblique-propagation plane wave *U*(*x*, *z*) = *e<sup>i</sup>* (*kx x* − *kzz*), the interference pattern at *z* = 0 reads:

$$I = |E(\varphi, z) + \mathcal{U}(\mathbf{x}, z)|^2 = 1 + I\_0 + 2\sqrt{I\_0}\cos(m\varphi + k\mathbf{x})\tag{20}$$

Taking the Fourier transform of the interference pattern, it is possible to obtain the generic transmittance function for a CGH in polar coordinates:

$$T(r,\varphi) = \exp\{i\alpha\cos(m\varphi + \frac{2\pi}{\Lambda}r\cos\varphi)\}\tag{21}$$

where Λ is the period of the hologram grating, and *α* the amplitude of phase modulation. In order to illustrate how a hologram actually works, let us consider a standard single-fork hologram (Δ*m* = 1), characterized by the transmittivity function:

$$T = \cos^4\left\{\frac{1}{2}(\varphi(\mathbf{x}, y) + \frac{2\pi}{\Lambda}\mathbf{x})\right\} \tag{22}$$

where *<sup>ϕ</sup>*(*x*, *<sup>y</sup>*) = *arctg <sup>x</sup> y* . When a generic wave *E*(*ρ*, *ϕ*) = *E*(*ρ*)*eim<sup>ϕ</sup>* impinges on the hologram, the output field is determined by <sup>√</sup>*TE*(*ρ*, *<sup>ϕ</sup>*), that gives rise to:

$$\frac{E(\rho)}{2}[1+\cos(\varphi(\mathbf{x},\mathbf{y})+\frac{2\pi}{\Lambda}\mathbf{x})]e^{im\boldsymbol{\rho}} = \frac{E(\rho)}{2}e^{im\boldsymbol{\rho}} + \frac{E(\rho)}{4}[e^{i(m+1)\boldsymbol{\rho}}e^{\frac{2\pi}{\Lambda}\mathbf{x}} + e^{i(m-1)\boldsymbol{\rho}}e^{\frac{-2\pi}{\Lambda}\mathbf{x}}] \tag{23}$$

Looking at the last term in eq.(23), it emerges how after the hologram the main part of the field is undiffracted and keeps the same value of the initial OAM. On the two first-order diffracted

Fig. 3. **a)** Schematic representation of a hologram and its action on an optical field.**b)**Action of a single-fork hologram on different input states. In the yellow box have been enlightened the modes that can be coupled to single mode fiber for analysis.

equal to *mh*¯, the global phase delay has to be an integer multiple of 2*π*, which means that the

Manipulation of Photonic Orbital Angular Momentum for Quantum Information Processing 87

*<sup>s</sup>* <sup>=</sup> *<sup>m</sup><sup>λ</sup>*

The main difficulty in the SPP preparation is to properly fix and design the height *s*. Let us now demonstrate how the SPP generates beams carrying OAM equal to *mh*¯. A beam is injected on the SPP in a certain position at distance *r* from the optical axis. The local azimuthal slope

*tan<sup>θ</sup>* <sup>=</sup> *<sup>s</sup>*

Since the beam crosses a dense medium with index of refraction *n*, it will be deflected by an

For small angles, both the sine as well as the tangent functions can be approximated by the angle itself, hence leading to *α* = *θ*(*n* − 1). After the refraction there will be a non-vanishing component of linear momentum transferred from the SPP in the azimuthal direction, equal to:

*<sup>λ</sup> <sup>α</sup>* <sup>=</sup> *<sup>h</sup>*¯

Recently *adjustable* spiral phase-plates have been developed, which allow to work for multiple wavelengths in the wavelength region where the material transmits (Rotschild et al. (2004)). In addition, various values of *m* can be achieved with the same plate by tilting it. They are created by twisting a piece of cracked Plexiglas and orienting the device so that one tab of the phase plate is perpendicular to the incident light. A beam injected at the end of the crack will then produce an optical vortex because of the azimuthally varying tilt around the center of

Most of the devices usually adopted in quantum optics experiments for the manipulation of OAM states, are not able, in general, to generate pure LG modes, but rather a superposition of modes with same azimuthal mode index. However in order to achieve a pure LG mode it is possible to exploit the relation between LG and HG modes (section 1.1.4) and the possibility to convert one into each other. Such conversion is made possible through the cylindrical lenses:

In section 3.3 we have shown that since both HG and LG modes constitute a complete orthogonal set that solve the paraxial equation, linear superpositions of one set can be used to describe the other (eq.12). A HG mode, whose axes is rotated by 45◦, can be expanded into

which demonstrates the generation of a beam carrying a well defined value of OAM

*Lz* <sup>=</sup> *rp<sup>ϕ</sup>* <sup>≈</sup> *rh*

*<sup>λ</sup>* Thus the interaction between the SPP and the beam leads to a transfer of a component

*s*(*n* − 1)

*<sup>n</sup>* <sup>−</sup> <sup>1</sup> (25)

<sup>2</sup>*π<sup>r</sup>* (26)

*<sup>λ</sup>* <sup>=</sup> *mh*¯ (28)

*sin*(*α* + *θ*) = *nsinθ* (27)

physical height of the step *s* has to be:

of the spiral surface is given by:

*<sup>p</sup><sup>ϕ</sup>* <sup>≈</sup> *<sup>h</sup><sup>α</sup>*

the phase plate.

**4.3 Cylindrical lens**

angle *α*, determinated by the Snell's law:

of orbital angular momentum given by:

Beijersbergen et al. (1993); Courtial & Padgett (1999).

mode, the beam carries a different value of OAM, equal to *m* ± 1. A schematic representation of the hologram dynamic is reported in Fig.(3).

Analogously to polarizers, the holograms are used in two ways: (i) for generating a given input quantum state of OAM; (ii) for analyzing a given OAM component of an arbitrary input quantum state. When using the holograms for generating one of the above OAM states, a TEM00 input mode is sent into the hologram and the first-order diffracted mode is used for output. The input beam must be precisely centered on the hologram pattern center. When using the holograms for analysis, the input mode, having unknown OAM quantum state, is sent through the hologram (with proper centering). The first-order diffracted output is then coupled to a single-mode fiber, which filters only the *m* = 0 state, before detection. One of the main advantage of holograms device is the possibility to generate and analyze different OAM states, not only LG modes but also their superposition. Unfortunately the efficiency of this device is still low, reaching ∼ 30% for phase holograms (higher transparency), and furthermore any measurement of OAM states in different basis implies a changing of the hologram itself, thus complicating experimental alignment.

#### **4.2 Spiral phase-plates**

The spiral phase plate (SPP) principle of function is to directly impose a phase shift on the incident light (Beijersbergen et al. (1994)). They are made by a transparent medium with index of refraction *n*, whose thickness *d* gradually varies with the azimuthal angle *ϕ*. A schematic representation of such device is reported in Fig.(4). The spiraling increasing thickness let the surface of the SPP look like a single period of helix. As the light beam crosses a dense medium, it is slowed down and hence with the increasing thickness of the plate, there will be a longer optical path corresponding to a wider phase shift *δ*. In particular each SPP introduces a phase

Fig. 4. Schematic representation of a spiral phase plate.

shift *δ* that is function of the working wavelength *λ* and of the height of the radial step *s*:

$$
\delta = \frac{(n-1)s}{\lambda} \varphi \tag{24}
$$

where we have considered a light beam that travels in the air as surrounding medium of the SPP. The phase shift introduced, due to its dependence to the azimuthal angle, induces a helical structure to the output phase front of the beam that is, a beam carrying orbital angular momentum. In order to design a device able to generate beams with a fixed value of OAM equal to *mh*¯, the global phase delay has to be an integer multiple of 2*π*, which means that the physical height of the step *s* has to be:

$$s = \frac{m\lambda}{n-1} \tag{25}$$

The main difficulty in the SPP preparation is to properly fix and design the height *s*. Let us now demonstrate how the SPP generates beams carrying OAM equal to *mh*¯. A beam is injected on the SPP in a certain position at distance *r* from the optical axis. The local azimuthal slope of the spiral surface is given by:

$$
tan\theta = \frac{s}{2\pi r}\tag{26}$$

Since the beam crosses a dense medium with index of refraction *n*, it will be deflected by an angle *α*, determinated by the Snell's law:

$$
\sin(\mathfrak{a} + \theta) = n \sin \theta \tag{27}
$$

For small angles, both the sine as well as the tangent functions can be approximated by the angle itself, hence leading to *α* = *θ*(*n* − 1). After the refraction there will be a non-vanishing component of linear momentum transferred from the SPP in the azimuthal direction, equal to: *<sup>p</sup><sup>ϕ</sup>* <sup>≈</sup> *<sup>h</sup><sup>α</sup> <sup>λ</sup>* Thus the interaction between the SPP and the beam leads to a transfer of a component of orbital angular momentum given by:

$$L\_z = rp\_\varphi \approx \frac{rh}{\lambda}\alpha = \hbar \frac{s(n-1)}{\lambda} = m\hbar\tag{28}$$

which demonstrates the generation of a beam carrying a well defined value of OAM

Recently *adjustable* spiral phase-plates have been developed, which allow to work for multiple wavelengths in the wavelength region where the material transmits (Rotschild et al. (2004)). In addition, various values of *m* can be achieved with the same plate by tilting it. They are created by twisting a piece of cracked Plexiglas and orienting the device so that one tab of the phase plate is perpendicular to the incident light. A beam injected at the end of the crack will then produce an optical vortex because of the azimuthally varying tilt around the center of the phase plate.

#### **4.3 Cylindrical lens**

12 Will-be-set-by-IN-TECH

mode, the beam carries a different value of OAM, equal to *m* ± 1. A schematic representation

Analogously to polarizers, the holograms are used in two ways: (i) for generating a given input quantum state of OAM; (ii) for analyzing a given OAM component of an arbitrary input quantum state. When using the holograms for generating one of the above OAM states, a TEM00 input mode is sent into the hologram and the first-order diffracted mode is used for output. The input beam must be precisely centered on the hologram pattern center. When using the holograms for analysis, the input mode, having unknown OAM quantum state, is sent through the hologram (with proper centering). The first-order diffracted output is then coupled to a single-mode fiber, which filters only the *m* = 0 state, before detection. One of the main advantage of holograms device is the possibility to generate and analyze different OAM states, not only LG modes but also their superposition. Unfortunately the efficiency of this device is still low, reaching ∼ 30% for phase holograms (higher transparency), and furthermore any measurement of OAM states in different basis implies a changing of the

The spiral phase plate (SPP) principle of function is to directly impose a phase shift on the incident light (Beijersbergen et al. (1994)). They are made by a transparent medium with index of refraction *n*, whose thickness *d* gradually varies with the azimuthal angle *ϕ*. A schematic representation of such device is reported in Fig.(4). The spiraling increasing thickness let the surface of the SPP look like a single period of helix. As the light beam crosses a dense medium, it is slowed down and hence with the increasing thickness of the plate, there will be a longer optical path corresponding to a wider phase shift *δ*. In particular each SPP introduces a phase

shift *δ* that is function of the working wavelength *λ* and of the height of the radial step *s*:

*<sup>δ</sup>* <sup>=</sup> (*<sup>n</sup>* <sup>−</sup> <sup>1</sup>)*<sup>s</sup>*

where we have considered a light beam that travels in the air as surrounding medium of the SPP. The phase shift introduced, due to its dependence to the azimuthal angle, induces a helical structure to the output phase front of the beam that is, a beam carrying orbital angular momentum. In order to design a device able to generate beams with a fixed value of OAM

*<sup>λ</sup> <sup>ϕ</sup>* (24)

Fig. 4. Schematic representation of a spiral phase plate.

**-**


of the hologram dynamic is reported in Fig.(3).

hologram itself, thus complicating experimental alignment.

**4.2 Spiral phase-plates**

Most of the devices usually adopted in quantum optics experiments for the manipulation of OAM states, are not able, in general, to generate pure LG modes, but rather a superposition of modes with same azimuthal mode index. However in order to achieve a pure LG mode it is possible to exploit the relation between LG and HG modes (section 1.1.4) and the possibility to convert one into each other. Such conversion is made possible through the cylindrical lenses: Beijersbergen et al. (1993); Courtial & Padgett (1999).

In section 3.3 we have shown that since both HG and LG modes constitute a complete orthogonal set that solve the paraxial equation, linear superpositions of one set can be used to describe the other (eq.12). A HG mode, whose axes is rotated by 45◦, can be expanded into

a phase difference between two successive terms equal to:

for lens with distance *d* = *f* /

*i*

lenses.

**4.4 Spatial light modulators**

*θ* = 2[*arctan*


/2**--** 1

polarization to left one by introducing a *π* shift between components.

*<sup>β</sup>* <sup>−</sup> *arctanβ*] , *<sup>β</sup>* <sup>=</sup>

with *<sup>f</sup>* focal length of the lenses. The condition *<sup>θ</sup>* <sup>=</sup> *<sup>π</sup>*/2 is fulfilled for *<sup>p</sup>* <sup>=</sup> <sup>−</sup><sup>1</sup> <sup>+</sup> <sup>√</sup>2 that is,

Manipulation of Photonic Orbital Angular Momentum for Quantum Information Processing 89

restrict the astigmatism only in the region between the two lenses leads to *zRx* = *pd* = *f* − *d* and *zRy* = *d*/*p* = *f* + *d*, as Rayleigh range for the input beam. As shown in Fig.(6-**a**), these conditions allow to implement experimentally the *π*/2-converter, which can be adopted also as converter from LG to HG modes simply injecting a LG beam and removing the

*<sup>k</sup>* factor. Analogously to the action of a quarter waveplate (QWP) on polarization states, converting linear polarized states in circular polarized ones by introducing a *π*/2 phase difference between the linear]y polarized components, the *π*/2-converter can be considered as the corresponding device for OAM states, converting a HG to LG by introducing a *π*/2 phase difference between the HG components. In order to implement the *π*-converter, it


Fig. 6. Schematic representation of **a**: *π*/2-converter and **b**: *π*-converter adopting cylindrical

is possible to set *θ* = *π* which leads to *p* = 0 and thus *d* = *f* and a collimated beam (*zRy* → ∞). The representation of this converter, that allows to change the sign of *m*, is reported in Fig.(6-**b**). Once again, a full analogy with a polarization device, the half waveplate (HWP), can be drawn. Indeed the *π*-converter is similar to the HWP which converts right-circular

The Spatial Light Modulator (SLM) is a device, based on liquid crystals, able to modulate the phase of a light beam or both the intensity and the phase simultaneously (Yoshida et al. (1996)). Besides applications for optical manipulation, SLM are extensively adopted in holographic data storage, and in display technology. A scratch of the SLM structure is reported in Fig.(7). The pure phase SLM adopted for quantum optics experiments are reflective devices in which liquid crystal (LC) is controlled by a direct and accurate voltage, and can modulate a wavefront of light beam (liquid crystal on silicon technology - LCOS). Tha main structure of a SLM is composed by a transparent glass coated with a transparent electrode, then a LC

**--**

 <sup>1</sup> <sup>−</sup> *<sup>d</sup> f* 1 + *<sup>d</sup> f*

<sup>√</sup>2. The imposition of the mode-matching condition in order to

(30)

the same constituent set:

$$u\_{\hbar,\hbar}^{\rm HG}(\frac{x+y}{\sqrt{2}},\frac{x-y}{\sqrt{2}},z) = \sum\_{k=0}^{N} b(n,m,k) u\_{\hbar+\hbar-k,k}^{\rm HG}(x,y,z) \tag{29}$$

where all terms are in phase. A device that is able to introduce the phase factor *i <sup>k</sup>*, a phase shift of *π*/2 between successive terms in eq.(29), will allow the experimental conversion of HG modes in LG modes. If after implementing this *<sup>π</sup>* <sup>2</sup> -converter, a *π* phase shift is introduced, then it is possible to change the reverse the helicity that is, the OAM sign. In Fig.(5) we report some examples of the transformation between Laguerre-Gauss and Hermite-Gauss modes. It has

Fig. 5. Examples of transformations between HG and LG modes.

been experimentally demonstrated that a system of cylindrical lenses can implement both the *π* <sup>2</sup> -converter and the *π*-converter: Alekseev et al. (1998); Beijersbergen et al. (1993); Courtial & Padgett (1999); Padgett & Allen (2002). Indeed it is possible to exploit the difference in Gouy phase shift between two HG astigmatic modes that take place in a HG mode focused in a cylindrical lens. When we deal with a non-astigmatic beam, the Gouy phase term is expressed by (*n*˜ + *m*˜ + 1)*ς*(*z*). However for astigmatic beams the astigmatism can be characterized in terms of different Rayleigh ranges on planes *xz* (*zRx* ) and *yz* (*zRy* ), thus leading to a Gouy phase:

$$(\mathfrak{n} + \frac{1}{2}) \arctan(\frac{z}{z\_{\mathcal{R}\_x}}) + (\mathfrak{m} + \frac{1}{2}) \arctan(\frac{z}{z\_{\mathcal{R}\_y}})$$

It has been demonstrated by Beijersbergen et al. (1993) that a system of two identical cylindrical lenses at distance ±*d* from the waist of the diagonal HG input beam introduces a phase difference between two successive terms equal to:

14 Will-be-set-by-IN-TECH

*N* ∑ *k*=0

shift of *π*/2 between successive terms in eq.(29), will allow the experimental conversion of HG

it is possible to change the reverse the helicity that is, the OAM sign. In Fig.(5) we report some examples of the transformation between Laguerre-Gauss and Hermite-Gauss modes. It has

**-**

 

**-**


been experimentally demonstrated that a system of cylindrical lenses can implement both the

<sup>2</sup> -converter and the *π*-converter: Alekseev et al. (1998); Beijersbergen et al. (1993); Courtial & Padgett (1999); Padgett & Allen (2002). Indeed it is possible to exploit the difference in Gouy phase shift between two HG astigmatic modes that take place in a HG mode focused in a cylindrical lens. When we deal with a non-astigmatic beam, the Gouy phase term is expressed by (*n*˜ + *m*˜ + 1)*ς*(*z*). However for astigmatic beams the astigmatism can be characterized in terms of different Rayleigh ranges on planes *xz* (*zRx* ) and *yz* (*zRy* ), thus leading to a Gouy

)+(*m*˜ +

It has been demonstrated by Beijersbergen et al. (1993) that a system of two identical cylindrical lenses at distance ±*d* from the waist of the diagonal HG input beam introduces

1 2 )*arctan*( *<sup>z</sup>*

*zRy* )

where all terms are in phase. A device that is able to introduce the phase factor *i*

*b*(*n*, *m*, *k*)*uHG*

> 

> 

*<sup>n</sup>*˜+*m*˜ <sup>−</sup>*k*,*k*(*x*, *<sup>y</sup>*, *<sup>z</sup>*) (29)

<sup>2</sup> -converter, a *π* phase shift is introduced, then

  *<sup>k</sup>*, a phase

the same constituent set:



*π*

phase:

*uHG <sup>n</sup>*˜,*m*˜ (

modes in LG modes. If after implementing this *<sup>π</sup>*

*x* + *y* <sup>√</sup><sup>2</sup> ,

(*n*˜ + 1 2 )*arctan*( *<sup>z</sup>*

*zRx*

Fig. 5. Examples of transformations between HG and LG modes.

 


*x* − *y* <sup>√</sup><sup>2</sup> , *<sup>z</sup>*) =

$$\theta = 2[\arctan\frac{1}{\beta} - \arctan\beta] \quad , \ \beta = \sqrt{\frac{1 - \frac{d}{f}}{1 + \frac{d}{f}}}\tag{30}$$

with *<sup>f</sup>* focal length of the lenses. The condition *<sup>θ</sup>* <sup>=</sup> *<sup>π</sup>*/2 is fulfilled for *<sup>p</sup>* <sup>=</sup> <sup>−</sup><sup>1</sup> <sup>+</sup> <sup>√</sup>2 that is, for lens with distance *d* = *f* / <sup>√</sup>2. The imposition of the mode-matching condition in order to restrict the astigmatism only in the region between the two lenses leads to *zRx* = *pd* = *f* − *d* and *zRy* = *d*/*p* = *f* + *d*, as Rayleigh range for the input beam. As shown in Fig.(6-**a**), these conditions allow to implement experimentally the *π*/2-converter, which can be adopted also as converter from LG to HG modes simply injecting a LG beam and removing the *i <sup>k</sup>* factor. Analogously to the action of a quarter waveplate (QWP) on polarization states, converting linear polarized states in circular polarized ones by introducing a *π*/2 phase difference between the linear]y polarized components, the *π*/2-converter can be considered as the corresponding device for OAM states, converting a HG to LG by introducing a *π*/2 phase difference between the HG components. In order to implement the *π*-converter, it

Fig. 6. Schematic representation of **a**: *π*/2-converter and **b**: *π*-converter adopting cylindrical lenses.

is possible to set *θ* = *π* which leads to *p* = 0 and thus *d* = *f* and a collimated beam (*zRy* → ∞). The representation of this converter, that allows to change the sign of *m*, is reported in Fig.(6-**b**). Once again, a full analogy with a polarization device, the half waveplate (HWP), can be drawn. Indeed the *π*-converter is similar to the HWP which converts right-circular polarization to left one by introducing a *π* shift between components.

#### **4.4 Spatial light modulators**

The Spatial Light Modulator (SLM) is a device, based on liquid crystals, able to modulate the phase of a light beam or both the intensity and the phase simultaneously (Yoshida et al. (1996)). Besides applications for optical manipulation, SLM are extensively adopted in holographic data storage, and in display technology. A scratch of the SLM structure is reported in Fig.(7). The pure phase SLM adopted for quantum optics experiments are reflective devices in which liquid crystal (LC) is controlled by a direct and accurate voltage, and can modulate a wavefront of light beam (liquid crystal on silicon technology - LCOS). Tha main structure of a SLM is composed by a transparent glass coated with a transparent electrode, then a LC

a piece of fabric. Once the q-plate is assembled, the molecules of the LC get oriented following

Manipulation of Photonic Orbital Angular Momentum for Quantum Information Processing 91

Fig. 8. **a)**Structure of a q-plate device and **b)** Photography of the singularity in a real q-plate device. The picture has been taken by inserting the q-plate between two polaroid films.

the pattern drawn on the glass. In this way the QP represents an optical device where for each point there is an optical axis in a different position. The specific pattern drawn on the q-plate defines the "charge" *q* of the singularity that characterize the q-plate. The value of *q* could be

Assuming the normal incidence for the beam of light that crosses the QP, the angle *α* that defines the local optical axis respect to the singularity of the q-plate is a linear function of the

where *q* is the topological charge, and *α*<sup>0</sup> a constant. The local direction of the optical axis expressed in eq.(31) enlightens the presence of a topological defect at *r* = 0 that is, at the plane

The working principle of the q-plate is based on the coupling between spin and orbital angular momenta in the NLC in order to exploit an effect similar to the "Pancharatnam-Berry phase" (PBP), already known for the polarization degree of freedom. Such effect is observed when a beam of light undergoes a continuous sequence of transformation on the Poincaré sphere following a closed path. As result, the final wave acquires a phase shift, known as Pancharatnam-Berry phase, that depends on the geometry of the closed path. It has been demonstrated that an analogous effect can be observed in the transverse plane when a wave undergoes a sequence of inhomogeneous polarization transformations having an initial and final homogeneous polarization state. As result, the final beam acquires an inhomogeneous geometrical phase reflected in an overall wave front reshaping, as shown by Marrucci et al.

In a single-photon quantum formalism, the QP implements the following quantum

−−→ |*R*�*π*|*m* + 2�*<sup>o</sup>*

*QP*

*QP*

where |·�*<sup>π</sup>* and |·�*<sup>o</sup>* stand for the photon quantum state 'kets' in the polarization and OAM degrees of freedom, and *L* and *R* denote the left and right circular polarization states,



*α*(*r*, *ϕ*) = *qϕ* + *α*<sup>0</sup> (31)

−−→ |*L*�*π*|*m* − 2�*<sup>o</sup>* (32)


**-**


(Marrucci et al. (2011))

azimuthal angle *ϕ*:

origin.

(2006b).

an integer or either a half-integer.

transformations on the single photon state:

**-**

Fig. 7. Schematic representation of a spatial light modulator.

layer and a LCOS backplane, connected to a controller driven by the computer. The LCOS backplane is composed by an array of aluminum pixels, which serve as both reflective mirrors and electrodes. Each electrode is an independently controllable pixel, so that a high-resolution phase modulating array is obtained. Optical modulation is achieved by applying a voltage across the LC layer from the backplane pixels to the transparent electrode on the cover glass. Switching on/off the different pixels composing the backplane, it is then possible to change the refractive index of the LC over them. In this way is possible to create different patterns, like the fork hologram, that can be dynamically modified on demand. The amount of the phase shift depends primarily on three factors: the extraordinary index of the LC material, the thickness of the LC layer, and the wavelength of the input light. As an electric field is applied to a nematic LC layer, there is a corresponding reduction in the extraordinary index of the LC material, and a reduction in the phase shift induced. The main advantage adopting the SLM is then the possibility to work with a dynamic hologram on the same setup. On the other hand, the main disadvantage is the low contrast actually achievable, which is reflected in the efficiency of the device, typically around 10% − 50%.

#### **4.5 The q-plate**

When a light beam interacts with matter, a transfer of angular momentum *J* can take place, obeying to the conservation of the global angular momentum of the system. In particular a photon absorbed by a medium can transfer only the spinorial component of angular momentum in *anisotropic* media, while the orbital component can be transferred in *inhomogeneous isotropic* transparent media: Beijersbergen et al. (1993; 1994); Beth (1936). A simultaneous exchange of both spinorial and orbital component of angular momentum is then expected to take place in a medium which is at the same time anisotropic and inhomogeneous. This is precisely the property of liquid crystals. Their structure is determined by the director axis, along witch all the molecules tend to be oriented. By applying electric or magnetic field, or by varying the temperature, it is possible to change the order parameter of the system, and hence change the properties of the LC.

A q-plate (QP) is a birefringent slab having a suitably patterned transverse optical axis, with a topological singularity at its center developed in Naples by Marrucci et al. (2006a). The device is composed by two thin transparent glasses, where in between are inserted some drops of a nematic liquid crystal. The inner surface of the glasses is coated with a polymide for planar alignment, so that it is possible to draw a desired pattern on the glass by simply pressing with 16 Will-be-set-by-IN-TECH

layer and a LCOS backplane, connected to a controller driven by the computer. The LCOS backplane is composed by an array of aluminum pixels, which serve as both reflective mirrors and electrodes. Each electrode is an independently controllable pixel, so that a high-resolution phase modulating array is obtained. Optical modulation is achieved by applying a voltage across the LC layer from the backplane pixels to the transparent electrode on the cover glass. Switching on/off the different pixels composing the backplane, it is then possible to change the refractive index of the LC over them. In this way is possible to create different patterns, like the fork hologram, that can be dynamically modified on demand. The amount of the phase shift depends primarily on three factors: the extraordinary index of the LC material, the thickness of the LC layer, and the wavelength of the input light. As an electric field is applied to a nematic LC layer, there is a corresponding reduction in the extraordinary index of the LC material, and a reduction in the phase shift induced. The main advantage adopting the SLM is then the possibility to work with a dynamic hologram on the same setup. On the other hand, the main disadvantage is the low contrast actually achievable, which is reflected

When a light beam interacts with matter, a transfer of angular momentum *J* can take place, obeying to the conservation of the global angular momentum of the system. In particular a photon absorbed by a medium can transfer only the spinorial component of angular momentum in *anisotropic* media, while the orbital component can be transferred in *inhomogeneous isotropic* transparent media: Beijersbergen et al. (1993; 1994); Beth (1936). A simultaneous exchange of both spinorial and orbital component of angular momentum is then expected to take place in a medium which is at the same time anisotropic and inhomogeneous. This is precisely the property of liquid crystals. Their structure is determined by the director axis, along witch all the molecules tend to be oriented. By applying electric or magnetic field, or by varying the temperature, it is possible to change the order parameter of the system, and

A q-plate (QP) is a birefringent slab having a suitably patterned transverse optical axis, with a topological singularity at its center developed in Naples by Marrucci et al. (2006a). The device is composed by two thin transparent glasses, where in between are inserted some drops of a nematic liquid crystal. The inner surface of the glasses is coated with a polymide for planar alignment, so that it is possible to draw a desired pattern on the glass by simply pressing with

**-** 

 

**-** 

Fig. 7. Schematic representation of a spatial light modulator.

in the efficiency of the device, typically around 10% − 50%.

hence change the properties of the LC.

**4.5 The q-plate**

**--**

**-**  a piece of fabric. Once the q-plate is assembled, the molecules of the LC get oriented following

Fig. 8. **a)**Structure of a q-plate device and **b)** Photography of the singularity in a real q-plate device. The picture has been taken by inserting the q-plate between two polaroid films. (Marrucci et al. (2011))

the pattern drawn on the glass. In this way the QP represents an optical device where for each point there is an optical axis in a different position. The specific pattern drawn on the q-plate defines the "charge" *q* of the singularity that characterize the q-plate. The value of *q* could be an integer or either a half-integer.

Assuming the normal incidence for the beam of light that crosses the QP, the angle *α* that defines the local optical axis respect to the singularity of the q-plate is a linear function of the azimuthal angle *ϕ*:

$$
\mathfrak{a}(r,\varphi) = \mathfrak{q}\varphi + \mathfrak{a}\_0 \tag{31}
$$

where *q* is the topological charge, and *α*<sup>0</sup> a constant. The local direction of the optical axis expressed in eq.(31) enlightens the presence of a topological defect at *r* = 0 that is, at the plane origin.

The working principle of the q-plate is based on the coupling between spin and orbital angular momenta in the NLC in order to exploit an effect similar to the "Pancharatnam-Berry phase" (PBP), already known for the polarization degree of freedom. Such effect is observed when a beam of light undergoes a continuous sequence of transformation on the Poincaré sphere following a closed path. As result, the final wave acquires a phase shift, known as Pancharatnam-Berry phase, that depends on the geometry of the closed path. It has been demonstrated that an analogous effect can be observed in the transverse plane when a wave undergoes a sequence of inhomogeneous polarization transformations having an initial and final homogeneous polarization state. As result, the final beam acquires an inhomogeneous geometrical phase reflected in an overall wave front reshaping, as shown by Marrucci et al. (2006b).

In a single-photon quantum formalism, the QP implements the following quantum transformations on the single photon state:

$$\begin{aligned} \langle |L\rangle\_{\pi} |m\rangle\_{o} & \xrightarrow{QP} |R\rangle\_{\pi} |m+2\rangle\_{o} \\ \langle |R\rangle\_{\pi} |m\rangle\_{o} & \xrightarrow{QP} |L\rangle\_{\pi} |m-2\rangle\_{o} \end{aligned} \tag{32}$$

where |·�*<sup>π</sup>* and |·�*<sup>o</sup>* stand for the photon quantum state 'kets' in the polarization and OAM degrees of freedom, and *L* and *R* denote the left and right circular polarization states,

possibility of adopting the QP as an interface between polarization and OAM of single photon

Manipulation of Photonic Orbital Angular Momentum for Quantum Information Processing 93

The single photon transformations applied by the QP, expressed by eq. (33), describe the coupling of the OAM *m* and the polarization *π* degrees of freedom. Interestingly, this property can be exploited to generate single-particle entanglement of *π* and *m* degrees of freedom. Indeed when an input photon in a *TEM*<sup>00</sup> mode and linear polarization is injected on a q-plate,

This is an entangled state between two qubits encoded in different degrees of freedom. In particular {| + 2�*m*, | − 2�*m*} is the basis for the OAM qubit which lies in the |*m*| = 2 subspace of the infinite dimensional Hilbert space of orbital angular momentum. The property of the q-plate presented in eq.(34) has been experimentally verified through the reconstruction of the density matrix of the output state emerging from the QP. Such reconstruction is possible by exploiting the quantum state tomography technique (James et al. (2001)), whose peculiarity is to determinate the different elements of the density matrix by analyzing the state in different basis. Hence for the two-qubit quantum state reported in eq.(34), we have performed measurements both in *π* and *m* degrees of freedom. Besides the normal {| +2�*m*, | −2�*m*} OAM basis, we had to carry out measurements also in the two superposition bases {|*a*�*m*, |*d*�*m*} and {|*h*�*m*, |*v*�*m*} by means of different computer-generated holograms Langford et al. (2004). We have considered as incoming states on the QP (**a**) |*H*�*π*|0�*m*,and (**b**) |*V*�*π*|0�*m*. As predicted by the transformation introduced by the QP, when the state |*H*�*π*|0�*<sup>m</sup>*

(|*L*�*π*| − 2�*<sup>m</sup>* ± |*R*�*π*| + 2�*m*) (34)

states.

**5.1 Single-photon entanglement**


is injected, the output state in the basis {|*L*, +2�, |*R*, −2�} reads:

*<sup>ρ</sup>*<sup>ˆ</sup> <sup>=</sup> <sup>1</sup> 2 ⎛

⎞

⎟⎟⎠

⎜⎜⎝

The experimental reconstruction is reported inf Fig.(10-**a**), with a concurrence of the state equal to *C* = (0.95 ± 0.02). Indeed the average experimental concurrence is *C* = (0.96 ± 0.02), while the average purity of the states is *<sup>P</sup>* <sup>=</sup> *Trρ*<sup>2</sup> = (0.94 <sup>±</sup> 0.02) (Nagali et al. (2009)).

Due to its peculiarities, the q-plate provides a convenient way to "interface" the photon OAM with the more easily manipulated spin degree of freedom. Hence as next step we have shown that such interface can be considered as a quantum "transferrer" device, which allows to transfer coherently the quantum information from the polarization *π* to the OAM *m* degree of freedom, and *vice versa*. In this Section, we present a complete description of two optical schemes, which have been shown in Nagali et al. (2009), that enable a qubit of quantum information to be transferred from the polarization to the OAM ( *transferrer π* → *o*2), from OAM to polarization ( *transferrer o*<sup>2</sup> → *π*). Moreover, we tested also the combination of these

� *QP* → 1 √2

the output state reads:

**5.2 Quantum transferrers**

respectively. In the following, whenever there is no risk of ambiguity, the subscripts *π* and *o* will be omitted for brevity. In Fig.(9) is reported a schematic representation of the QP action. Any coherent superposition of the two input states given in Eq. (32) is expected

Fig. 9. Schematic representation of the action of a q-plate device with *δ* = *π* and *q* = 1, like the ones adopted in our experiments (Marrucci et al. (2011)).

to be preserved by the QP transformation, leading to the equivalent superposition of the corresponding output states, see Nagali et al. (2009). Explicitly, we have

$$a|L\rangle\_{\pi}|m\rangle\_{o} + \beta|R\rangle\_{\pi}|m\rangle\_{o} \xrightarrow{QP} a|R\rangle\_{\pi}|m+2\rangle\_{o} + \beta|L\rangle\_{\pi}|m-2\rangle\_{o} \tag{33}$$

These equations completely define the ideal behavior of the QP on the OAM and polarization subspaces of the photon.

Let us observe that the efficiency of the q-plate device, close to 85%, is related to the birefringent retardation *δ* introduced by the q-plate itself. An ideal q-plate should have *δ* uniform across the device and equal to *δ* = *π*, in order to act on the polarization degree of freedom as a perfect half-waveplate. As have been shown, once a liquid crystal QP is assembled, the birefringent retardation can be tuned either by mechanical compression (exploiting the elasticity of the spacers that fix the thickness of the liquid crystal cell) or by temperature control Karimi et al. (2009), in order to reach the desired value of *δ*. From eq.(33) it is possible to conclude that for *δ* = *π*, a QP modifies the OAM state *m* of a light beam crossing it, imposing a variation Δ*m* = ±2*q* whose sign depends on the input polarization, positive for left-circular and negative for right-circular. The handedness of the output circular polarization is also inverted, i.e. the optical spin is flipped (Calvo & Picón (2007)). In the experiments that will be presented in the following, we have adopted only QPs with charge *q* = 1 and *δ* � *π*. Hence, an input TEM00 mode (having *m* = 0) is converted into a beam with *m* = ±2.

#### **5. Manipulation of polarization and OAM of single photons**

Here we briefly review the experimental results on the adoption of the q-plate device in the single-photon regime.The spin-orbit coupling has been exploited in order to demonstrate the possibility of adopting the QP as an interface between polarization and OAM of single photon states.

#### **5.1 Single-photon entanglement**

18 Will-be-set-by-IN-TECH

respectively. In the following, whenever there is no risk of ambiguity, the subscripts *π* and *o* will be omitted for brevity. In Fig.(9) is reported a schematic representation of the QP action. Any coherent superposition of the two input states given in Eq. (32) is expected


*-*


 *-*

Fig. 9. Schematic representation of the action of a q-plate device with *δ* = *π* and *q* = 1, like

to be preserved by the QP transformation, leading to the equivalent superposition of the

These equations completely define the ideal behavior of the QP on the OAM and polarization

Let us observe that the efficiency of the q-plate device, close to 85%, is related to the birefringent retardation *δ* introduced by the q-plate itself. An ideal q-plate should have *δ* uniform across the device and equal to *δ* = *π*, in order to act on the polarization degree of freedom as a perfect half-waveplate. As have been shown, once a liquid crystal QP is assembled, the birefringent retardation can be tuned either by mechanical compression (exploiting the elasticity of the spacers that fix the thickness of the liquid crystal cell) or by temperature control Karimi et al. (2009), in order to reach the desired value of *δ*. From eq.(33) it is possible to conclude that for *δ* = *π*, a QP modifies the OAM state *m* of a light beam crossing it, imposing a variation Δ*m* = ±2*q* whose sign depends on the input polarization, positive for left-circular and negative for right-circular. The handedness of the output circular polarization is also inverted, i.e. the optical spin is flipped (Calvo & Picón (2007)). In the experiments that will be presented in the following, we have adopted only QPs with charge *q* = 1 and *δ* � *π*. Hence, an input TEM00 mode (having *m* = 0) is converted into a beam with

Here we briefly review the experimental results on the adoption of the q-plate device in the single-photon regime.The spin-orbit coupling has been exploited in order to demonstrate the

*QP*

+

+

−−→ *α*|*R*�*π*|*m* + 2�*<sup>o</sup>* + *β*|*L*�*π*|*m* − 2�*<sup>o</sup>* (33)



*-*

**-** 

*-* the ones adopted in our experiments (Marrucci et al. (2011)).

*α*|*L*�*π*|*m*�*<sup>o</sup>* + *β*|*R*�*π*|*m*�*<sup>o</sup>*

subspaces of the photon.

*m* = ±2.

corresponding output states, see Nagali et al. (2009). Explicitly, we have

**5. Manipulation of polarization and OAM of single photons**

The single photon transformations applied by the QP, expressed by eq. (33), describe the coupling of the OAM *m* and the polarization *π* degrees of freedom. Interestingly, this property can be exploited to generate single-particle entanglement of *π* and *m* degrees of freedom. Indeed when an input photon in a *TEM*<sup>00</sup> mode and linear polarization is injected on a q-plate, the output state reads:

$$
\begin{pmatrix} |H\rangle\_{\pi} \ |0\rangle\_{m} \\ |V\rangle\_{\pi} \ |0\rangle\_{m} \end{pmatrix} \stackrel{QP}{\rightarrow} \frac{1}{\sqrt{2}} (|L\rangle\_{\pi}|-2\rangle\_{m} \pm |R\rangle\_{\pi}|+2\rangle\_{m})\tag{34}
$$

This is an entangled state between two qubits encoded in different degrees of freedom. In particular {| + 2�*m*, | − 2�*m*} is the basis for the OAM qubit which lies in the |*m*| = 2 subspace of the infinite dimensional Hilbert space of orbital angular momentum. The property of the q-plate presented in eq.(34) has been experimentally verified through the reconstruction of the density matrix of the output state emerging from the QP. Such reconstruction is possible by exploiting the quantum state tomography technique (James et al. (2001)), whose peculiarity is to determinate the different elements of the density matrix by analyzing the state in different basis. Hence for the two-qubit quantum state reported in eq.(34), we have performed measurements both in *π* and *m* degrees of freedom. Besides the normal {| +2�*m*, | −2�*m*} OAM basis, we had to carry out measurements also in the two superposition bases {|*a*�*m*, |*d*�*m*} and {|*h*�*m*, |*v*�*m*} by means of different computer-generated holograms Langford et al. (2004). We have considered as incoming states on the QP (**a**) |*H*�*π*|0�*m*,and (**b**) |*V*�*π*|0�*m*. As predicted by the transformation introduced by the QP, when the state |*H*�*π*|0�*<sup>m</sup>* is injected, the output state in the basis {|*L*, +2�, |*R*, −2�} reads:

$$
\hat{\rho} = \frac{1}{2} \begin{pmatrix} 0 \ 0 \ 0 \ 0 \\ 0 \ 1 \ 1 \ 0 \\ 0 \ 1 \ 1 \ 0 \\ 0 \ 0 \ 0 \ 0 \end{pmatrix}.
$$

The experimental reconstruction is reported inf Fig.(10-**a**), with a concurrence of the state equal to *C* = (0.95 ± 0.02). Indeed the average experimental concurrence is *C* = (0.96 ± 0.02), while the average purity of the states is *<sup>P</sup>* <sup>=</sup> *Trρ*<sup>2</sup> = (0.94 <sup>±</sup> 0.02) (Nagali et al. (2009)).

#### **5.2 Quantum transferrers**

Due to its peculiarities, the q-plate provides a convenient way to "interface" the photon OAM with the more easily manipulated spin degree of freedom. Hence as next step we have shown that such interface can be considered as a quantum "transferrer" device, which allows to transfer coherently the quantum information from the polarization *π* to the OAM *m* degree of freedom, and *vice versa*. In this Section, we present a complete description of two optical schemes, which have been shown in Nagali et al. (2009), that enable a qubit of quantum information to be transferred from the polarization to the OAM ( *transferrer π* → *o*2), from OAM to polarization ( *transferrer o*<sup>2</sup> → *π*). Moreover, we tested also the combination of these


**-**

Manipulation of Photonic Orbital Angular Momentum for Quantum Information Processing 95

Fig. 11. **Right Side -** Experimental density matrices *ρ* (the left column shows the real part and right column the imaginary part) measured for the output of the *π* → *o*<sup>2</sup> qubit transfer, for each of the three different predicted output states shown in the upper left corner of each row.**Left Side -** Experimental density matrices *ρ* (the left column shows the real part and right column the imaginary part) measured for the output of the *o*<sup>2</sup> → *π* qubit transfer, for each of the three different predicted output states shown in the upper left corner of each row.

The input arbitrary qubit is written in the polarization using two waveplates, as discussed previously. The experimental results for three specific choices of the input state are shown in Fig. (11). We find a good agreement with theory as demonstrated by the fidelity parameter, with an average fidelity value between the experimental states and the theoretical predictions

Thus, we have demonstrated experimentally that the initial information encoded in an input TEM00 state can be coherently transferred to the OAM degree of freedom, thanks to the *π* → *o*<sup>2</sup> converter, giving rise to the preparation of a qubit in the orbital angular momentum. As the initial information has been stored in the orbital part of the qubit wave-function, new information can be stored in the polarization degree of freedom, allowing the transportation

Let us now show that the reverse process can be realized as well, by transferring a qubit initially encoded in the OAM subspace *o*<sup>2</sup> into the polarization space. We therefore consider


in a single photon of a higher amount, at least two qubits, of information.

as initial quantum state of the photon the following one:


**-**

(Nagali et al. (2009))

equal to *F* = (97.7 ± 0.2)%.

**5.2.2 Quantum transferrer** *o*<sup>2</sup> → *π*

[] []

 **02** -

[] []


**02**

Fig. 10. Experimental density matrices (real and imaginary parts) for the single photon entangled state (Nagali et al. (2009)). The computational values {0, 1} are associated to the {|*R*�, |*L*�} polarization states, and to {| + 2�, | − 2�} for the orbital angular momentum *m* for the first and the second qubit, respectively. The incoming state on the QP is (**a**) |*H*�*π*|0�*m*,and (**b**) |*V*�*π*|0�*m*.

two schemes, thus realizing the *bidirectional transfer* polarization-OAM-polarization ( *π* → *o*<sup>2</sup> → *π*). The latter demonstration is equivalent to demonstrate quantum communication using OAM for encoding the message. In other words, the qubit is initially prepared in the polarization space, then passed to OAM in a transmitting unit (Alice), sent to a receiving unit (Bob), where it is transferred back to polarization for further processing or detection.
