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

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 as initial quantum state of the photon the following one:

$$|\Psi\rangle\_{\rm in} = |H\rangle\_{\pi}|\varphi\rangle\_{\vartheta\_2} = |H\rangle(\mathfrak{a}|+\mathfrak{2}\rangle+\mathfrak{z}|-\mathfrak{2}\rangle) \tag{39}$$

through the *π* → *o*<sup>2</sup> transferrer.

i.e. √ 1 2

state:

BBO

**(A)**

kA

**6.1 Experimental apparatus and generation of hybrid states**

1 √2

(|*H*�*<sup>A</sup>*

*<sup>π</sup>* <sup>|</sup> <sup>+</sup> <sup>2</sup>�*<sup>B</sup>*

kB

CP

**(B) (C)**

**Hybrid Entangled state Generation**

QP QP

Manipulation of Photonic Orbital Angular Momentum for Quantum Information Processing 97

**Analysis**

QWP

Fig. 12. Experimental setup adopted for the generation and characterization of hybrid *π*-OAM entangled states (Nagali & Sciarrino (2005)).**(A)** Generation of polarization

entangled photons on modes *kA* and *kB*.**(B)** Projection on the OAM state with *m* = 0 through the coupling on a single mode fiber (SMF).**(C)**Encoding of the state in the OAM subspace *o*<sup>2</sup>

Let us now describe the experimental layout shown in Fig.(12). A 1.5*mm* thick *β*-barium borate crystal (BBO) cut for type-II phase matching Kwiat et al. (1995), is pumped by the second harmonic of a Ti:Sa mode-locked laser beam, and generates via spontaneous parametric fluorescence polarization entangled photon pairs on modes *kA* and *kB* with wavelength *λ* = 795 nm, and pulse bandwidth Δ*λ* = 4.5 nm, as determined by two interference filters

a 0.75 mm thick BBO crystal on each output mode *kA* and *kB* Kwiat et al. (1995). Thus the source generates photon pair in the singlet entangled state encoded in the polarization,

standard polarization analysis setup and then coupled to a single mode fiber connected to the single-photon counter modules (SPCM) *DA*. The photon generated on mode *kB* is coupled to a single mode fiber, in order to collapse its transverse spatial mode into a pure TEM00, corresponding to OAM *m* = 0. After the fiber output, two waveplates compensate (CP) the polarization rotation introduced by the fiber. To transform the polarization entangled pairs into an hybrid entangled state the photon *B* is sent through the quantum transferrer *π* → *o*2, which transfers the polarization quantum states in the OAM degree of freedom. After the transferrer operation the polarization entangled state is transformed into the hybrid entangled

*<sup>o</sup>*<sup>2</sup> − |*V*�*<sup>A</sup>*

In order to analyze with high efficiency the OAM degree of freedom, we exploited the *o*<sup>2</sup> → *π* transferrer. By this approach any measurement on the OAM state is achieved by measuring the polarization after the transferrer device, as shown in Fig.12. Finally the photon has been coupled to a single mode fiber and then detected by *DB* connected to the coincidence box (CB), which records the coincidence counts between [*DA*, *DB*]. We observed a final coincidence

*<sup>π</sup>* | − <sup>2</sup>�*<sup>B</sup>*

*<sup>o</sup>*<sup>2</sup> )|0�*<sup>A</sup>*

*<sup>o</sup>* <sup>|</sup>*H*�*<sup>B</sup>*

(|*H*�*A*|*V*�*<sup>B</sup>* − |*V*�*A*|*H*�*B*). The photon generated on mode *kA* is sent through a

(IF). The spatial and temporal walk-off is compensated by inserting a *<sup>λ</sup>*

PBS

HWP

PBS PBS

QWP

HWP

**OAM Analysis**

**CB**

DB

<sup>2</sup> waveplate and

*<sup>π</sup>* (42)

DA

By injecting the state |Ψ�*in* in the q-plate device, and then rotating the output state by means of a pair of waveplates, we obtain the following state:

$$\frac{1}{2}\{\mathfrak{a}|V\rangle|+4\rangle+\mathfrak{a}|H\rangle|0\rangle+\mathfrak{z}|V\rangle|0\rangle+\mathfrak{z}|H\rangle|-4\rangle\}\tag{40}$$

Now, by coupling the beam to a single mode fiber, only the states with *m* = 0 that is, the TEM00 modes, will be efficiently transmitted. Of course, this implies that a probabilistic process is obtained again, since we discard all the contributions with *m* �= 0 (ideally, again *p* = 50%). After the fiber, the output state reads:

$$|\Psi\rangle\_{out} = (\mathfrak{a}|H\rangle + \mathfrak{z}|V\rangle)|0\rangle = |\mathfrak{q}\rangle\_{\pi}|0\rangle\_{\mathfrak{o}} \tag{41}$$

which demonstrates the successful conversion from the OAM degree of freedom to the polarization one. The experimental results for three cases are shown in Fig.(11). We find again a good agreement with theory, with an average fidelity *F* = (97.3 ± 0.2)%.

#### **6. Hybrid entanglement**

Hybrid entangled states exhibit entanglement between different degrees of freedom of a particle pair. The generation of such states can be useful for asymmetric optical quantum network where the different communication channels adopted for transmitting quantum information exhibit different properties. In such a way one could adopt the suitable degree of freedom with larger robustness along the channel. From a fundamental point of view, the observation of non-locality with hybrid systems proves the fundamental independence of entanglement from the physical realization of the adopted Hilbert space. Very recently the hybrid entanglement of photon pairs between the path (linear momentum) of one photon and the polarization of the other photon has been reported by two different techniques (Ma et al. (2009); Neves et al. (2009)). Nevertheless, the capability of generating hybrid-entangled state encoded in the polarization and OAM of single photons could be advantageous since it could allow the engineering of qubit-qudit entangled states, related to the different Hilbert space dimensionality of the two degrees of freedom. It has been pointed out that such states are desiderable for quantum information and communication protocols, as quantum teleportation, and for the possibility to send quantum information through an optical quantum network composed by optical fiber channels and free-space (Chen & She (2009); Neves et al. (2009)).

In this section we review the realization of hybrid polarization-OAM entangled states, by adopting the deterministic polarization-OAM transferrer described in the previous Section. Polarization entangled photon pairs are created by spontaneous parametric down conversion, the spatial profile of the twin photons is filtered through single mode fibers and finally the polarization is coherently transferred to OAM state for one photon. A complete characterization of the hybrid entangled quantum states has been carried out by adopting the quantum state tomography technique. This result, together with the achieved generation rate, the easiness of alignment and the high quality of the generated state, can make this optical source a powerful tool for advanced quantum information tasks and has been presented in Nagali & Sciarrino (2005).

22 Will-be-set-by-IN-TECH

By injecting the state |Ψ�*in* in the q-plate device, and then rotating the output state by means

Now, by coupling the beam to a single mode fiber, only the states with *m* = 0 that is, the TEM00 modes, will be efficiently transmitted. Of course, this implies that a probabilistic process is obtained again, since we discard all the contributions with *m* �= 0 (ideally, again *p* = 50%).

which demonstrates the successful conversion from the OAM degree of freedom to the polarization one. The experimental results for three cases are shown in Fig.(11). We find

Hybrid entangled states exhibit entanglement between different degrees of freedom of a particle pair. The generation of such states can be useful for asymmetric optical quantum network where the different communication channels adopted for transmitting quantum information exhibit different properties. In such a way one could adopt the suitable degree of freedom with larger robustness along the channel. From a fundamental point of view, the observation of non-locality with hybrid systems proves the fundamental independence of entanglement from the physical realization of the adopted Hilbert space. Very recently the hybrid entanglement of photon pairs between the path (linear momentum) of one photon and the polarization of the other photon has been reported by two different techniques (Ma et al. (2009); Neves et al. (2009)). Nevertheless, the capability of generating hybrid-entangled state encoded in the polarization and OAM of single photons could be advantageous since it could allow the engineering of qubit-qudit entangled states, related to the different Hilbert space dimensionality of the two degrees of freedom. It has been pointed out that such states are desiderable for quantum information and communication protocols, as quantum teleportation, and for the possibility to send quantum information through an optical quantum network composed by optical fiber channels and free-space (Chen & She (2009);

In this section we review the realization of hybrid polarization-OAM entangled states, by adopting the deterministic polarization-OAM transferrer described in the previous Section. Polarization entangled photon pairs are created by spontaneous parametric down conversion, the spatial profile of the twin photons is filtered through single mode fibers and finally the polarization is coherently transferred to OAM state for one photon. A complete characterization of the hybrid entangled quantum states has been carried out by adopting the quantum state tomography technique. This result, together with the achieved generation rate, the easiness of alignment and the high quality of the generated state, can make this optical source a powerful tool for advanced quantum information tasks and has been presented in

again a good agreement with theory, with an average fidelity *F* = (97.3 ± 0.2)%.

{*α*|*V*�| + 4� + *α*|*H*�|0� + *β*|*V*�|0� + *β*|*H*�| − 4�} (40)


of a pair of waveplates, we obtain the following state:

1 2

After the fiber, the output state reads:

**6. Hybrid entanglement**

Neves et al. (2009)).

Nagali & Sciarrino (2005).

Fig. 12. Experimental setup adopted for the generation and characterization of hybrid *π*-OAM entangled states (Nagali & Sciarrino (2005)).**(A)** Generation of polarization entangled photons on modes *kA* and *kB*.**(B)** Projection on the OAM state with *m* = 0 through the coupling on a single mode fiber (SMF).**(C)**Encoding of the state in the OAM subspace *o*<sup>2</sup> through the *π* → *o*<sup>2</sup> transferrer.

#### **6.1 Experimental apparatus and generation of hybrid states**

Let us now describe the experimental layout shown in Fig.(12). A 1.5*mm* thick *β*-barium borate crystal (BBO) cut for type-II phase matching Kwiat et al. (1995), is pumped by the second harmonic of a Ti:Sa mode-locked laser beam, and generates via spontaneous parametric fluorescence polarization entangled photon pairs on modes *kA* and *kB* with wavelength *λ* = 795 nm, and pulse bandwidth Δ*λ* = 4.5 nm, as determined by two interference filters (IF). The spatial and temporal walk-off is compensated by inserting a *<sup>λ</sup>* <sup>2</sup> waveplate and a 0.75 mm thick BBO crystal on each output mode *kA* and *kB* Kwiat et al. (1995). Thus the source generates photon pair in the singlet entangled state encoded in the polarization, i.e. √ 1 2 (|*H*�*A*|*V*�*<sup>B</sup>* − |*V*�*A*|*H*�*B*). The photon generated on mode *kA* is sent through a standard polarization analysis setup and then coupled to a single mode fiber connected to the single-photon counter modules (SPCM) *DA*. The photon generated on mode *kB* is coupled to a single mode fiber, in order to collapse its transverse spatial mode into a pure TEM00, corresponding to OAM *m* = 0. After the fiber output, two waveplates compensate (CP) the polarization rotation introduced by the fiber. To transform the polarization entangled pairs into an hybrid entangled state the photon *B* is sent through the quantum transferrer *π* → *o*2, which transfers the polarization quantum states in the OAM degree of freedom. After the transferrer operation the polarization entangled state is transformed into the hybrid entangled state:

$$\frac{1}{\sqrt{2}}(|H\rangle\_{\pi}^{A}|+2\rangle\_{o\_{2}}^{B}-|V\rangle\_{\pi}^{A}|-2\rangle\_{o\_{2}}^{B})|0\rangle\_{o}^{A}|H\rangle\_{\pi}^{B}\tag{42}$$

In order to analyze with high efficiency the OAM degree of freedom, we exploited the *o*<sup>2</sup> → *π* transferrer. By this approach any measurement on the OAM state is achieved by measuring the polarization after the transferrer device, as shown in Fig.12. Finally the photon has been coupled to a single mode fiber and then detected by *DB* connected to the coincidence box (CB), which records the coincidence counts between [*DA*, *DB*]. We observed a final coincidence

 **-**

*S* = *E*(**a**, **b**) + *E*(**a'**, **b**) + *E*(**a**, **b'**) − *E*(**a'**, **b'**) (43)

<sup>8</sup> )<sup>|</sup> <sup>+</sup> <sup>2</sup>� <sup>+</sup> *cos*( *<sup>π</sup>*

<sup>8</sup> )| − 2�}. Experimentally we obtained the following value

**-**

Fig. 14. Coincidence rate [*DA*, *DB*] measured as a function of the angle *θ* of the half wave plate on the arm *kA* for OAM detected state **(a)** | + 2� and **(b)** |*h*�*o*<sup>2</sup> (Nagali & Sciarrino (2005)). either **a** or **a'** while Bob measures **b** or **b'**, where the outcomes of each measurement are either +1 or −1. For any couple of measured observables (*A* = {**a**, **a'**}, *B* = {**b**, **b'**}), we define the

for the number of events in which the observables *A* and *B* have been found equal to the dichotomic outcomes *i* and *j*. Finally we define the parameter *S* which takes into account the

Assuming a local realistic theory, the relation |*S*| ≤ *SCHSH* = 2 holds. To carry out a non-locality test in the hybrid regime, we define the two sets of dichotomic observables for A and B. For Alice the basis **a** and **a'** correspond, respectively, to the linear polarization basis {|*H*�*π*, |*V*�*π*} and {|+�*π*, |−�*π*}. For Bob the basis **b** and **b'** correspond, respectively, to the

by carrying out a measurement with a duration of 60*s* and an average statistics per setting equal to about 1500 events: *S* = (2.51 ± 0.02). Hence a violation by more than 25 standard deviation over the value *SCHSH* = 2 is obtained. This experimental value is in good agreement with an experimental visibility of *V* = (0.930 ± 0.007) which should lead to *S* = (2.57 ± 0.02).

Among all degrees of freedom offered by single photons, the orbital angular momentum (OAM) has a great potential in the quantum information field, as it provides a natural choice for implementing single-photon qudits, the units of quantum information in a higher dimensional space. This can be an important practical advantage, as it enables higher security in quantum cryptographic protocols, as well as implications in fundamental quantum mechanics theory. Moreover, the combined use of different degrees of freedom of a photon,

The authors acknowledge the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission, under FET-Open

such as OAM and spin, enables the implementation of entirely new quantum tasks.

<sup>8</sup> )| − <sup>2</sup>�, <sup>−</sup>*sin*( *<sup>π</sup>*

 **-**

correlations for the different observables

<sup>8</sup> )<sup>|</sup> <sup>+</sup> <sup>2</sup>� − *sin*( *<sup>π</sup>*

<sup>8</sup> )<sup>|</sup> <sup>+</sup> <sup>2</sup>� − *cos*( *<sup>π</sup>*

OAM basis {*cos*( *<sup>π</sup>*

**7. Conclusion**

<sup>8</sup> )| − <sup>2</sup>�,*sin*( *<sup>π</sup>*

Grant No. 255914, PHORBITECH.

*sin*( *<sup>π</sup>*

**-**

following correlation function *E*(*A*, *B*) = *<sup>N</sup>*(+,+)+*N*(−,−)−*N*(+,−)−*N*(−,+)

Manipulation of Photonic Orbital Angular Momentum for Quantum Information Processing 99

*<sup>N</sup>*(+,+)+*N*(−,−)+*N*(+,−)+*N*(−,+) where *<sup>N</sup>*(*i*, *<sup>j</sup>*) stands

<sup>8</sup> )| − <sup>2</sup>�} and {*cos*( *<sup>π</sup>*

<sup>8</sup> )| + 2� +

Fig. 13. Experimental density matrix of the hybrid entangled state generated after the transferrer transformation on photons on *kB* mode. Each measurement setting lasted 15s (Nagali & Sciarrino (2005)).

rate equal to *C* = 100*Hz* within a coincidence window of 3 ns. This experimental data is in agreement with the expected value, determined from *Csource* = 6*kHz* after taking into account two main loss factors: hybrid state preparation probability *pprep*, and detection probability *pdet*. *pprep* depends on the conversion efficiency of the q-plate (0.80 ± 0.05) and on the probabilistic efficiency of the quantum transferrer *π* → *o*<sup>2</sup> (0.5), thus leading to *pprep* = 0.40 ± 0.03. The detection efficiency includes the q-plate conversion efficiency (0.8), the transferrer *o*<sup>2</sup> → *π* (0.5), and the single mode fiber coupling (0.2). Hence *pdet* = 0.08.

#### **6.2 Characterization of the state**

To completely characterize the state in Eq. (42) we reconstructed the density matrix of the quantum state. The tomography reconstruction requires the estimation of 16 operators James et al. (2001) through 36 separable measurements on the polarization-OAM subspaces. We carried out the reconstruction of the density matrix *<sup>ρ</sup>A*,*<sup>B</sup> <sup>π</sup>*,*o*<sup>2</sup> after the polarization-OAM conversion. The experimental results are reported in Fig.13, with the elements of the density matrices expressed in the polarization and OAM basis {|*H*, +2�, |*H*, −2�, |*V*, +2�, |*V*, −2�}. The fidelity with the singlet states <sup>|</sup>Ψ−� has been evaluated to be *<sup>F</sup>*(|Ψ−�, *<sup>ρ</sup>A*,*<sup>B</sup> <sup>π</sup>*,*o*<sup>2</sup> )=(0.957 <sup>±</sup> 0.009), while the experimental linear entropy of the state reads *SL* = (0.012 ± 0.002). A more quantitative parameter associated to the generated polarization-entangled states is given by the concurrence *C* = (0.957 ± 0.002). These values demonstrate the high degree of hybrid entanglement generation.

To further characterize the hybrid quantum states, the violation of Bell's inequalities with the two photon system have been addressed. First, we measured the photon coincidence rate as a function of the orientation of the half-wave plate on Alice arm for two different OAM basis analysis, namely {| + 2�*o*<sup>2</sup> , | − 2�*o*<sup>2</sup> } and {|*h*�*o*<sup>2</sup> , |*v*�*o*<sup>2</sup> }. The variation of the number of coincidences *N*(*θ*) with the angle *θ* is in agreement with the one expected for entangled states such as *N*(*θ*) = *N*0(1 + *cosθ*): Fig.14. The coincidence fringe visibility reaches the values *V* = (0.966 ± 0.001) and *V* = (0.930 ± 0.007). Hence, a non-locality test, the CHSH one (Clauser et al. (1969)), has been carried out. Each of two partners, A (Alice) and B (Bob) measures a dichotomic observable among two possible ones, i.e. Alice randomly measures

Fig. 14. Coincidence rate [*DA*, *DB*] measured as a function of the angle *θ* of the half wave plate on the arm *kA* for OAM detected state **(a)** | + 2� and **(b)** |*h*�*o*<sup>2</sup> (Nagali & Sciarrino (2005)).

either **a** or **a'** while Bob measures **b** or **b'**, where the outcomes of each measurement are either +1 or −1. For any couple of measured observables (*A* = {**a**, **a'**}, *B* = {**b**, **b'**}), we define the following correlation function *E*(*A*, *B*) = *<sup>N</sup>*(+,+)+*N*(−,−)−*N*(+,−)−*N*(−,+) *<sup>N</sup>*(+,+)+*N*(−,−)+*N*(+,−)+*N*(−,+) where *<sup>N</sup>*(*i*, *<sup>j</sup>*) stands for the number of events in which the observables *A* and *B* have been found equal to the dichotomic outcomes *i* and *j*. Finally we define the parameter *S* which takes into account the correlations for the different observables

$$S = E(\mathbf{a}, \mathbf{b}) + E(\mathbf{a}', \mathbf{b}) + E(\mathbf{a}, \mathbf{b}') - E(\mathbf{a}', \mathbf{b}') \tag{43}$$

Assuming a local realistic theory, the relation |*S*| ≤ *SCHSH* = 2 holds. To carry out a non-locality test in the hybrid regime, we define the two sets of dichotomic observables for A and B. For Alice the basis **a** and **a'** correspond, respectively, to the linear polarization basis {|*H*�*π*, |*V*�*π*} and {|+�*π*, |−�*π*}. For Bob the basis **b** and **b'** correspond, respectively, to the OAM basis {*cos*( *<sup>π</sup>* <sup>8</sup> )<sup>|</sup> <sup>+</sup> <sup>2</sup>� − *sin*( *<sup>π</sup>* <sup>8</sup> )| − <sup>2</sup>�, <sup>−</sup>*sin*( *<sup>π</sup>* <sup>8</sup> )<sup>|</sup> <sup>+</sup> <sup>2</sup>� <sup>+</sup> *cos*( *<sup>π</sup>* <sup>8</sup> )| − <sup>2</sup>�} and {*cos*( *<sup>π</sup>* <sup>8</sup> )| + 2� + *sin*( *<sup>π</sup>* <sup>8</sup> )| − <sup>2</sup>�,*sin*( *<sup>π</sup>* <sup>8</sup> )<sup>|</sup> <sup>+</sup> <sup>2</sup>� − *cos*( *<sup>π</sup>* <sup>8</sup> )| − 2�}. Experimentally we obtained the following value by carrying out a measurement with a duration of 60*s* and an average statistics per setting equal to about 1500 events: *S* = (2.51 ± 0.02). Hence a violation by more than 25 standard deviation over the value *SCHSH* = 2 is obtained. This experimental value is in good agreement with an experimental visibility of *V* = (0.930 ± 0.007) which should lead to *S* = (2.57 ± 0.02).

#### **7. Conclusion**

24 Will-be-set-by-IN-TECH

H, 2-H, 2-

0.5 0.25 0 0.25 0.5

rate equal to *C* = 100*Hz* within a coincidence window of 3 ns. This experimental data is in agreement with the expected value, determined from *Csource* = 6*kHz* after taking into account two main loss factors: hybrid state preparation probability *pprep*, and detection probability *pdet*. *pprep* depends on the conversion efficiency of the q-plate (0.80 ± 0.05) and on the probabilistic efficiency of the quantum transferrer *π* → *o*<sup>2</sup> (0.5), thus leading to *pprep* = 0.40 ± 0.03. The detection efficiency includes the q-plate conversion efficiency (0.8), the transferrer *o*<sup>2</sup> → *π* (0.5), and the single mode fiber coupling (0.2). Hence *pdet* = 0.08.

To completely characterize the state in Eq. (42) we reconstructed the density matrix of the quantum state. The tomography reconstruction requires the estimation of 16 operators James et al. (2001) through 36 separable measurements on the polarization-OAM subspaces. We carried out the reconstruction of the density matrix *<sup>ρ</sup>A*,*<sup>B</sup> <sup>π</sup>*,*o*<sup>2</sup> after the polarization-OAM conversion. The experimental results are reported in Fig.13, with the elements of the density matrices expressed in the polarization and OAM basis {|*H*, +2�, |*H*, −2�, |*V*, +2�, |*V*, −2�}. The fidelity with the singlet states <sup>|</sup>Ψ−� has been evaluated to be *<sup>F</sup>*(|Ψ−�, *<sup>ρ</sup>A*,*<sup>B</sup> <sup>π</sup>*,*o*<sup>2</sup> )=(0.957 <sup>±</sup> 0.009), while the experimental linear entropy of the state reads *SL* = (0.012 ± 0.002). A more quantitative parameter associated to the generated polarization-entangled states is given by the concurrence *C* = (0.957 ± 0.002). These values demonstrate the high degree of hybrid

To further characterize the hybrid quantum states, the violation of Bell's inequalities with the two photon system have been addressed. First, we measured the photon coincidence rate as a function of the orientation of the half-wave plate on Alice arm for two different OAM basis analysis, namely {| + 2�*o*<sup>2</sup> , | − 2�*o*<sup>2</sup> } and {|*h*�*o*<sup>2</sup> , |*v*�*o*<sup>2</sup> }. The variation of the number of coincidences *N*(*θ*) with the angle *θ* is in agreement with the one expected for entangled states such as *N*(*θ*) = *N*0(1 + *cosθ*): Fig.14. The coincidence fringe visibility reaches the values *V* = (0.966 ± 0.001) and *V* = (0.930 ± 0.007). Hence, a non-locality test, the CHSH one (Clauser et al. (1969)), has been carried out. Each of two partners, A (Alice) and B (Bob) measures a dichotomic observable among two possible ones, i.e. Alice randomly measures

Fig. 13. Experimental density matrix of the hybrid entangled state generated after the transferrer transformation on photons on *kB* mode. Each measurement setting lasted 15s

H, 2-

H, 2- V, 2-V, 2-

V, 2-

V, 2-

V

H, 2-

0.5 0.25 0 0.25 0.5

(Nagali & Sciarrino (2005)).

**6.2 Characterization of the state**

entanglement generation.

H, 2-

H, 2-

V, 2-V, 2-

,

V, ,**Re[] Im[]**

H, 2-

V, 2-

V, 2-

V

Among all degrees of freedom offered by single photons, the orbital angular momentum (OAM) has a great potential in the quantum information field, as it provides a natural choice for implementing single-photon qudits, the units of quantum information in a higher dimensional space. This can be an important practical advantage, as it enables higher security in quantum cryptographic protocols, as well as implications in fundamental quantum mechanics theory. Moreover, the combined use of different degrees of freedom of a photon, such as OAM and spin, enables the implementation of entirely new quantum tasks.

The authors acknowledge the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission, under FET-Open Grant No. 255914, PHORBITECH.

Harke, B., Keller, J., Ullal, C., Westphal, V., Sch

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**Part 2** 

**Photonic Materials** 


**Part 2** 

**Photonic Materials** 

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Hermite-Gaussian or Laguerre-Gaussian modes and the variable-phase mode

**5** 

*Russia* 

**New Nanoglassceramics Doped with Rare** 

Aseev Vladimir, Kolobkova Elena and Nikonorov Nikolay

*St. Petersburg State University of Information Technologies* 

*Mechanics and Optics, St. Petersburg* 

**Earth Ions and Their Photonic Applications** 

Glass-crystalline materials (or glassceramics) are heterophase composite materials, usually consisting of a glassy matrix (glass phase) and micro- or nano-sized dielectric or semiconductor crystals (crystalline phase), or metallic particles distributed in it. Glassceramics are formed by the growth of crystalline phase inside of a glass matrix. The crystal growth can occur as a result of spontaneous thermal crystallization of glass, as in case of heat treatment. For example, there are following typical representatives of the spontaneous crystallization: glasses doped with microcrystals of CdS, CdSe, CdTe, PbS, PbSe (non-linear media) [1-7], glasses doped with semiconductor microcrystals AgBr, AgCl, CuBr, CuCl (photochromic media) [8], glasses doped with dielectric microcrystals Li2O-SiO2 (photoetchable media – FOTURAN TM, FOTOFORM TM, PEG TM [9-11]. The crystal growth can occur as a result of photo-thermo-induced crystallization caused by UV photoirradiation and subsequent heat treatment. In this case, UV radiation generates centers of nucleation and the thermal treatment results in the growth of microcrystals in irradiated area of the glass host. Glasses doped with complicated microcrystals of NaF-AgBr (polychromatic glasses [12] and photo-thermo-refractive - PTR glasses [13]) are typical representatives of the

One of the major drawbacks of glass ceramic materials is a high light scattering occurring at the boundary of crystalline phase and glass phase. That is why current research in the development of optical glass-crystalline materials is aimed at decreasing of light scattering by means of formation of nanosize (5-30 nm) crystals or nanoparticles in the glass matrix. Only the nanoscale nature of the crystalline phase can significantly reduce the light scattering in heterophase composites (where the extinction coefficient can reach less than 0.01 cm-1) and classify these materials as optical. Fig.1 illustrates transition from millimetersize crystals to micrometer-size crystals and finally nano-size crystals in the glass host. The transition to the nanoscale crystalline phase not only leads to changes in physical, chemical and optical properties of glassceramics, but is also a cause for fundamentally new and

**1. Introduction** 

**1.1 Why glassceramics?** 

photo-thermo-induced crystallization.

**1.2 Why nanostructured glassceramics?** 
