## How a Dove prism transforms the orbital angular momentum of a light beam

Optics Express, Vol. 14, Issue 20, pp. 9093-9102 (2006)

http://dx.doi.org/10.1364/OE.14.009093

Acrobat PDF (139 KB)

### Abstract

It is generally assumed that a light beam with orbital angular momentum (OAM) per photon of *lh̄*, is transformed, when traversing a Dove prism, into a light beam with OAM per photon of -*lh̄*. In this paper, we show theoretically and experimentally that this *OAM transformation rule* does not apply for highly focused light beams. This result should be taken into account when designing classical and quantum algorithms that make use of Dove prims to manipulate the OAM of light.

© 2006 Optical Society of America

## 1. Introduction

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

*ilφ*), carries an OAM per photon of

*lh̄*. In general, in the paraxial approximation, light beams can be represented as superpositions of Laguerre-Gaussian (LG) beams, or alternatively, as superposition of spiral harmonics. The weights of the superposition determine the corresponding angular momentum content of the light beam [2

2. G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum,” Phys. Rev. Lett. **88**, 013601 (2002). [CrossRef] [PubMed]

3. Graham Gibson, Johannes Courtial, Miles J. Padgett, Mikhail Vasnetsov, Valeriy Pasko, Stephen M. Barnett, and Sonja Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express **12**, 5448 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-22-5448 [CrossRef] [PubMed]

5. Lluis Torner, Juan P. Torres, and Silvia Carrasco, “Digital spiral imaging,” Opt. Express **13**, 873 (2005). [CrossRef] [PubMed]

2. G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum,” Phys. Rev. Lett. **88**, 013601 (2002). [CrossRef] [PubMed]

6. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentumstates of photons,” Nature **412**, 313 (2001). [CrossRef] [PubMed]

7. A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. **89**, 240401 (2002). [CrossRef] [PubMed]

8. G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger,” Experimental Quantum Coin Tossing,” Phys. Rev. Lett. **94**, 040501 (2005). [CrossRef] [PubMed]

9. Julio T. Barreiro, Nathan K. Langford, Nicholas A. Peters, and Paul G. Kwiat, “Generation of Hyperentangled Photon Pairs,” Phys. Rev. Lett. **95**, 260501 (2005). [CrossRef]

11. A. N. de Oliveira, S. P. Walborn, and C H Monken, “Implementing the Deutsch algorithm with polarization and transverse spatial modes,” J. Opt. B: Quantum Semiclass. Opt. **7**, 288–292 (2005). [CrossRef]

12. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the Orbital Angular Momentum of a Single Photon,” Phys. Rev. Lett. **88**, 257901 (2002). [CrossRef] [PubMed]

13. R. Zambrini and S. M. Barnett, “Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum,” Phys. Rev. Lett. **96**, 113901 (2006). [CrossRef] [PubMed]

15. K. T. Kapale and J. P. Dowling, “Vortex phase qubit: Generating arbitrary, counterrrotating, coherent superpositions in Bose-Einstein condensates via optical angular momentum beams,” Phys. Rev. Lett. **95**173601 (2005). [CrossRef] [PubMed]

*lh̄*, i.e., with spatial shape in cylindrical coordinates at the beam waist

*A*

_{in}=

*A*

_{0}(

*ρ*)exp(

*ilφ*), traverse a Dove prim, it is generally assumed that the output beam has a well defined OAM per photon of -

*lh̄*, i.e., with spatial shape

*A*

_{out}=

*A*

_{0}(

*ρ*)exp(-

*ilφ*)exp(-

*ilγ*), where

*γ*/2 is the angle of rotation of the Dove prism. The time dependence of the angle of rotation, and therefore the phase shift

*lγ*, makes possible the observation of the rotational frequency shift of light beams [16

16. J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational Frequency Shift of a Light Beam,” Phys. Rev. Lett. **81**, 4828 (1998). [CrossRef]

19. J. Lekner, “Polarization of tightly focused beams,” J. Opt. A: Pure Appl. Opt. bf 5, 6 (2003). [CrossRef]

20. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam,” Phys. Rev. Lett. **88**, 053601 (2002). [CrossRef] [PubMed]

*lh̄*⇒-

*lh̄*is not valid for highly focused light beams, since Dove prisms inherently introduce astigmatism, and therefore further OAM changes. Light beams with a well defined value of the OAM per photon, after traversing the Dove prism, are transformed into a superposition of states with well defined OAM. The violation of the rule

*lh̄*⇒-

*lh̄*, turns out to be more severe for highly focused light beams. We will provide a quantitative study of the properties of the Dove prism, by making use of the geometrical optics properties of the Dove prism, and we will verify experimentally the validity of our theoretical results in a series of experiments with a commercially available Dove prism.

## 2. ABCD law for a Dove prism

*x*

_{2},

*y*

_{2}) and angle (

*o*

_{x},

*o*

_{y}) of a ray, and the input position (

*x*

_{1},

*y*

_{1}) and angle (

*i*

_{x},

*i*

_{y}) are given by (see appendix)

*L*is the length of the base of the Dove prism,

*n*is the refractive index of the material, α is the base angle, and

*L*=63mm, α=45° and

*n*=1.51. We use a CW He-Ne laser (wavelength 633nm). The output beam of the laser is conveniently shaped so that at the input plane of the Dove prism, the beam width is

*w*

_{0}≃560

*µ*m. The beam is directed to the Dove prism by means of two mirrors to accurately control the angle and position of the beam at the input plane. The beam at the output plane of the system is demagnified to fit on a CCD camera with an appropriate imaging system.

*x*

_{1}=0,

*y*

_{1}=0), propagates with different angles (

*i*

_{x}) at the input plane of the Dove prism. Similarly, Fig. 2(b) corresponds to the case of changing the angle

*i*

_{y}. The experimentally measured values agree well with the theoretical predictions as given by Eqs. (1) and (2).

## 3. Ellipticity induced by a Dove prism

*y*direction, Eqs. (1) also show that the Dove prism modifies the beam waist position of the beam, (

*z*

_{x}and

*z*

_{y},

*z*

_{x}=

*z*

_{y}), differently in both transverse dimensions. The new beam waist positions (

*z̄*

_{x}and

*z̄*

_{y}) read

22. J. Visser and G. Nienhuis, “Orbital AngularMomentum of General AstigmaticModes,” Phys. Rev. A **70**, 013809 (2004). [CrossRef]

*w*

_{0}is the width of the beam at the input plane and

*z*

_{0}is the corresponding Rayleigh range.

*e*=(

*w̄*

_{x}/

*w̄*

_{y})

^{2}. The inset of Fig. 3 shows how the output elliptical beam rotates when the Dove prism rotates.

*m*=2, with two different beam widths. For very large beam widths, (a) and (b), the astigmatism induced by the Dove prism is not relevant, contrary to the case of highly focused beams, as shown in (c) and (d).

## 4. OAM transformation rule of the Dove prism

*k*is the wavenumber,

*N*is the normalization factor and the wavefront radius of curvature reads

*R̄*

_{x,y}=

*z̄*

_{x,y}[1+

*z*

_{0}/

*z̄*

_{x,y})

^{2}].

*l*, but a superposition of spiral harmonics that can be written as [2

2. G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum,” Phys. Rev. Lett. **88**, 013601 (2002). [CrossRef] [PubMed]

*a*

_{m}(

*ρ*)=1/(2

*π*)

^{1/2}∫

*dφA*

_{out}(

*ρ*,

*φ*)exp(-

*imφ*). The weight of the

*m*-harmonic is given by

*C*

_{m}=∫

*ρdρ*|

*a*

_{m}(

*ρ*)|

^{2}. We thus obtain [23] that the weights of the OAM superposition {

*C*

_{m}} that describes the light beam, after traversing the Dove prism, is given by

*l*+

*m*)/2 is an integer and

*C*

_{m}=0 otherwise. In the formula above

*J*

_{m}is the Bessel function of the first kind and order

*m*, and the parameter

*s*reads

*l*=1 vortex input beam. In all cases, the OAM decomposition of the output beam is centered at -

*l*.

*l*to -

*l*. For highly focused light beams, such as it is the case of Figs. 5(a) and (c), the Dove prism transform a pure LG beam into a superposition of spiral harmonics with different OAM index.

*C*

_{m}is determined by Eq. (6). For highly focused light beams, the OAM decomposition shows many modes. For larger beam widths values, the usual transformation

*l*⇒-

*l*holds. From Fig. 6, we notice that, for a given value of the input beam width, the weight of the central mode of the OAM superposition is smaller for the case of the input vortex beam than for the gaussian beam.

## 5. Conclusions

11. A. N. de Oliveira, S. P. Walborn, and C H Monken, “Implementing the Deutsch algorithm with polarization and transverse spatial modes,” J. Opt. B: Quantum Semiclass. Opt. **7**, 288–292 (2005). [CrossRef]

12. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the Orbital Angular Momentum of a Single Photon,” Phys. Rev. Lett. **88**, 257901 (2002). [CrossRef] [PubMed]

13. R. Zambrini and S. M. Barnett, “Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum,” Phys. Rev. Lett. **96**, 113901 (2006). [CrossRef] [PubMed]

15. K. T. Kapale and J. P. Dowling, “Vortex phase qubit: Generating arbitrary, counterrrotating, coherent superpositions in Bose-Einstein condensates via optical angular momentum beams,” Phys. Rev. Lett. **95**173601 (2005). [CrossRef] [PubMed]

## 6. Appendix: Derivation of the ABCD matrix for a Dove prism

*x*

_{1},

*y*

_{1})→(

*x′*,

*y′*). Secondly, we let the ray traverse the Dove prism (

*x′*,

*y′*)→(

*x″*,

*y″*), and finally, we calculate the ray trajectory from the output face of the prism to the output plane (

*x″*,

*y″*)→(

*x*

_{2},

*y*

_{2}). The first and last steps are straightforward free-space propagations, which in our case just means finding the crossings in the three dimensional space of a straight line with a plane.

*x*

_{1},

*y*

_{1};

*i*

_{x},

*i*

_{y}), with those at the output plane (

*x*

_{2},

*y*

_{2};

*o*

_{x},

*o*

_{y}) in the following way

*i′*

_{x},

*i′*

_{y})=(arcsin(sin(

*i*

_{x})/

*n*),arcsin(sin(

*π*/2-

*α*-

*i*

_{y})/

*n*), which are shown in Fig. 1. Next, we perform a Taylor expansion to first order in the angles of these equation, since we consider the paraxial approximation regime. The result of this approximation are Eqs.(1) and (2), which we repeat here to ease the following discussion

*y*

_{2}, in order to clarify the following discussion.

*i*

_{x},

*i*

_{y})), in the linearized equations the two transverse dimensions are completely decoupled. This allows a simplification for the ABCD law, which otherwise would become a larger matrix [21]. Nevertheless, this simplification is only valid within the paraxial approximation, i.e. to first order in the incoming angles.

*h*

_{0}, which is explicitly written in Eq.(2). It can be easily checked from the equation for

*y*

_{2}, that in the case of incidence angle parallel to the base of the Dove prism (

*i*

_{y}=0),

*h*

_{0}/2 is exactly the position where the Dove prim has no effect over the ray (

*y*

_{2}=

*y*

_{1}=

*h*

_{0}/2).

*q*=(

*z*-

*z*

_{0})-

*iλ*/(

*πw*

^{2}

^{0}), where

*z*is the actual longitudinal position of the beam,

*z*

_{0}the position of the beam waist of the beam, λ the wavelength of the light and

*w*

_{0}the beam width at the waist position. The beam can have a different complex radius of curvature for each dimension (

*q*

_{x},

*q*

_{y}). The transformation through an optical system gives

*i*∈{

*x*,

*y*}, for each dimension. We can write it in this simple way, because Eqs.(10) are decoupled for the two transversal directions.

## Acknowledgments

## References and links

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

2. | G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum,” Phys. Rev. Lett. |

3. | Graham Gibson, Johannes Courtial, Miles J. Padgett, Mikhail Vasnetsov, Valeriy Pasko, Stephen M. Barnett, and Sonja Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express |

4. | R. J. Voogd, M. Singh, S. Pereira, A. van de Nes, and J. Braat, “The use of orbital angular momentum of light beams for super-high density optical data storage,” OSA Annual Meeting, (Optical Society of America, 2004) paper FTuG14. |

5. | Lluis Torner, Juan P. Torres, and Silvia Carrasco, “Digital spiral imaging,” Opt. Express |

6. | A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentumstates of photons,” Nature |

7. | A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. |

8. | G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger,” Experimental Quantum Coin Tossing,” Phys. Rev. Lett. |

9. | Julio T. Barreiro, Nathan K. Langford, Nicholas A. Peters, and Paul G. Kwiat, “Generation of Hyperentangled Photon Pairs,” Phys. Rev. Lett. |

10. | M. Born and E. Wolf, |

11. | A. N. de Oliveira, S. P. Walborn, and C H Monken, “Implementing the Deutsch algorithm with polarization and transverse spatial modes,” J. Opt. B: Quantum Semiclass. Opt. |

12. | J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the Orbital Angular Momentum of a Single Photon,” Phys. Rev. Lett. |

13. | R. Zambrini and S. M. Barnett, “Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum,” Phys. Rev. Lett. |

14. | W. Chan, J.P. Torres, and J.H. Eberly, “Entanglement Migration of Biphotons in Spontaneous Parametric Down-conversion,” in CLEO/QELS 2006 Technical Digest (Optical Society of America, Long Beach, California, May 2006. |

15. | K. T. Kapale and J. P. Dowling, “Vortex phase qubit: Generating arbitrary, counterrrotating, coherent superpositions in Bose-Einstein condensates via optical angular momentum beams,” Phys. Rev. Lett. |

16. | J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational Frequency Shift of a Light Beam,” Phys. Rev. Lett. |

17. | M. J. Padgett and J. P. Lesso, “Dove prisms and polarized light,” J. Mod. Opt. |

18. | C. Cohen-Tannoudji, J. Dupont-Roc, and G Grynberg, “Atom-Photon Interactions : Basic Processes and Applications,” (Wiley Science Paperback Series, 1992) |

19. | J. Lekner, “Polarization of tightly focused beams,” J. Opt. A: Pure Appl. Opt. bf 5, 6 (2003). [CrossRef] |

20. | A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam,” Phys. Rev. Lett. |

21. | A. E. Siegman, |

22. | J. Visser and G. Nienhuis, “Orbital AngularMomentum of General AstigmaticModes,” Phys. Rev. A |

23. | I. S. Gradshteyn and I. M. Ryzhik, |

**OCIS Codes**

(080.0080) Geometric optics : Geometric optics

(090.1970) Holography : Diffractive optics

(230.5480) Optical devices : Prisms

**History**

Original Manuscript: July 11, 2006

Revised Manuscript: September 8, 2006

Manuscript Accepted: September 12, 2006

Published: October 2, 2006

**Citation**

N. González, G. Molina-Terriza, and J. P. Torres, "How a Dove prism transforms the orbital angular momentum of a light beam," Opt. Express **14**, 9093-9102 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-9093

Sort: Year | Journal | Reset

### References

- L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 458185 (1992). [CrossRef] [PubMed]
- G. Molina-Terriza, J. P. Torres anf L. Torner, "Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum," Phys. Rev. Lett. 88, 013601 (2002). [CrossRef] [PubMed]
- Graham Gibson, Johannes Courtial, Miles J. Padgett, Mikhail Vasnetsov, Valeriy Pasko, Stephen M. Barnett, and Sonja Franke-Arnold, "Free-space information transfer using light beams carrying orbital angular momentum," Opt. Express 12, 5448 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-22-5448 [CrossRef] [PubMed]
- R. J. Voogd, M. Singh, S. Pereira, A. van de Nes, and J. Braat, "The use of orbital angular momentum of light beams for super-high density optical data storage," OSA Annual Meeting, (Optical Society of America, 2004) paper FTuG14.
- Lluis Torner, Juan P. Torres, and Silvia Carrasco, "Digital spiral imaging," Opt. Express 13, 873 (2005). [CrossRef] [PubMed]
- A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, "Entanglement of the orbital angular momentumstates of photons," Nature 412, 313 (2001). [CrossRef] [PubMed]
- A. Vaziri, G. Weihs and A. Zeilinger, "Experimental two-photon, three-dimensional entanglement for quantum communication," Phys. Rev. Lett. 89, 240401 (2002). [CrossRef] [PubMed]
- G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger," Experimental Quantum Coin Tossing," Phys. Rev. Lett. 94, 040501 (2005). [CrossRef] [PubMed]
- JulioT. Barreiro, Nathan K. Langford, Nicholas A. Peters, and Paul G. Kwiat, "Generation of Hyperentangled Photon Pairs," Phys. Rev. Lett. 95, 260501 (2005). [CrossRef]
- M. Born and E. Wolf, Principles of Optics, Pergamon Press, 1993.
- A. N. de Oliveira, S. P. Walborn and C H Monken, "Implementing the Deutsch algorithm with polarization and transverse spatial modes," J. Opt. B: Quantum Semiclass. Opt. 7, 288-292 (2005). [CrossRef]
- J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the Orbital Angular Momentum of a Single Photon," Phys. Rev. Lett. 88, 257901 (2002). [CrossRef] [PubMed]
- R. Zambrini and S. M. Barnett, "Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum," Phys. Rev. Lett. 96, 113901 (2006). [CrossRef] [PubMed]
- <jrn>. W. Chan, J.P. Torres, and J.H. Eberly, "Entanglement Migration of Biphotons in Spontaneous Parametric Downconversion," in CLEO/QELS 2006 Technical Digest (Optical Society of America, Long Beach, California, May 2006.</jrn>
- K. T. Kapale and J. P. Dowling, "Vortex phase qubit: Generating arbitrary, counterrrotating, coherent superpositions in Bose-Einstein condensates via optical angular momentum beams," Phys. Rev. Lett. 95173601 (2005). [CrossRef] [PubMed]
- J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational Frequency Shift of a Light Beam," Phys. Rev. Lett. 81, 4828 (1998). [CrossRef]
- M. J. Padgett and J. P. Lesso, "Dove prisms and polarized light," J. Mod. Opt. 46, 175-179 (1999)
- C. Cohen-Tannoudji, J. Dupont-Roc, and G Grynberg, "Atom-Photon Interactions : Basic Processes and Applications," (Wiley Science Paperback Series, 1992)
- <jrn>. J. Lekner, "Polarization of tightly focused beams," J. Opt. A: Pure Appl. Opt. bf 5, 6 (2003).</jrn> [CrossRef]
- A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam," Phys. Rev. Lett. 88, 053601 (2002). [CrossRef] [PubMed]
- A. E. Siegman, Lasers, University Science Books, 1986.
- J. Visser and G. Nienhuis, "Orbital AngularMomentum of General AstigmaticModes," Phys. Rev. A 70, 013809 (2004). [CrossRef]
- I. S. Gradshteyn and I. M. Ryzhik, Tables of series, integrals and products, Academic Press, 1980. We make use of some useful properties of series of Bessel functions in chapters 8-9 about Special functions.

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.