## Negative experimental evidence for magneto-orbital dichroism |

Optics Express, Vol. 21, Issue 4, pp. 3941-3945 (2013)

http://dx.doi.org/10.1364/OE.21.003941

Acrobat PDF (1441 KB)

### Abstract

A light beam can carry both spin angular momentum (SAM) and orbital angular momentum (OAM). SAM is commonly evidenced by circular dichroism (CD) experiments *i. e.* differential absorption of left and right-handed circularly polarized light. Recent experiments, supported by theoretical work, indicate that the corresponding effect with OAM instead of SAM is not observed in chiral matter. Isotropic materials can show CD when subjected to a magnetic field (MCD). We report a set of experiments, under well defined conditions, searching for magnetic orbital dichroism (MOD), differential absorption of light as a function of the sign of its OAM. We experimentally demonstrate that this effect, if any, is smaller than a few 10^{−4} of MCD for the Nd:YAG ^{4}*I*_{9/2} → ^{4}*F*_{5/2} transition. This transition is essentially of electric dipole nature. We give an intuitive argument suggesting that the lowest order of light matter interaction leading to MOD is the electric quadrupole term.

© 2012 OSA

## 1. Introduction

*ω*and wave vector

**. These can be then interpreted in terms of photons of well definite energy**

*k**h̄ω*, momentum

**=**

*p**h̄*

**and spin**

*k***. The right-handed and left-handed circular polarization states correspond to photons having their spin parallel or anti-parallel to their momentum. These two configurations are clearly mirror images of each other with respect to a plane perpendicular to the direction of motion. Photons are thus chiral particles and optical activity can then be simply interpreted as differential interaction of a chiral probe with a chiral material.**

*S*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**, 8185–8189 (1992). [CrossRef] [PubMed]

*LG*, exhibits a exp(

_{ℓ}*iℓϕ*) phase factor where

*ϕ*and

*ℓ*denote the azimuthal angle and index. As a consequence, such beams are often referred to as

*helical beams*or

*optical vortices*. One can then establish a proportionality between the total energy flux

*ℱ*and the angular momentum flux

*M*=

*M*+

^{spin}*M*through a transverse plane [1

^{orbital}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**, 8185–8189 (1992). [CrossRef] [PubMed]

*ℱ*/

*h̄ω*and we get: where

*σ*= 0, ±1 for linearly or circularly polarized light. This proportionality relationship holds beyond the paraxial approximation [2

2. S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B: Quantum Semiclass. Opt. **4**, S7–S16 (2002). [CrossRef]

*σ*identifies with the

*helicity*which is the projection of the spin state of the associated photons onto the direction of motion. For massless particles like photons, the direction of motion cannot be reversed by change of reference frame so helicity and chirality are equivalent concepts. Equations (1) can then be interpreted as

*h̄σ*being the SAM per photon [3

3. K. Y. Bliokh and F. Nori, “Characterizing optical chirality,” Phys. Rev. A **83**, 021803(R) (2011). [CrossRef]

4. A. M. Stewart, “Angular momentum of the electromagnetic field: the plane wave paradox resolved,” Eur. J. Phys. **26**, 635–641 (2005). [CrossRef]

*h̄ℓ*units of OAM ‘per photon’ following Eq. (1b) should perhaps be considered with care. The definition of appropriate quantities to describe the angular momentum associated with optical polarization is still a matter of theoretical investigations [5

5. R. P. Cameron, S. M. Barnett, and A. M. Yao, “Optical helicity, optical spin and related quantities in electromagnetic theory,” New J. Phys. **14**, 053050 (2012). [CrossRef]

6. F. Araoka, T. Verbiest, K. Clays, and A. Persoons, “Interactions of twisted light with chiral molecules: an experimental investigation,” Phys. Rev. A **71**, 055401 (2005). [CrossRef]

7. W. Löffler, D. J. Broer, and J. P. Woerdman, “Circular dichroism of cholesteric polymers and the orbital angular momentum of light,” Phys. Rev. A **83**, 065801 (2011). [CrossRef]

8. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. **3**, 161–204 (2011). [CrossRef]

9. M. M. Coles and D. L. Andrews, “Chirality and angular momentum in optical radiation,” Phys. Rev. A **85**, 063810 (2012). [CrossRef]

*magneto-orbital dichroism*(MOD) is at most a few 10

^{−4}of MCD for the transition we study. This transition is essentially of electric dipole nature and, as stated in [10

10. M. Babiker, C. R. Bennett, D. L. Andrews, and L. C. Dávila Romero, “Orbital angular momentum exchange in the interaction of twisted light with molecules,” Phys. Rev. Lett. **89**, 143601 (2002). [CrossRef] [PubMed]

6. F. Araoka, T. Verbiest, K. Clays, and A. Persoons, “Interactions of twisted light with chiral molecules: an experimental investigation,” Phys. Rev. A **71**, 055401 (2005). [CrossRef]

7. W. Löffler, D. J. Broer, and J. P. Woerdman, “Circular dichroism of cholesteric polymers and the orbital angular momentum of light,” Phys. Rev. A **83**, 065801 (2011). [CrossRef]

*B*–field introduces a time odd-term in the interaction which thus involves the time-odd part of the molecular tensor whereas natural CD couples with its time-even part [11

11. L. D. Barron, *Molecular Light Scattering and Optical Activity* (Cambridge University Press, 2004). [CrossRef]

*LG*beam is modulated between left and right-handed circular states and SAM is thus superimposed on OAM. Here we modulate the

_{ℓ}*B*-field and use linearly polarized light. We can then compare directly different (

*S*= 0;

*L*=

*ℓh̄*) signals. Furthermore, we use an almost parallel beam to avoid mixing of SAM and OAM that occurs in non-paraxial beams [12

12. S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. **110**, 670–678 (1994). [CrossRef]

^{4}

*I*

_{9/2}→

^{4}

*F*

_{5/2}transition of Nd

^{3+}ions in a yttrium aluminium garnet (YAG) host.

## 2. Experimental setup

*B*= 330 mT

_{RMS}at

*f*= 85.75 Hz.

_{B}13. A 6 pages PDF file of supplemental information is available at http://arxiv.org/abs/1208.4227.

## 3. Results and discussion

^{19}≈ 5 × 10

^{5}samples of duration

*τ*= 5 ms. It represents about

*T*= 44 min acquisition time each. After numerical Fast Fourier Transformation (FFT) we get spectra of 0.4 mHz resolution over a 100 Hz span. In Fig. 2(a) we show the region around the modulation frequency for signals recorded with

*LG*

_{0}and

*LG*

_{1}beams (red and blue curves).

*ℰ*is concentrated in the 4 frequency bins around

*f*that defines our analysis band. Spectra are then normalized by

_{B}*ℰ*and plotted in dB units.

*η*

_{0}= 1.2 × 10

^{−7}and

*η*

_{1}= 1.9 × 10

^{−7}for the

*LG*

_{0}and

*LG*

_{1}beams. The optical power is however proportional to the amplitude of the photodiode signal. The corresponding amplitude ratios are (

*η*

_{0})

^{1/2}= 3.5×10

^{−4}and (

*η*

_{1})

^{1/2}= 4.3×10

^{−4}. We can then conclude that the difference between the absorption of the

*LG*

_{1}and

*LG*

_{0}beams, is at most on the order of a few 10

^{−4}the MCD signal. To get a better estimate we carry out a more elaborate numerical treatment of the data.

*B*–field so should appear in phase with it. On the contrary a pickup artefact, proportional to

*∂B/∂t*, is in quadrature. We thus do a numerical post acquisition phase sensitive detection. The

*B*–field recorded during the experiment is fitted by a

*cosine*function to generate an in-phase signal commonly labeled

*X*. With the same parameters we create a

*sine*function that defines the quadrature signal

*Y*. We then compute the cross-correlation with

*X*and

*Y*and normalize with the MCD signal amplitude. The result is depicted in Fig. 2(b). The observed difference between the

*LG*

_{0}and

*LG*

_{1}in-phase signals is only 1.7 ppm relative to MCD. This very low value should however be compared to the dispersion of the measurements.

*LG*

_{0}and

*LG*

_{1}beams. Each individual sample corresponds to a

*τ*integration time and we calculate the variance

*σ*

_{1τ}over the whole set of

*N*samples. Then we compute the mean of each pair of two successive samples. We get a set of

*N*/2 samples simulating a 2

*τ*integration time on which the variance

*σ*

_{2τ}is evaluated. The procedure is repeated recursively and stopped when the set contains too few samples so that no reliable variance can be calculated.

*σ*= 63 ppm which is plotted as error bars in Fig. 2(b). We notice first that both measurements are compatible with 0. Secondly, the variance on

_{T}*δ*is √2

*σ*= 90 ppm so, at a 95% confidence level, we conclude that MOD is lower than 1.8 × 10

_{T}^{−4}of MCD under the well defined experimental conditions described above.

13. A 6 pages PDF file of supplemental information is available at http://arxiv.org/abs/1208.4227.

*h̄*to 10

*h̄*. No significant signature was found at the 10

^{−4}level with respect to the MCD signal.

## 4. Outlook and conclusion

^{−4}level with respect to the magnetic circular dichroism of Nd:YAG for the

^{4}

*I*

_{9/2}→

^{4}

*F*

_{5/2}transition.

*iℓϕ*) phase factor, that confers this

*LG*beam a non-zero OAM.

_{ℓ}*a*

_{0}is much smaller than the wavelength of light

*λ*. The interaction is usually expanded in power series of

*a*

_{0}/

*λ*. The lowest order is the electric dipole approximation. It is the 0

*order in*

^{th}*a*

_{0}/

*λ*: the spatial variations of the electric field over the atomic wavefunction are neglected. The field strength and phase are evaluated at the position of the center of mass of the atom. At such an approximation level, the phase relationship of the field at two nearby points cannot be taken into account. The electric dipole interaction is thus insensitive to OAM. This picture is in accordance with the theoretical prediction of Babiker

*et al.*[10

10. M. Babiker, C. R. Bennett, D. L. Andrews, and L. C. Dávila Romero, “Orbital angular momentum exchange in the interaction of twisted light with molecules,” Phys. Rev. Lett. **89**, 143601 (2002). [CrossRef] [PubMed]

14. C. T. Schmiegelow and F. Schmidt-Kaler, “Light with orbital angular momentum interacting with trapped ions,” Eur. Phys. J. D **66**, 157–165 (2012). [CrossRef]

## 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. | S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B: Quantum Semiclass. Opt. |

3. | K. Y. Bliokh and F. Nori, “Characterizing optical chirality,” Phys. Rev. A |

4. | A. M. Stewart, “Angular momentum of the electromagnetic field: the plane wave paradox resolved,” Eur. J. Phys. |

5. | R. P. Cameron, S. M. Barnett, and A. M. Yao, “Optical helicity, optical spin and related quantities in electromagnetic theory,” New J. Phys. |

6. | F. Araoka, T. Verbiest, K. Clays, and A. Persoons, “Interactions of twisted light with chiral molecules: an experimental investigation,” Phys. Rev. A |

7. | W. Löffler, D. J. Broer, and J. P. Woerdman, “Circular dichroism of cholesteric polymers and the orbital angular momentum of light,” Phys. Rev. A |

8. | A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. |

9. | M. M. Coles and D. L. Andrews, “Chirality and angular momentum in optical radiation,” Phys. Rev. A |

10. | M. Babiker, C. R. Bennett, D. L. Andrews, and L. C. Dávila Romero, “Orbital angular momentum exchange in the interaction of twisted light with molecules,” Phys. Rev. Lett. |

11. | L. D. Barron, |

12. | S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. |

13. | A 6 pages PDF file of supplemental information is available at http://arxiv.org/abs/1208.4227. |

14. | C. T. Schmiegelow and F. Schmidt-Kaler, “Light with orbital angular momentum interacting with trapped ions,” Eur. Phys. J. D |

**OCIS Codes**

(050.1930) Diffraction and gratings : Dichroism

(260.5430) Physical optics : Polarization

(050.4865) Diffraction and gratings : Optical vortices

**ToC Category:**

Physical Optics

**History**

Original Manuscript: July 19, 2012

Revised Manuscript: August 6, 2012

Manuscript Accepted: August 6, 2012

Published: February 11, 2013

**Citation**

Renaud Mathevet, Bruno Viaris de Lesegno, Laurence Pruvost, and Geert L. J. A. Rikken, "Negative experimental evidence for magneto-orbital dichroism," Opt. Express **21**, 3941-3945 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-4-3941

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### 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. A45, 8185–8189 (1992). [CrossRef] [PubMed]
- S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B: Quantum Semiclass. Opt.4, S7–S16 (2002). [CrossRef]
- K. Y. Bliokh and F. Nori, “Characterizing optical chirality,” Phys. Rev. A83, 021803(R) (2011). [CrossRef]
- A. M. Stewart, “Angular momentum of the electromagnetic field: the plane wave paradox resolved,” Eur. J. Phys.26, 635–641 (2005). [CrossRef]
- R. P. Cameron, S. M. Barnett, and A. M. Yao, “Optical helicity, optical spin and related quantities in electromagnetic theory,” New J. Phys.14, 053050 (2012). [CrossRef]
- F. Araoka, T. Verbiest, K. Clays, and A. Persoons, “Interactions of twisted light with chiral molecules: an experimental investigation,” Phys. Rev. A71, 055401 (2005). [CrossRef]
- W. Löffler, D. J. Broer, and J. P. Woerdman, “Circular dichroism of cholesteric polymers and the orbital angular momentum of light,” Phys. Rev. A83, 065801 (2011). [CrossRef]
- A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon.3, 161–204 (2011). [CrossRef]
- M. M. Coles and D. L. Andrews, “Chirality and angular momentum in optical radiation,” Phys. Rev. A85, 063810 (2012). [CrossRef]
- M. Babiker, C. R. Bennett, D. L. Andrews, and L. C. Dávila Romero, “Orbital angular momentum exchange in the interaction of twisted light with molecules,” Phys. Rev. Lett.89, 143601 (2002). [CrossRef] [PubMed]
- L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge University Press, 2004). [CrossRef]
- S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun.110, 670–678 (1994). [CrossRef]
- A 6 pages PDF file of supplemental information is available at http://arxiv.org/abs/1208.4227 .
- C. T. Schmiegelow and F. Schmidt-Kaler, “Light with orbital angular momentum interacting with trapped ions,” Eur. Phys. J. D66, 157–165 (2012). [CrossRef]

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