1. Introduction

The polarization state expresses some fundamental symmetry properties of the electromagnetic field. Materials have symmetry properties at different levels such as molecular chirality, crystalline structure, mesoscopic order in liquid crystals. . . Interaction of chiral matter with polarized light gives rise to a full set of effects commonly referred to as optical activity. We will concentrate in the following on circular dichroism (CD) which is the differential absorption of left and right-handed circularly polarized light by a material system.

From a theoretical point of view, a light beam can be decomposed into plane waves of well defined frequency ω and wave vector k. These can be then interpreted in terms of photons of well definite energy h̄ω, momentum p = h̄ k and spin S. 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.

But light beams can carry not only spin angular momentum (SAM) but also orbital angular momentum (OAM) associated with their spatial phase distribution [1]. In particular, the field of a Laguerre-Gaussian beam LG_ℓ, exhibits a exp(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^orbital through a transverse plane [1]. We introduce for convenience Φ = ℱ/h̄ω and we get:

For plane waves 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]. However plane waves have a null OAM [4] and only coherent superpositions like helical beams can have non zero OAM. Therefore, assigning 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].

From the most basic symmetry point of view, nothing distinguishes SAM and OAM which are moreover of the same order of magnitude if nonzero. Thereby, one can wonder if the interaction of a light beam with a material system is also dependent on its OAM state.

To our knowledge, up to now, two experiments [6, 7] concluded that the effect, if any, is suppressed by at least 2 resp. 3 orders of magnitude with respect to CD in chiral molecular samples. In a recent review article, Yao and Padgett conclude [8]: ‘optically active media do not interact with the OAM’ in accordance with theoretical support [9].

CD can also be induced in an isotropic medium by an external magnetic field parallel to the beam propagation direction (MCD). We experimentally show in the following that what could be called by analogy 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], this might be the reason of all the negative experimental results reported so far.

Our configuration has several differences with respect to the previously reported experiments [6, 7]. First, the 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]. Secondly, we avoid a SAM contribution to the raw signals. In the earlier reports, a photo-elastic modulator is used for phase sensitive detection. The polarization of a given LG_ℓ beam is modulated between left and right-handed circular states and SAM is thus superimposed on OAM. Here we modulate the 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]. Finally, we probe a well defined optical transition, namely the ⁴I_9/2 → ⁴ F_5/2 transition of Nd³⁺ ions in a yttrium aluminium garnet (YAG) host.

2. Experimental setup

The experiment depicted in Fig. 1(a) was made as simple as possible for maximum reliability. Light from a laser diode is coupled into a 10 m-long polarization maintaining monomode fiber for spatial mode filtering. It is then directed onto a spatial light modulator (SLM) that imprints the desired helical phase map onto the wavefront. The diffracted Laguerre-Gaussian beam is then linearly re-polarized and slightly focussed towards the sample. Transmitted light is collected on a photodiode whose current is amplified and fed into a recorder for subsequent computer manipulation.

 

Fig. 1 Left: experimental Setup. LD: laser diode, FC: fiber couplers, PMF: polarization maintaining monomode fiber, SLM: spatial light modulator, L: lens, P: polarizer, QW: quarter-wave plate, S: sample, B: AC longitudinal B–field, PD: photodiode, WG: waveform generator, TIA: transimpedance amplifier, PA: power amplifier, REC: recorder. Right: Absorbance (black) and MCD (red) spectra of Nd:YAG around 809 nm.

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The sample is a 3 mm in diameter, 2 mm-long Nd:YAG rod with a concentration of ∼ 1 at.%. It is located in the ∼ 3 mm gap of an electromagnet. We operate typically around B = 330 mT_RMS at f_B = 85.75 Hz.

For the sake of quantitative comparison, MCD experiments are performed placing a quarter-wave plate just after the polarizer. Further experimental details are available as supplemental information [13].

3. Results and discussion

The data presented here consist of recordings of 2¹⁹ ≈ 5 × 10⁵ 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 LG₀ and LG₁ beams (red and blue curves).

 

Fig. 2 Left: LG₀ (red) and LG₁ (blue) and simulated noiseless MCD (black) power spectra. LG’s signals are normalized with respect to MCD signal and taken at its optimum wavelength (809.5 nm). Right: Corresponding phase sensitive analysis. The MOD would appear as a difference in the in-phase components (x). Both signals are compatible with 0 and MOD is below 1.8 × 10⁻⁴ of MCD at 95% confidence level.

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These two spectra are at the noise floor of our experiment and cannot be distinguished from each other. The MOD, if any, is lower than the sensitivity of our experiment. To get a quantitative value of the corresponding upper limit for the effect, we proceed in the following way.

A purely sinusoidal function of the same amplitude and frequency as the MCD signal is generated and FFT is performed. This makes a noiseless reference (black curve in Fig 2-left). More than 99.95% of its energy ℰ is concentrated in the 4 frequency bins around f_B that defines our analysis band. Spectra are then normalized by ℰ and plotted in dB units.

After integration over the analysis band, we find that the power ratios with the MCD signal are η₀ = 1.2 × 10⁻⁷ and η₁ = 1.9 × 10⁻⁷ for the LG₀ and LG₁ beams. The optical power is however proportional to the amplitude of the photodiode signal. The corresponding amplitude ratios are (η₀)^1/2 = 3.5×10⁻⁴ and (η₁)^1/2 = 4.3×10⁻⁴. We can then conclude that the difference between the absorption of the LG₁ and LG₀ 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.

The MOD effect should be proportional to the 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 LG₀ and LG₁ 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.

To evaluate it, we perform an Allan variance analysis on the temporal series recorded with the LG₀ and LG₁ 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 2τ 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.

We observe a classical inverse square root dependence of the variance with respect to the simulated integration time. We get accordingly an estimated variance σ_T = 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 δ is √2σ_T = 90 ppm so, at a 95% confidence level, we conclude that MOD is lower than 1.8 × 10⁻⁴ of MCD under the well defined experimental conditions described above.

However, MOD could have a different lineshape than MCD as different parts of the molecular tensor are involved. We thus checked if any signal could be found on both sides of the MCD maximum where MCD signal is roughly zero and absorption is maximum or minimum (see Fig. 1(b)). The reader is referred to supplemental material [13] for these spectra and a comprehensive set of other ones for OAM values ranging from −10h̄ to 10h̄. No significant signature was found at the 10⁻⁴ level with respect to the MCD signal.

4. Outlook and conclusion

Our experiments exclude magnetic orbital dichroism at least at the 10⁻⁴ level with respect to the magnetic circular dichroism of Nd:YAG for the ⁴I_9/2 → ⁴ F_5/2 transition.

We propose the following intuitive interpretation for this negative result. Let us consider two beams with the same polarization. The first one is an helical beam whereas the second one comes from a properly shaped classical source. They can have the same intensity distribution but they differ then in their spatial coherence: contrary to the classical one, the helical beam has well defined phase differences at different positions of the wavefront. And it is this peculiar phase pattern, here the exp(iℓϕ) phase factor, that confers this LG_ℓ beam a non-zero OAM.

In the optical domain, the typical atomic length scale a₀ is much smaller than the wavelength of light λ. The interaction is usually expanded in power series of a₀/λ. The lowest order is the electric dipole approximation. It is the 0^th order in a₀/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]: “internal ‘electronic-type’ motion does not participate in any exchange of orbital angular momentum in a dipole transition.” We reach the same conclusion that the electric quadrupole term is the lowest order which could give rise to MOD. It describes the interaction of the atomic or molecular system with the electric field gradient and is thus sensitive to the phase coherence of the wavefront. We have undertaken theoretical investigations to find a good couple of material and transition line that would allow for experimental confirmation of the effect as in the proposal [14] for trapped ions.