In numerous approaches, the properties of a light field are solely connected to the properties of its electric field. This restriction still holds in most works of quantum optics, including all works following Glauber’s theory about the quantum properties of optical fields and the detection of photons. Indeed, in his seminal paper [1], Glauber justifies that in optics, one can restrict in most cases to a detector only sensitive to the electric field amplitude. The detector he considers, assumed to be of negligible size and extra wide frequency band, is implicitly based upon electric dipole (E1) transitions. Although Glauber explicitly mentions that the possibility of correlation functions for the magnetic field could prove “someday useful”, we are not aware of further works along this direction.

Here, we establish that the consideration of the electric field alone is restrictive when the spatial structure of optical fields becomes very complicated. In particular, we show that with suitable detectors, simply based upon atomic absorption on a magnetic dipole transition (M1) or on an electric quadrupole transition (E2), photons should be detectable in space regions where a standard E1 detector is unable to perform the detection. There are now numerous realizations of optical fields bearing a complicated spatial structure as found with the optical fields near nanostructures. However, connecting quantum properties of light with the structure of the field is simpler for a light field freely propagating in vacuum, than when the properties of a neighboring material medium have to be taken into account. In the present work, we concentrate on beams exhibiting a spiral phase such as Bessel beams, and Laguerre-Gauss (LG) beams.

To be more specific about the way a M1 or E2 detector operates, we consider that a click of the detector only occurs after a specific atomic excitation, through the observation of a deexcitation process, such as an elementary emission on a strong line. The excitation probability of the detecting atom or molecule can be found from usual Fermi’s golden rule, using a multipole expansion of the interaction. Assuming coherent narrow band optical fields, one gets a quasiclassical expression

where d_i, m_i and Q_ij are matrix elements of electric dipole (E1), magnetic dipole (M1) and electric quadrupole (E2) operators between ground and excited states respectively and where δω and Γ stand for characteristic excitation detuning and transition linewidth respectively. For quantum detector with arbitrary internal structure the matrix elements d_i, m_i and Q_ij have 3, 3 and 5 independent non zero components respectively. In Eq. (1) the main contribution to detector interaction with light is usually due to (E1) absorption processes. However, these transitions give no contribution to excitation of molecules in regions where there is no electric field, so that (M1) and (E2) transitions give the predominant contribution to excitation rate in this situation. Moreover, when the detector is simply an elementary atom, the different E1, M1 and E2 processes occur for different specific resonances, and even the behavior of different Zeeman components can be selectively addressed by the choice of the light field frequency, i.e. through spectroscopy. Note, that we neglect the atomic motion when deriving Eq. (1) and everywhere in this paper. If one takes into consideration the quantum state of motion of the absorbing atom, new interesting possibilities for (E1) absorption processes arise even in regions where there is no electric field [2].

Now, we consider electric fields satisfying Maxwell equations and including an azimuthal phase factor exp(imϕ) (with m integer), such as Bessel beams or LG beams. We show that these beams are often unduly considered to be hollow beams. Indeed, the exact (non paraxial) shape of the solution is established to be [3]:

with r, φ, z the cylindrical coordinates, J_m the Bessel function of order $h=\sqrt{k^2-{\kappa }^2}$, k=ω/c with ω the light (circular) frequency, α and β being polarization components, and g(κ) an arbitrary function.

Clearly, on axis, for r=0, the electric field is null, except for m=0 (the field just exhibits a standard Gaussian structure) and for m=1 (only a longitudinal field exists on-axis, resulting from the wave front curvature i.e. the E_z term vanishes for a quasi-plane wave structure, κ→0). However, if one considers the associated magnetic field, as derived from the Faraday’s law of induction, ik B=[∇×E], one deduces that for m=±2 the magnetic field on axis is non zero, although the electric field is strictly null

One notes that this on-axis magnetic field exhibits a transverse structure and a circular polarization independent of the polarization as defined by α, β. In a similar manner, but with a higher complexity as due to its tensorial nature, one can demonstrate that the gradient of the electric field is non zero on axis for m=±2 (and so on, for arbitrary m, if one considers successive derivations of the field at the adequate order). A clear consequence of these non zero values on axis is that a detector of the magnetic field (M1 transition), or of the gradient of the electric field (E2 transition) yields an on-axis response for a beam with m=±2.

Now, to be more specific and to obtain quantitative estimates, we concentrate on the practical situation of LG beams, which are leading order approximations to Bessel beams for κ→0 region. LG beams have attracted a lot of interest owing to their orbital angular momentum [4,5]. Although the most common expressions to define a LG beam are valid only in the limit of the paraxial approximation, they can be recovered through an expansion in Bessel beams such as in Eq. (2) [3]. In order to limit ourselves only to E1, M1 and E2 transitions, we consider an expansion of the electric field that includes a longitudinal component, and for which the paraxial approximation is performed only at first order. Hence, the electric field of a LG beam can be represented as [6]

where $C_p^{\mid m\mid }=\sqrt{2p!⁄\pi \left(p+\mid m\mid \right)!}$ is the normalization constant, E₀ is the electric field amplitude, $w\left(z\right)=w_0\sqrt{1+z^2⁄z_R^2}$ is the beam radius at z, w₀ is the beam waist, L ^|m| _p(x) is the generalized Laguerre polynomial, and Z_R=kw²₀/2 is the Rayleigh range of the beam. The number of nodes of the field in the radial direction is p+1 and unit vector (α,β) describes the polarization of the beam. As can be understood from the general discussion on Bessel beams, we concentrate on the m=2 situation. We take α=1/√2, β=i/√2 (σ- polarization) as symmetry remains more apparent with circular polarization.

Considering specifically a LG beam with m=2, one finds that, although the electric field is strictly null, the magnetic energy density, I_M, on the axis is

Eq. (5) shows that this on-axis magnetic field quickly increases with focusing, and also with the number of radial zeroes. Fig. 1 shows that, for strongly focused beams, the magnetic energy on axis becomes comparable to the maximum of the electric energy density.

 

Fig. 1. Radial dependence of the electric and magnetic fields of LG beam in the waist plane (kw₀=10, p=6, m=2)

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An important property of LG beams is the fact that they can carry both spin and orbital angular momentum. Their z-component of the total momentum per photon is given by [4]

where mħ is the orbital momentum carried by the beam along its propagation direction and σ is spin [3,4]. Our choice for m, α, β, corresponds to a circularly polarized LG beam, with spin σ-. This implies anyhow that a LG beam with m=2, known to carry an orbital angular momentum 2ħ [3], exhibits only a z-component j_z=(m-1)ħ for the total angular momentum.

To detect the effects of this electromagnetic energy lying in the region where the beam is hollow for E field, a M1 or a E2 detector can be used. For brevity, we assume here that the detector deals with a S-D transition in a spherically symmetric atom; quadrupole matrix elements are equal to Q^M_ij=Q^R∫sin θdθdφY ^*0 ₀(3n_in_j-r ² δ_ij)Y^M ₂(θ,φ), where Y are spherical harmonics, M=0,±1,±2 characterizes z-component of orbital momentum of an atom and n=(sinθcosφ,sinθsinφ,cosθ)). Q^R characterizes strength of transition.

The on-axis quadrupole transition amplitude T ^2,1 _Q for a LG beam with m=2 is given by

With our choice of circular polarization (σ^-), only the M=1 sublevel is reached from the initial S level. Here again, a strong focusing is needed for the transition to occur on-axis with the LG beam with m=2. Note also that the gain of one unit of z-component of angular momentum associated to T ^2,1 _Q term has been obtained in spite of the negative spin of the considered LG beam, so that two units of angular momentum have been taken from the orbital momentum of the beam. Also a significant original conclusion, implied by these nonzero transition rates, is that photons carrying orbital angular momentum do exist on the beam axis, confirming that the LG beam is not hollow. For a LG beam carrying a linear polarization the possible exchange of 2ħ of orbital moment in the absorption of a single photon is even more obvious in spite of more complex calculation. This idea of a transfer of the orbital angular momentum through a quadrupole interaction has been discussed in [7], but our present result shows that it can occur and be dominant on-axis. This transition amplitude can be compared to the one for the trivial m=0 LG beam, where only the M=-1 sublevel is reached. The transition amplitude is given by

It corresponds to the quadrupole transition governed by the longitudinal gradient of the field that comes with propagation of any nearly plane wave beam. In addition to the difference present in the angular momentum selection rule, a significant distinct dependence appears with respect to the focusing of the beam.

The off axis spatial distribution of the excitation rate for an E2 or a M1 transition also exhibits interesting features, connected to the very complicated (see Fig. 2) spatial structure of the LG beam. A typical result is given in Fig. 3, where one sees that the excitation rate to M=+1 sublevel (transition amplitude T ^2,1 _Q) has a well pronounced maximum at the beam axis. Conversely, the rate of excitation to M=-1 (transition amplitude T ^2,-1 _Q), is null on-axis. Furthermore the spatial structure of the excitation rates - made of rings as expected from the cylindrical symmetry- does not coincide with the structure of the LG beam energy (note in particular the relatively large peak on the edges). This behavior opens the way to a spatially selective excitation of atoms or molecules located near the axis, with a sub-wavelength resolution reminiscent of confocal microscopy.

 

Fig. 2. Space distribution in the waist plane of the real part of the electric field (left) and of the magnetic field (right) in a Laguerre-Gauss beam (kw₀=6,p=6,m=2). Each distribution rotates counterclockwise at the optical frequency.

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Fig. 3. Normalized radial distribution of excitation rates of an E2 (M=±1) transitions of an atom placed in the waist plane of LG beams with (p=2, m=2,kw0=6).

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Such a spatially selective excitation, may allow, close to the E field hollow region, the selective manipulation of chiral molecules - a fascinating possibility apparently offered with LG beams (the optical activity is usually associated to a coupled E1-M1 transition). Although a negative experimental result for this optical activity was obtained in [8], our investigation of the local properties of the electromagnetic field suggests that it may be a too large spatial averaging that has rendered unobservable such a possible manipulation.

From the above results, we can discuss now on general grounds our suggestion of using specific type of atomic transitions detectors, which are sensitive to the gradients of electric fields or to the magnetic fields, in order to detect photons with a complicated space structure. Our detection scheme clearly offers the possibility of detecting processes that have been intrinsically neglected in all quantum optics approaches relying only on E1 -type detector. For optical experiments relying specifically on an E2 or M1 transition, an issue is certainly the low oscillator strength of these transitions, usually considered as nearly forbidden transitions. However the feasibility of detecting an E2 transition has already been established [9] with an evanescent optical wave, which is precisely another type of e.m. field with a complicated structure [10].

We also establish, through our direct calculation of excitation rates, that the high order angular momentum of LG beam can be transferred in an elementary exchange with a quantum system, hence breaking the usual (actually E1) selection rules. If particle physics considerations have shown long ago that electromagnetic fields can bear a large angular momentum, which is involved in high-order multipole transition (like in nuclear physics, see [11]), the coherent production of large number of identical spiral photons is a recent achievement, up to now limited to the optical domain, and whose ultimate possibilities of focusing remain a challenge for opticians. It is however susceptible to open new frontiers in quantum optics (e.g. quantum limits to spatial correlation,…). The investigation of the intimate nature of spiral photons such as generated with LG beams has remained until now extremely limited, although the only experiment performed to date [12] at the quantum level has attracted much attention due to the opening of new sets of variables in entanglement. In particular, it should be noted that most of the experimental investigations involving LG beams and their specific angular momentum [4, 5, 12-14] have been integrated on at least a one micron-size volume, instead of using a detector much smaller than the optical wavelength. (For recent investigation of LG beam with subwavelength E1 detectors see [15,16]).

At last, an interesting result of our semiclassical derivation is that the specificity of the photons carried by a LG or a singular beam appears enhanced under strong focusing. This regime of sharply focused propagating beams opens a natural connection with the blossoming domain of nano-optics, where on the one hand, it is known that the relative strength of E2 transition might be enhanced [17], but where on the other hand, quantized photons are actually attached to a material interface, instead of simply originating from a freely propagating field. With the development of nanotechnologies, it becomes conceivable to produce suitable non-E1 detectors, such as artificial nanoparticles of special shape (nanoantennas) designed to be sensitive to gradients of electric fields. In particular, because the lower sensitivity of E2 or M1 transition is actually due to the small size of electronic orbit relatively to optical wavelength, building up more sensitive non-E1 detector could be feasible with long molecules, such as twisted or bio-molecules. In return, these detectors should benefit to the very contemporary need of characterizing quantum optics effects with nanooptical fields for which singularities of electric field can be predicted [18].