1. Introduction

Laser beams are commonly modeled in a scalar and paraxial approximation by Hermite-Gaussian or Laguerre-Gaussian beams [1, 2]. However it has been known for a long time that the electromagnetic field near a focal point has a complex polarization structure [3]. Moreover, it can easily be shown that a beam of finite transverse extent and transverse polarization is inconsistent with Maxwell’s equations [4]. Lax et al. first proposed a coherent method describing vector Gaussian beams beyond the paraxial approximation. In particular, they outlined that a longitudinal E-field component is necessary. This seminal work induced since then a lot of theoretical developments [5].

Meanwhile, several experiments have been performed to investigate the vector nature of light fields, for example in applications like confocal microscopy (see e.g. [6]) where effects are large due to the high apertures of the objectives. More specifically, non-zero crossed polarization components have been revealed by analyzing with crossed polarizers the light generated by lasers of different technologies and emitting different Gaussian modes [7–12]. As for the longitudinal component, it has been evidenced using different methods in the case of annular illumination [13], axially symmetric polarized beams [14–17] or in the vicinity of a metallic tip [18] where it is particularly intense.

Recently, Yang and Cohen proposed to use magnetic circular dichroism (MCD), i.e. circular dichroism induced in an isotropic sample by an external quasi-static magnetic field, to investigate the non trivial polarisation state in the focal region of a gaussian beam [19]. We report below a realization of their proposal with a twofold purpose. Firstly we have chosen very common experimental conditions: a beam emerging from the output coupler of a monomode fiber passes through a linear polarizer and is focused by a low aperture doublet. We obtain a quantitative agreement with theory showing that the longitudinal field can play a significant role in broad areas of physics where laser beams are used like, for example, in experiments very sensitive to the field polarization state such like trapping multilevel atoms in optical tweezers [20,21]. For a more specialized community concerned with the characterization and tailoring of optical fields, the sensitive experiment presented here could be developed in a complementary technique as it is a measurement of the coherence between the longitudinal and transverse field components rather than their intensities. Secondly, and from a more theoretical point of view, we will recall the definition of optical chirality and chirality flux. These two field quantities are associated respectively to natural circular dichroism (NCD) in chiral samples and MCD. We will show that such a common laser beam, despite its elliptical polarization, is an example of a light field having no optical chirality a but finite chirality flux. It can thus produce MCD but not NCD emphasizing the fundamental difference between these two kind of optical activity.

The paper is organized as follows. We give in Sec. 2 a straightforward derivation of the expected signal. The experimental setup and the techniques we implemented are described in Sec. 3. We then present our findings in Sec. 4 before we introduce in Sec. 5 chirality and chirality flux to distinguish between NCD and MCD. After a concluding paragraph we give in an appendix some more details on theory and experiments.

2. Theoretical background

We first give here a simple, first order, derivation of the polarization state of a focused gaussian beam and refer the reader to [5] for more rigorous and systematic approaches. Let us consider the complex electrical field E of a monochromatic Gaussian beam, of wavenumber k = 2π/λ = ω/c, propagating along the z-axis, with a beam waist w₀ located in z = 0 (Fig. 1(a)). Its divergence angle and Rayleigh range are θ = λ/πw₀ and $z_R=\pi w_0^2/\lambda$. We suppose moreover that the beam has traveled through a linear polarizer that eliminates the y-component of E. As we use a low aperture lens, we assume furthermore that E_y remains negligibly small. We model E_x by:

In the far field |z| ≫ z_R and E_z ≈ −(x/z)E_x: the two components are in phase so the polarization is linear and perpendicular to the ray as sketched in Fig. 1(a).

 

Fig. 1 a) Longitudinal cross-section of a light beam asymptotically linearly polarized. (b) Sketch of the polarization evolution across the beam. (c) Beam transverse cross-section in the waist plane. Color scale measures the circular intensity V from green to magenta. Black arrows point toward the extrema of V where the circular fraction is ±θ.

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On the other hand, in the focal region |z| ≪ z_R, we have E_z ≈ i(x/z_R)E_x. The two components are in quadrature and the polarization is elliptical. Moreover, E_z is an odd function of x so the ellipticity is of opposite sign on both sides of the x = 0 plane (Fig. 1(c)). Finally, following the Stokes formalism [22] one can define the circular fraction $V/I=2\mathit{Im}\left(E_x^{*}{E}_z\right)/\left({\left|E_x\right|}^2+{\left|E_z\right|}^2\right)$ of a light field. At the symmetrical points (±w₀/2, 0, 0), the circular intensity V is maximum and the circular fraction is simply ±θ. The longitudinal component is indeed of first order in θ as shown by Lax et al.

According to [19], the magnetic field-dependent (MFD) contribution to the absorption rate of a molecule (or an ion in our case), i.e. the MCD strength, reads:

In Eq. (4), p oscillates at optical frequencies. Its time average involves ${〈\mathbf{E}\times \dot{\mathbf{E}}〉}_t=\frac{1}{2}\text{Re}\left[\mathit{E}\times {(i\omega \mathit{E})}^{*}\right]$. From Eq. (3) we get:

We can now compute the differential MFD absorption between the upper and lower halves of the beam. In principle it is superimposed on a much stronger regular absorption which will be discarded in the following as a phase sensitive detection will allow us to extract only signals proportional to B_QS.

We denote n_v the ion density and assume that the MFD absorption is so small that the beam is negligibly absorbed. In a thin slice of width dz centered in z the MFD absorption rate of the upper half of the crystal (x > 0) is:

We get the differential MFD absorption rate ΔP integrating Eq. (7) over the crystal length, that is for z varying from −L/2 to L/2 assuming the beam is focused in the center of the crystal. We finally normalise by the incident power to get the differential MFD absorption:

When the beam aperture is changed, the signal varies because of two opposite tendencies: the circular fraction in the waist plane is proportional to θ but the Rayleigh range scales as θ⁻². ΔA first increases linearly with θ, reaches a maximum when 2z_R ≃ L and then decreases.

As we shall see below, a conventional MCD experiment is carried in parallel on a separate part of the beam for calibration purposes and phase sensitive detection. On this reference beam we measure ΔA_MCD = (P_σ+ − P_σ−)/P₀ the differential MFD absorption of collimated right and left-handed circularly polarized beams. We can compute it following the process of Eqs. (4)–(8) but using z-independent polarisation states E = E₀(1, ±i, 0)/√2 instead. We get $\mathrm{\Delta }{\text{A}}_{\text{MCD}}=\sqrt{\frac{\pi }{2}}\eta n_v{w}_0^{2}L$ so:

In summary, the expected effect should manifest itself as a differential absorption between the upper and lower half part of the beam, linear in magnetic field, material MCD coefficient α′ and beam divergence θ at low apertures.

3. Experimental techniques and setup

Our sample is a 3 mm diameter, 2 mm long rod of Nd:YAG (∼ 1 at.%, refractive index n = 1.82) that exhibits a strong MCD line around λ = 808 nm. The experimental setup is depicted in Fig. 2. Light from a laser diode is spatially filtered in a 20 cm-long monomode fiber. The beam is then divided in two parts on a beam splitter.

 

Fig. 2 Experimental setup. LD: laser diode, FC_1,2: monomode fiber couplers, D: iris diaphgram, BS: beam splitter, HWP: half-wave plate, P: dichroic polarizer, L_1–3: lenses, TS: translation stage, PEM: photo-elastic modulator, Ph_MCD: photodiode, Ph_2Q: split photodiode.

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A small fraction is used in a conventional MCD measurement which provides the MCD reference signal using a photo-elastic modulator (Hinds PEM-90) and another Nd:YAG sample located in a 0.3 mT longitudinal static B-field.

The main part of the beam first passes through an half-wave plate and a linear polarizer in order to set the polarization. The beam is then focused into the sample located in the 4-mm gap of an electromagnet with a vertical magnetic field alternating at 270 Hz. We use a doublet, 50 mm in focal length. This results in a low aperture beam θ ∼ 35 mrad, and consequently, small signal. However, this insures a good quality beam (see Appendix B) and the lens is located far from the electromagnet. It reduces unwanted effects that appear in the stray field. The doublet is attached to a translation stage to scan the focal point across the sample. Transmitted light is then imaged on a split photodetector by lens labeled L₃ in Fig. 2 (Appendix C). The photodetector’s outputs are then connected to the differential input of a lock-in amplifier whose reference signal is provided by an Hall probe glued near the sample. Only the in-phase first harmonic labeled X_1f is further considered. Many spurious effects, always present at such low level signals, are rejected. For example, electronic pickup, proportional to ∂B_QS/∂t, is in quadrature and non linear terms in B_QS, such as vibrations induced in ferromagnetic parts ( $\propto B_{\mathit{QS}}^2$), give rise to higher order harmonics.

We first used our conventional MCD setup to obtain a spectrum of the MCD coefficient α′ around 808 nm (Fig. 3(a)). ΔA/B is the differential absorption of the right and left handed circular polarisation in a unit magnetic field which we calibrated with help of a commercial MCD spectrometer (JASCO J-810CD). A dispersive lineshape is observed [23].

 

Fig. 3 a) Spectroscopy of the Nd:YAG MCD response. ΔA/B is the differential absorption of the right and left handed circular polarisation in a unit magnetic field. b) Top panel: reference MCD signal recorded as the laser wavelength is modulated in the linear part of the MCD lineshape mark out by vertical dotted lines in the left panel. Bottom panel: raw data from lock-in amplifier with 1 s integration time, horizontal (red) or vertical (blue) polarization.

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When we slowly modulate the laser wavelength using the temperature modulation input of the laser diode driver in the linear region marked out by vertical dotted lines in Fig. 3(a), a sinusoidal MCD signal is recorded (top panel Fig. 3(b)). In the focused beam experiment, the expected effect is synchronous with that reference signal. We improve so the signal to noise ratio and reduce other spurious effects like laser frequency drifts.

4. Experimental results

The bottom panel of Fig. 3(b) is a typical example of raw data. Depending on whether the polarisation of the incoming beam is perpendicular (red) or parallel (blue) to the B-field one observes a strong or insignificant signal in phase with the MCD reference (top panel). This clearly demonstrates the expected effect. Its amplitude (see red curve) is ΔX_1f = 2.20 ± 0.01 μV_pp. This small differential signal sits on top of a V_t ≃ 1.30 V mean value associated to the transmitted power. The incident power P₀ corresponds to a voltage V₀ = 3.6 V taking into account the average absorption and reflection losses (Appendix D). The measured differential absorption is thus on the order of ΔA_exp. = ΔX_1f/V₀ ≃ 0.61 10⁻⁶.

We can compare it with the model presented before. In the region of interest, ΔA/B varies with the wavelength at rate s = d(ΔA/B)/dλ = −1.44 × 10⁻³ T⁻¹nm⁻¹ (dashed line in Fig. 3(a)). Using a small pick-up coil, we measured B_QS at the crystal location whose amplitude is B₀ = 150 ± 20 mT_RMS. Scanning the wavelength between λ₁ = 807.61 nm and λ₂ = 808.51 nm (dotted lines) we thus expect a peak to peak amplitude variation of the differential absorption:

We also investigated the role of the geometrical factors. Using the translation stage labeled TS in Fig. 2, we have varied the position of the focal point along the crystal. The resulting signal, normalized by the B-field value as evaluated with a small pick-up coil in an independent measurement, is shown in Fig. 4(a). Error bars have been estimated repeating 8 acquisitions with the same parameters. The solid line is a simple model to guide the eye with smooth edges whose width equals the Rayleigh range and the crystal length is reduced to account for refraction (Fig. 4(b)).

 

Fig. 4 a) Signal as a function of the focus position inside the crystal. Solid line results from a simple model taking into account the refraction of the beam (see right panel) and the fact that the signal builds up over the Rayleigh range. b) Refraction of the beam as it enters the crystal: the divergence angle θ₀ is reduced to θ₀/n and the focal point is displaced by Δz = (1 − 1/n)L = 0.9 mm. The actual length of the crystal L = 2 mm is reduced to L/n = 1.1 mm in the model.

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Finally we studied the signal dependence on the beam divergence θ. In order not to modify the hard-won optimized alignement, we reduced the beam diameter by inserting a diaphragm as far as possible upstream from the crystal to minimize edge diffraction effects (labeled ”D” in Fig. 2). We present in Fig. 5 the signal relative to the full aperture one (blue point). The beam divergence has been estimated as the ratio of the diaphragm’s radius to the focal length of lens L₁. Strictly speaking, the beam is no longer Gaussian after the diaphragm and one should consider these points cautiously. However, we can globally fit the data using the model of the differential MFD absorption Eq. (8) with an overall amplitude as a single free parameter.

 

Fig. 5 Beam divergence θ is varied using diaphragm (red points) and signal is normalized to the full aperture one (blue point). Black line is a fit corresponding to Eq. (8).

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5. MCD vs NCD

Tang and Cohen showed both theoretically [25] and then experimentally [26] that natural circular dichroism (NCD), i.e. differential absorption of left and right handed circularly polarized light in a chiral material, results from the coupling of the material chirality with the electromagnetic quantity

We compute the B-field of the gaussian beam considered above starting from

6. Conclusion and outlook

Following a proposal by Yang and Cohen we have experimentally studied the polarization in the Rayleigh zone of an asymptotically linearly polarized gaussian beam. We deliberately chose a low aperture beam as very commonly used in many experiments and shown that the polarization is elliptical around its focal point. It results from a longitudinal polarization component, which is necessary because of the finite transverse size of the beam, in quadrature with the transverse one. A more detailed analysis shows that such a simple beam is indeed a typical example of a light field with elliptical polarisation but zero optical activity and highlights the fundamental difference between NCD and MCD.

The experiment reported here is sensitive, weakly perturbs the beam propagation and no specially prepared sample is required. It could be further developed in a novel technique to investigate the local polarization of light fields. As it probes the coherence 〈E_xE_z〉_t it is complementary to previously reported ones that usually measure their intensities ( $\propto {〈E_x^2〉}_t$ or ${〈E_z^2〉}_t$).

Appendices

A. Alternate derivation of the expected signal

The total differential absorption rate can be obtained in an alternate way. Let

(15)$$\begin{array}{lll}\mathit{\rho }(z)\hfill & =\hfill & \left(x(z),y(z)\right)\hfill \\ \hfill & =\hfill & \left(x_0,y_0\right){\left(1+{\left(z/z_R\right)}^2\right)}^{\frac{1}{2}}\hfill \end{array}$$

be a parametrization of a ray Γ passing at distance ${\rho }_0={\left(x_0^2+y_0^2\right)}^{1/2}$ from the axis in z = 0 and l a curvilinear abscissa along Γ.

Let us denote I_Γ(r) the beam intensity at point r:

(16)$${\text{I}}_{\mathrm{\Gamma }}(\mathbf{r})={\text{I}}_0{\left(\frac{w_0}{w(z)}\right)}^2\text{exp}(-2r^2/w{(z)}^2),$$

with and $w(z)=w_0{\left(1+{(z/z_R)}^2\right)}^{\frac{1}{2}}$.

We consider a flux tube around Γ whose infinitesimal cross section reads dS(z) = dS₀ 1 + (z/z_R)². In the absence of absorbers, ${\text{I}}_{\mathrm{\Gamma }}(\mathbf{r})dS={\text{I}}_{0}d{S}_0\text{exp}\left(-2r_0^2/w_0^2\right)$ is a conserved quantity.

In the case of absorption, the intensity decreases and we replace I₀ by I_Γ(l). Energy conservation reads:

(17)$$\frac{d\left({\text{I}}_{\mathrm{\Gamma }}dS\right)}{dl}=-n_v\sigma \left({\text{I}}_{\mathrm{\Gamma }}dS\right),$$

where $\sigma ={\sigma }_0\frac{x/z_R}{1+{\left(z/z_R\right)}^2}$ with σ₀ = 2α′ωB/ε₀c the MCD cross section and $\frac{x/z_R}{1+{\left(z/z_R\right)}^2}$ comes from the imaginary part in Eq. (5). We then get a Beer-Lambert law, along the ray

(18)$$\frac{d({\text{I}}_{\mathrm{\Gamma }})}{dl}=-n_v\sigma ({\text{I}}_{\mathrm{\Gamma }}).$$

The intensities along the ray beyond and before the crystal are thus related according to:

(19)$$\begin{array}{lll}{\text{I}}_{\mathrm{\Gamma }}(+\infty )\hfill & =\hfill & {\text{I}}_{\mathrm{\Gamma }}(-\infty )\text{exp}\left[-\underset{-L/2}{\overset{+L/2}{\int }}du{n}_v{\sigma }_0\frac{x_0{(1+u^2)}^{1/2}}{1+u^2}\right]\hfill \\ \hfill & =\hfill & {\text{I}}_{\mathrm{\Gamma }}(-\infty )\text{exp}\left[-2n_v{\sigma }_0{x}_0{\text{sinh}}^{-1}(L/2z_R)\right],\hfill \end{array}$$

with u = z/z_R and identifying dl ≃ dz at first order in the beam divergence θ.

As stated before, absorption is small and Eq. (19) can be linearized to give ΔI = I_Γ(−∞) − I_Γ(∞) ≃ −κx₀I₀ with κ = 2n_vσ₀ sinh⁻¹ (L/2z_R). In the plane of the detector located in z_d > L/2:

(20)$$\mathrm{\Delta }I(x_d,y_d;z_d)=-\kappa \frac{x_d}{\sqrt{1+{(z_d/z_R)}^2}}{\text{I}}_0.$$

Integration over the upper half plane x_d > 0 gives the absorbed power:

(21)$$\begin{array}{lll}{\text{P}}_{+}\hfill & =\hfill & \underset{0}{\overset{+\infty }{\int }}d{x}_d\underset{-\infty }{\overset{+\infty }{\int }}d{y}_d(-\mathrm{\Delta }I(x_d,y_d;z_d))\hfill \\ \hfill & =\hfill & \frac{\kappa }{\sqrt{1+{(z_d/z_R)}^2}}\sqrt{\frac{\pi }{32}}{w}_0^3\sqrt{1+{(z_d/z_R)}^2}{\text{I}}_0\hfill \\ \hfill & =\hfill & \sqrt{\frac{\pi }{32}}\kappa w_0^3{\text{I}}_0,\hfill \end{array}$$

from which Eq. (8) can be easily recovered.

B. Beam characterization

We have determined the relevant beam parameters recording the beam profile as a function of position z on a CCD camera. For each position, a vertical and horizontal cross section is made and fitted with a Gaussian profile from which we extract the horizontal and vertical widths plotted in Fig. 6.

 

Fig. 6 Beam widths as a fonction of the position beyond the focusing lens labeled L₁ in Fig. 2 (arbitrary origin). Black squares/red circles: vertical/horizontal cross sections. Straight lines are fits from which the beam parameters can be extracted.

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The data points are then fitted by the theoretical expression $w(z)=w_0{\left(1+{(z-z_0)}^2/z_R^2\right)}^{1/2}$ with w₀, z_R and z₀ as free parameters. We get w_0h = 7.5 ± 1 μm and w_0v = 7.9 ± 1.5 μm, z_Rh = 208±4 μm and z_Rv = 231±4 μm, z_0h = 11.98±0.015 μm and z_0v = 11.99±0.018 μm. The corresponding divergence angles are θ_h = 36 ± 4 mrad and θ_h = 34 ± 4 mrad.

The two beam waists are compatible within the error bars with their directly measured value on the CCD camera 6.6 μm and 7.4 μm. The beam is thus slightly elliptical as confirmed with the two significantly different z_R values. The error bars on the fitted waists are relatively large because very few points near the focus contribute to the fitting process. We will thus take the direct measurements as a better estimates of the actual waists and take w₀ = 7.0 ± 0.4 μm as a typical value and conservative error bars covering both measured values.

On the contrary, many points contribute to the evaluation of z_R whose values are well defined by the fitting process and we take z_R = 220 ± 16 μm.

We can then evaluate the geometrical factor w₀ sinh⁻¹ (L/2n²z_R)/L to 3.9 ± 0.3 10⁻³.

In principle, w₀ and neither z_R nor θ are independent parameters as for a diffraction limited beam $z_R=\pi w_0^2/\lambda$ and θ = λ/πw₀. One often introduce the M² factor defined as θ = M²λ/πw₀ to quantify the deviation from such an ideal beam for which M² = 1. Repeating the fitting procedure with M² instead of z_R as free parameter gives $M_h^2=1.06\pm 0.18$ and $M_v^2=1.04\pm 0.16$ showing that the beam is practically diffraction limited. This is a prerequisite of the study presented here as the field in the focal region is very sensitive to the beam quality.

C. Collimating optics

The signal consists in a two-fold structure inside the beam profile (Fig. 1(c)). As a consequence, its characteristic dimension is typically half the beam waist and thus emerge in the far field at twice the beam divergence angle. The collecting lens (L₃ in Fig. 2) must have at least twice the numerical aperture (NA) of that of the incident beam. The one we use is 50 mm in diameter and 250 mm in focal length so NA_L3 ≈ 0.1 significantly greater than 2θ_h,v ≃ 70 mrad as determined in the previous section.

D. Evaluation of the voltage corresponding to the effective incident power

The incident power P₀ to be considered should be distinguished from the actual incident power on the crystal P_i. Indeed, we use an uncoated sample to allow precise positioning of the beam focus with respect to the crystal. We thus have to correct for the reflection losses: P₀ = (1 − R)²P_i where R = 0.085 is the reflection coefficient per face for a refractive index n = 1.82.

We have measured the overall transmission T = P_t/P_i, taking into account absorption and reflection. It varies between 0.26 at λ₁ = 807.61 nm, 0.21 at 808.0 nm and 0.36 at λ₂ = 808.51 nm. We take T ≃ 0.30 as a typical value. We have thus V₀ = V_t (1 − R)²/T ≃ 3.6 V with the measured V_t ≃ 1.30 V.

Acknowledgments

We thank Charles-Louis Serpentini (IMRCP) for providing us with the MCD spectra and Christian Beurthe (Institut d’Optique) for sample polishing. Referees are sincerely acknowledged for their fruitful remarks and bibliographic suggestions. This work was supported by Université Paul Sabatier through its AO1 program and the ANR project PhotonImpulse.

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