1. Introduction

The degree and state of polarization of optical fields play an essential role in all photonic systems where polarization-dependent elements or structures exist [1, 2]. Conventionally, the polarization properties of a random beam field are measured by inserting a wave plate and a polarizer into the path of the beam and detecting the transmitted intensity with various relative orientations of these elements [3]. In this way the polarization distribution across the beam is obtained at a spatial resolution limited by the detector’s pixel size. While this is sufficient in many situations, in the context of nano-scale photonics a method of higher resolution may be needed. This is achieved by using nanoscopic probe scatterers to transfer local information about the field at the probe position to the field in the far zone.

The principle of probing electromagnetic fields with nanoparticles has been employed in scanning near-field optical microscopy (SNOM) [4] where measurements so far have focused mainly on the near-field intensity distributions. In particular, it has been shown that a gold nanoparticle attached to the end of an optical fiber tip can be used for near-field intensity profiling [5] (see also Ref. [6]). Instead, the detection of the polarization properties of optical near fields has not been considered much, even though probing three-component light fields has been demonstrated [7] and used to observe, e.g., longitudinal field modes [8], polarization rings [9], and Möbius strips of polarization [10].

In this work we present an experimental demonstration of our previous theoretical result [11] that the polarization characteristics of a partially polarized electromagnetic beam, i.e., its degree and state of polarization, can be measured by the nanoprobe technique. We use a gold nanocube deposited on silicon substrate to scatter the beam and detect the polarization properties of the scattered far field. Unlike traditional intensity measurements, the observed far-field polarization allows us to deduce the polarization properties of the beam at the probe site. The spatial resolution is effectively determined by the probe size and the technique can, in principle, be extended to the detection of electromagnetic near fields [11].

2. Field scattered by a cubic nanoparticle

Consider a random electromagnetic beam propagating along the z axis with the electric field realization at position r and frequency ω represented by column vector E(r,ω). Into the field at position r ₀ we place a subwavelength, cubic nanoparticle composed of material with complex permittivity ε _p, as is depicted in Fig. 1. The x and y components of the incident electric field are orthogonal to the side faces of the cube which is deposited on an (absorbing) substrate with permittivity ε _s and thickness large enough allowing us to ignore reflections from the back interface. The particle can be considered as a polarizable dipole with a dipole moment p(r ₀,ω) = α(ω)E _tot(r ₀,ω), where α (ω) is the polarizability matrix of the nanocube [11, 12] and E _tot(r ₀,ω) is the (total) electric field at the particle site. The matrix α (ω) is proportional to the unit matrix as the Cartesian field components are orthogonal to the faces of the cube [12]. The field E _tot(r ₀,ω) differs from E(r ₀,ω) as a result of backscattering of the incident field and the dipole-radiated field from the substrate surface. These effects can be incorporated into an effective polarizability in the form p(r ₀,ω) = α _eff(ω)E(r ₀,ω). Since the particle is of nanometer scale we may employ the quasi-static approximation and write the effective polarizability matrix as [13]

 

Fig. 1 Illustration of the polarization-state measurement of a partially polarized electromagnetic beam field E(r,ω) with a nanoscatterer. A cubic nanoparticle deposited on a substrate is placed in the beam and the polarization properties of the field scattered by the particle are measured in the far zone with a conventional quarter-wave plate, polarizer, and detector combination. The observation direction is in the yz plane and makes an angle of 45° with respect to the y axis.

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Above, h is the distance from the cube center to the substrate, c is the vacuum speed of light, μ ₀ is the magnetic permeability of free space, and r _p denotes the quasi-static (high-spatial-frequency) limit of the Fresnel reflection coefficient [4] for p-polarized light at the nanoparticle-substrate interface. Equation (1) holds equally well for other particles whose polarizability is proportional to a diagonal matrix. In this work a cubic nanoparticle was chosen owing to fabrication reasons.

The particle scatters the incoming beam and the total electric field far from the cube is a sum of the directly and indirectly (via substrate reflection) scattered contributions. The far field is given as [11, 13, 14] (see Appendix)

The polarization properties of an electromagnetic field are specified by the polarization matrix [3] written as Φ(r,ω) = 〈E ^*(r,ω)E ^T(r,ω)〉, where the asterisk denotes complex conjugate. The far field is locally planar (i.e., the electric-field components are orthogonal to the propagation direction) and we may represent the far-field polarization matrix in a 2 × 2 form in the spherical polar coordinates [11]. Employing Eq. (2), we obtain the following relation between the far-field polarization matrix, ϕ(r,ω), and the 3×3 dipole-site polarization matrix, Φ(r ₀,ω):

Using the conventional wave-plate and polarizer setup [3, 11], the full far-field polarization matrix can be obtained experimentally with a total of four measurements. By straightforward developments of Eq. (3) the polarization matrix at the dipole site can be solved in terms of the far-field polarization matrix [11]. Thus the polarization properties of a random light beam can be deduced by probing the beam with a nanoprobe and detecting the polarization state of the scattered far field.

3. Probing a beam with dipolar nanocubes

To demonstrate the functioning of the measurement system we first need to construct a light beam with a known degree and state of polarization, together with a means to control these properties. This is, in principle, achieved by dividing a laser beam with a beam splitter and a delay line into two uncorrelated, orthogonally polarized x and y components, adjusting the ratio of the two intensities with a neutral density filter, and subsequently recombining the components into a single beam [15]. In this way we obtain a partially polarized field whose fully polarized part is linearly polarized in the direction of stronger component. Since the resulting polarization matrix is diagonal, the intensities I_x and I_y are also the eigenvalues of the matrix and therefore the related degree of polarization [3] can be written as

However, in practice it is challenging to align and maintain the two beams at the probe site in such a way that the spatial non-uniformities present in the components would not alter the polarization properties of the total superposition field. Since our goal is primarily to demonstrate a proof-of-principle measurement, we approach the problem slightly differently. Instead of combining the beams, we perform the scattering measurement for the x- and y-polarized incident components separately and add together the two far-field polarization matrices to obtain the polarization matrix associated with the incident field with both x and y components present simultaneously. This method is justified since the two components are uncorrelated and it allows us to employ just a single beam in the whole measurement, where we simply rotate the x polarization into y polarization using a half-wave plate. The technique minimizes the errors in beam preparation due to misalignments of the field components and their non-uniformities.

The setup for probing the degree and state of polarization of optical beams is depicted in Fig. 2, where the scatterer is a cubic gold particle with sides approximately 100 nm long. The nanocube is deposited on a silicon surface using e-beam lithography (Raith EBPG-5000 ES+HR), which allows a controllable deployment of nanoparticles. A He-Ne laser beam is prepared with a polarizer, neutral density filter, and a half-wave plate as explained above. The beam is directed onto the probe cube using a lens (focal length 7.5 cm) which focuses the beam to a spot smaller than 100 μm. This will increase the field intensity at the particle site resulting in a scattered field that can be detected in the far zone. A long focal length retains the beam-like character of the field and avoids too tight a focus which would alter the polarization properties of the incident field leading, for instance, to polarization rings [9]. The light scattered from the nanoprobe is analyzed at an angle of 45° in relation to the incident beam with a standard polarimetric setup including a quarter-wave plate, a linear polarizer, and a highly sensitive detector (Thorlabs PDF10A) able to measure powers down to femtowatt level. The power of the scattered field [11, 12] after the polarizer (2.5 cm in diameter) is approximately 10 pW and it is focused onto the active area (size 1.2 mm²) of the detector. The collecting lens has a focal length of 3.5 cm and it is located at a distance of 8.5 cm from the probe cube. In our setup the ratio between the power at the detector and that incident on to the particle is approximately 0.0048, corresponding to 25 W/cm² incident intensity. The minimum power that the detector can observe is 6.3 fW implying that the incident intensity for this nanoprobe must be at least 16 mW/cm².

 

Fig. 2 Setup for the measurement of the polarization properties of beams with a nanoscatterer. A linearly polarized He-Ne laser beam (wavelength 632.8 nm, power 5 mW) is prepared with a linear polarizer (LP), a neutral density filter (ND), and a half-wave plate (HWP), and directed to a convex lens L_A (focal length 7.5 cm) that focuses the beam onto a 100 nm gold nanocube (the inset shows a scanning electron microscope image of the particle). The field scattered from the nanoparticle is analyzed in the far zone with a quarter-wave plate (QWP) and a linear polarizer (LP) that are located at the distance of approximately 6 cm from the cube. Due to low scattered power, light emerging from the waveplate-polarizer system is focused with a lens L_B (focal length 3.5 cm, located at approximately 8.5 cm from the particle) onto a detector with picowatt-range sensitivity.

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In the measurements we first set the incident beam as y polarized and acquire the full 2 × 2 far-field polarization matrix ϕ(r,ω) with four intensity measurements corresponding to various relative orientations of the polarizer and the quarter-wave plate [3]. We denote the angles that the polarization axis of the polarizer and the fast axis of the wave plate make with respect to û_φ (detection direction is in the yz plane, see Fig. 1) as ψ _p and ψ _q, respectively. The angle is positive counterclockwise when the observer looks into the field. The four pairs of angles are (ψ _q,ψ _p) = (0,0), (0,π/4), (π/4,π/4), and (π/2,π/2), and in each case the transmitted intensity is measured. Next, we repeat the measurement to obtain the far-field polarization matrix for the x-polarized incident beam prepared by rotating the y-polarized beam with a half-wave plate and keeping the intensity unchanged. Adding the measured polarization matrices pertaining to the x-polarized and y-polarized beams results in a far-field polarization matrix corresponding to an unpolarized incident beam. Using this 2 × 2 polarization matrix we can solve Eq. (3) for the polarization matrix Φ(r ₀,ω) related to the incident beam at the particle site which yields the (intensity) normalized Stokes parameters [11] with the values s ₁ ≈ −0.0029, s ₂ ≈ 0.0267, and s ₃ ≈ −0.0058 (s ₀ = 1 since it represents the intensity). Consequently, the degree of polarization defined as [3, 11]

 

Fig. 3 Degree of polarization (P) and the normalized Stokes parameters (s ₁,s ₂,s ₃) of a light beam as a function of the intensity difference δ = (I_y − I_x)/I_y of the x and y field components. Values of P obtained by probing with a nanocube are shown with red circles, while the Stokes parameters are presented with triangles: blue (s ₁), magenta (s ₂), and green (s ₃) (note the negative axis). The red dashed line represents the theoretical curve of s ₁. The black curve depicts the degree of polarization obtained with a traditional polarimetric measurement.

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Next we measure the far-field polarization for the x-polarized incident beam again, but with decreased intensity, and add the ensuing polarization matrix with that obtained previously for the y-polarized beam. The resulting degrees of polarization and the normalized Stokes parameters obtained on gradually reducing the x-polarized incident contribution are shown in Fig. 3. We see that increasing the intensity difference of the orthogonal incident components increases P and decreases s ₁, while the other two Stokes parameters remain near zero. The values of s ₁ are negative since this parameter characterizes the intensity difference of the x- and y-polarized components, explicitly s ₁ = (I_x − I_y)/(I_x + I_y). More precisely, the polarized part of the combined (incident) light beam is y polarized and its intensity increases. When the intensity difference in Fig. 3 achieves 100 percent, the beam is fully polarized (in the y direction) with P ≈ 0.97.

To evaluate the performance of the nanoprobe method, we also measure for reference the degree of polarization using standard polarimetric tools. Here we observe the intensities of the whole beam while making changes in the degree of polarization in similar manner as with the particle measurements. The results are shown in Fig. 3 with the black curve. We see that the values obtained for the degree of polarization by the two approaches are in good agreement. They also match very well to the theoretical curve P = −s ₁ = δ/(2−δ), with δ = (I_y − I_x)/I_y, plotted with red dashed line. The slight differences are due to accumulation of errors originating from imperfections in wave plates and polarizers, beam focusing, and particle deformation. The background noise was taken into account by subtracting the signal obtained without a laser beam and the experiments were carried out in near dark conditions.

4. Conclusions

We have successfully demonstrated the measurement of the state and degree of polarization of optical beams by probing them with a subwavelength dipolar nanoparticle. Information on the beam polarization at the dipole site is deduced from the polarization properties of the far field scattered by the particle. The immediate advantage of our approach is the possibility to obtain the polarization distribution across the beam with a very high spatial resolution, provided a system capable of scanning the whole beam with a nanoscatterer is realized. In contrast to current devices able to measure the Stokes-parameter distributions of optical beams with the resolution fixed by the pixel size of the detector, in our approach the resolution is limited mainly by the size of the nanoparticle. The method has obvious applications on beam characterization.

Our work demonstrates a practical realization of polarization probing of optical beams. It was recently pointed out theoretically that similar polarization probing could be realized also for electromagnetic near fields in which three electric-field components are present. A near-field polarization detection necessitates several observation directions so as to access the in total nine independent polarization parameters pertaining to electromagnetic near fields [11]. In practice, such a measurement could be performed by means of scanning near-field optical microscopy modified for multiple far-field measurement directions. Besides polarization, it has been shown theoretically that two-point spatial coherence properties and the related transverse coherence length of light beams could be measured by probing the field with two nanoscatterers [16].

Appendix

Consider a polarizable dipole scatterer at a position r ₀ in front of a slab of material. The field emitted by the dipole into a far-zone observation point r consists of directly and indirectly scattered contributions. The direct contribution is given as [11, 14]

(6)$${\mathbf{E}}_{\mathrm{d}}\left(\mathbf{r},\omega \right)=\frac{{\mu }_0{\omega }^2{e}^{ik\left|\mathbf{r}-{\mathbf{r}}_0\right|}}{4\pi \left|\mathbf{r}-{\mathbf{r}}_0\right|}\left(\mathbf{I}-{\widehat{\mathbf{u}}}_r{\widehat{\mathbf{u}}}_r^{\mathrm{T}}\right){\mathit{\alpha }}_{\text{eff}}\left(\omega \right)\mathbf{E}\left({\mathbf{r}}_0,\omega \right),$$

where E(r ₀,ω) is the incident beam field at the dipole location, α _eff(ω) is the effective polarizability given in Eq. (1) of the main text, û _r is the unit vector in the direction from r ₀ to r (see Fig. 4), and I is the 3 × 3 unit matrix.

 

Fig. 4 Illustration of the coordinate system and the unit vectors in treating the reflection of the field emitted into the far zone by the point dipole. The particle center is at the origin and the substrate surface is marked with a vertical line. The figure is drawn for a 45° reflection corresponding to the actual measurements.

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The indirect part is constructed as follows. The dipole far field propagating in the direction û _{r +} of half space z > 0, and which after reflection would propagate in the û _r direction (see Fig. 4), is of the form of Eq. (6) and given as

(7)$${\mathbf{E}}_{\mathrm{i}}\left({\mathbf{r}}^{\prime },\omega \right)=\frac{{\mu }_0{\omega }^2{e}^{ik\left|{\mathbf{r}}^{\prime }-{\mathbf{r}}_0\right|}}{4\pi \left|{\mathbf{r}}^{\prime }-{\mathbf{r}}_0\right|}\left(\mathbf{I}-{\widehat{\mathbf{u}}}_{r+}{\widehat{\mathbf{u}}}_{r+}^{\mathrm{T}}\right){\mathit{\alpha }}_{\text{eff}}\left(\omega \right)\mathbf{E}\left({\mathbf{r}}_0,\omega \right).$$

We next make use of the fact that ${\widehat{\mathbf{u}}}_{r+}{\widehat{\mathbf{u}}}_{r+}^{\mathrm{T}}+{\widehat{\mathbf{u}}}_{\phi +}{\widehat{\mathbf{u}}}_{\phi +}^{\mathrm{T}}+{\widehat{\mathbf{u}}}_{\theta +}{\widehat{\mathbf{u}}}_{\theta +}^{\mathrm{T}}=\mathbf{I}$, where the unit vectors are ${\widehat{\mathbf{u}}}_{\phi +}=\widehat{\mathbf{z}}\times {\widehat{\mathbf{u}}}_{r+}/\left|\widehat{\mathbf{z}}\times {\widehat{\mathbf{u}}}_{r+}\right|$ and û _{θ +} = û _{φ +} × û _{r +}, and write Eq. (7) in the form

(8)$${\mathbf{E}}_{\mathrm{i}}\left({\mathbf{r}}^{\prime },\omega \right)=\frac{{\mu }_0{\omega }^2{e}^{ik\left|{\mathbf{r}}^{\prime }-{\mathbf{r}}_0\right|}}{4\pi \left|{\mathbf{r}}^{\prime }-{\mathbf{r}}_0\right|}\left[{\widehat{\mathbf{u}}}_{\phi +}{\widehat{\mathbf{u}}}_{\phi +}^{\mathrm{T}}+{\widehat{\mathbf{u}}}_{\theta +}{\widehat{\mathbf{u}}}_{\theta +}^{\mathrm{T}}\right]{\mathit{\alpha }}_{\text{eff}}\left(\omega \right)\mathbf{E}\left({\mathbf{r}}_0,\omega \right).$$

The reflection is treated by separating the extra path length Δr of the indirect contribution in relation to the directly scattered field and noting that the field amplitudes in the û _{φ +} and û _{θ +} directions, i.e., the s-polarized and p-polarized components, are changed in accordance with the corresponding Fresnel reflection coefficients R _s and R _p, respectively. In addition, the direction of the p-polarized field component is changed from û _{θ +} to û _θ. Noticing further that û _{θ +} = −û _−θ, we replace ${\widehat{\mathbf{u}}}_{\theta +}{\widehat{\mathbf{u}}}_{\theta +}^{\mathrm{T}}$ with $-{\widehat{\mathbf{u}}}_{\theta }{\widehat{\mathbf{u}}}_{-\theta }^{\mathrm{T}}$ in Eq. (8). Thereby we write the indirect field contribution at the observation point r as

(9)$${\mathbf{E}}_{\mathrm{i}}\left(\mathbf{r},\omega \right)=\frac{{\mu }_0{\omega }^2{e}^{ik\left(\left|\mathbf{r}-{\mathbf{r}}_0\right|+\mathrm{\Delta }r\right)}}{4\pi \left(\left|\mathbf{r}-{\mathbf{r}}_0\right|+\mathrm{\Delta }r\right)}\left[R_{\mathrm{s}}{\widehat{\mathbf{u}}}_{\phi +}{\widehat{\mathbf{u}}}_{\phi +}^{\mathrm{T}}-R_{\mathrm{p}}{\widehat{\mathbf{u}}}_{\theta }{\widehat{\mathbf{u}}}_{-\theta }^{\mathrm{T}}\right]{\mathit{\alpha }}_{\text{eff}}\left(\omega \right)\mathbf{E}\left({\mathbf{r}}_0,\omega \right).$$

For a nanoscale particle Δr is very small and we may neglect it in the denominator while retaining it in the spatial phase term of the numerator. Taking into account that û _{φ +} = û _φ, since the direction of the s-polarized field component does not change on reflection, and that $\mathbf{I}-{\widehat{\mathbf{u}}}_r{\widehat{\mathbf{u}}}_r^{\mathrm{T}}={\widehat{\mathbf{u}}}_{\phi }{\widehat{\mathbf{u}}}_{\phi }^{\mathrm{T}}+{\widehat{\mathbf{u}}}_{\theta }{\widehat{\mathbf{u}}}_{\theta }^{\mathrm{T}}$ in Eq. (6), the total field at point r can be written as

(10)$$\begin{array}{l}{\mathbf{E}}_{\mathrm{f}}\left(\mathbf{r},\omega \right)={\mathbf{E}}_{\mathrm{d}}\left(\mathbf{r},\omega \right)+{\mathbf{E}}_{\mathrm{i}}\left(\mathbf{r},\omega \right)\\ =\frac{{\mu }_0{\omega }^2{e}^{ik\left|\mathbf{r}-{\mathbf{r}}_0\right|}}{4\pi \left|\mathbf{r}-{\mathbf{r}}_0\right|}\left[A{\widehat{\mathbf{u}}}_{\phi }{\widehat{\mathbf{u}}}_{\phi }^{\mathrm{T}}+{\widehat{\mathbf{u}}}_{\theta }{\widehat{\mathbf{u}}}_{\theta }^{\mathrm{T}}+B{\widehat{\mathbf{u}}}_{\theta }{\widehat{\mathbf{u}}}_{-\theta }^{\mathrm{T}}\right]{\mathit{\alpha }}_{\text{eff}}\left(\omega \right)\mathbf{E}\left({\mathbf{r}}_0,\omega \right),\end{array}$$

which is Eq. (2) of the paper with A and B given in the main text.

Acknowledgments

This work was partly funded by the Academy of Finland (projects 268705 and 268480) and by Dean’s special support for coherence research at the University of Eastern Finland (project 930350).

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