Optical nanoantennas made of two metals are proposed to produce a Seebeck voltage proportional to the Stokes parameters of a light beam. The analysis is made using simulations in the electromagnetic and thermal domains. Each Stokes parameter is independently obtained from a dedicated nanoantenna configuration. S1 and S2 rely on the combination of two orthogonal dipoles. S3 is given by arranging two Archimedian spirals with opposite orientations. The analysis also includes an evaluation of the error associated with the Seebeck voltage, and the crosstalk between Stokes parameters. The results could lead to the conception of polarization sensors having a receiving area smaller than 10λ2. We illustrate these findings with a design of a polarimetric pixel.
© 2014 Optical Society of America
Nanoantennas and resonant structures are at the core of new advances in photonics due to their capability to enhance and manipulate optical fields in the nanometric scale. Moreover, they allow the design and fabrication of new nanophotonic devices [1, 2]. When using optical antennas to sense and detect optical radiation in the infrared, we take advantage of the selective properties in polarization, direction, and wavelength of the resonant structures. These basic properties, specially the selectivity to the polarization state and directivity are preserved when moving to the optical range. This is because they are mostly related with the geometry, meanwhile spectral selectivity is affected by the dispersive properties of materials at optical frequencies. Also, optical antennas have a very small footprint with a reception area in the order of λ2 [3, 4]. This makes it possible to obtain a high spatial resolution in the detection of light. Unfortunately, some drawbacks appear when dealing with optical antennas working as light detectors. Transduction mechanisms have to deal with very low currents or voltages generated by the incident light, and therefore signals are also very weak. Two main transduction mechanisms have been used so far. One of them is the rectification of the currents built in the antenna structure by metal-insulator-metal (MIM) junctions. This mechanism provides a signal proportional to the irradiance on the device . The bolometric effect is also routinely used in antenna-coupled detectors . It is driven by the change in temperature caused by the currents flowing through the antenna. A bolometric device may contain a nano-bolometer located at the feed point of the antenna, or the whole resonant structure is used to produce the desired change in resistance in distributed bolometric devices [7–9]. Due to the low signal involved in these mechanisms, synchronous detection is required. Furthermore, bolometric and symmetric MIM junctions require external biasing to detect the response of the device.
Antenna-coupled thermopiles have been analized previously in the infrared region [10, 11]. The Seebeck effect relies on a temperature difference to produce a voltage . In this contribution we use optical antennas to produce this difference in temperature. Seebeck nanoantennas are composed of two metals that configure a junction that, in its simplest configuration, is located at the feed point of the antenna. When light illuminates the structure at a certain wavelength and at a certain polarization state the induced currents heat up the antenna and the junction. Metals exhibit a moderate Seebeck effect described by their Seebeck coefficient. However, an appropriate choice of materials can be made to increase the performance of such device. However, moving towards nanometric devices may require a full characterization and measurement of metals at the nanoscale, and the analysis of the thermal behavior of nanometric structures [11, 13, 14].
Polarimetric imaging in the infrared is used in astronomy , thermal imaging systems [16, 17], and optical characterization of light and media [18, 19]. These systems use several strategies to detect certain characteristic parameters of the incoming light [15, 16, 20]. In the infrared, the options to manufacture a polarization sensitive pixel are reduced and require the addition of filters and auxiliary elements . As an alternative solution, the polarization selective properties of optical antennas are used in this contribution to define the hot and cold junctions of a thermocouple that is selective to polarization. Combining antenna configurations having different response to the polarization of the light, we can obtain a signal that is proportional to the difference in response of these antennas. This has been proved previously using bolometric devices coupled to orthogonal dipoles , but it requires the combination of the signals obtained from the individual elements. Actually, the rise in temperature at the feed point of the antenna is proportional to the optical power incident on the device that is selectively resonating according with its state of polarization. An array of at least two optical antennas properly designed to respond to orthogonal polarization states can be used to obtain a signal proportional to the difference in power carried out by the chosen polarization states. This is what we need to detect the Stokes parameters. Furthermore, the capability of optical antennas and thermocouples to be combined in series can be used to multiply the signal by increasing the number of elements. On the other hand, when placing together antenna elements selective to the polarization states that define the Stokes parameters, it is possible to configure a polarimetric pixel that provides the vector Stokes of a light beam. This pixel also takes advantage of the small receiving area of optical antennas, having itself a lateral dimension of very few wavelengths and allowing a high spatial resolution.
Section 2 is devoted to the presentation and evaluation of the proposed arrangements sensible to the Stokes parameters. An analysis in the electromagnetic and thermal domains is made using COMSOL Multiphysics (COMSOL Inc., Burlington, MA 01803, USA). The reliability of this approach has been tested in previous contributions where the simulated results have been compared with actual values from fabricated devices [7, 9]. Once the designs worked as expected, we have tested its response by evaluating the signal obtained from them when illuminated by a variety of states of polarization. This is explained in section 3. We have also included in this section a possible design of a four-element pixel for polarimetric measurements. Finally section 4 summarizes the main findings of this paper.
2. Design of Seebeck nanoantennas for polarimetric measurements
When characterizing the polarization state of a given light beam, it is commonplace to look for the value of the Stokes parameters. They form a set of 4 numbers that fully describe the polarization of a light beam, even for incoherent, or partially polarized light beams . In this paper we will use the following form of the Stokes parameters:Eqs. (2)–(4) can be seen as the difference in power for three choices in this decomposition. S1 works for a decomposition into two linear polarizations aligned along the x and y directions. S2 is also for two linear polarizations now aligned along the 45° and −45° directions. Finally, S3, represents the difference in power when decomposing the light into Right-handed Circular Polarization (RCP, dextro) and Left-handed Circular Polarization (LCP, levo). The aim of the designs proposed here is to detect S1, S2, and S3 independently.
Therefore, to measure the value of the Stokes parameters we should obtain a difference in power between two well defined and orthogonal states of polarization. This is where Seebeck nanoantennas can be useful. Fortunately, the selectivity in polarization is linked with the geometry of the antenna . The Seebeck voltage is proportional to the difference in temperature reached at two locations where the junction between two materials are placed. One of them acts as a hot spot and the other works as a cold one. Nanoantennas are used to couple the electromagnetic radiation having a specific state of polarization to the location of the junction that is now placed at the feed point of a nanoantenna made of two different metals. As far as S1 and S2 are related to linear polarization states, the antenna design used for the measurement of these parameters should be selective to linear polarization. The simplest element that can accomplish this task is a dipole antenna. Figure 1(a) shows a couple of orthogonal dipoles. The dipoles are designed using two different metals in the two half-dipole arms at each side of the feed point (located at the center of the dipole). These two arms are different in length to account for the difference in their response to electromagnetic radiation at infrared frequencies . These two orthogonal dipoles are connected between them and to the outer circuit using a lead line. This design has been tested previously for a single dipole and provides a good thermal response [6, 7]. Light incident on each one of the orthogonal dipoles produces a current on the dipole that heats, through the Joule effect, each dipole. This localized heating is proportional to the power incident on each dipole at the corresponding linear polarization direction. Since the Seebeck effect is proportional to the difference in temperature between the junctions (hot and cold), and the temperature at each junction is proportional to the power coupled to the nanoantena, the proposed device becomes a natural candidate to measure S1. The same type of arrangement working for the S1 parameter can be used for the S2 just by rotating it 45°. To measure S3 we propose the use of two Archimedian spirals having opposite rotation directions (see Fig. 1(b)). The two components (dextro and levo) are coupled selectively to each one of the spirals. Then, the Seebeck signal is proportional to the difference in power carried out by each one of the components.
The detailed design of the selective thermocouple begins with the choice of two available metals showing an appropriate difference in the values of their Seebeck coefficients. In this analysis we have chosen nickel and titanium having sNi = −19.5μV/K, and sTi = 7.19μV/K . The antennas are placed on a semi-infinite Si substrate having a SiO2 coating layer (200 nm in thickness) that works as a very efficient thermal insulator. The resonant structures are optimized for a wavelength of λ0 = 10.6μm incident from vacuum. The incoming irradiance has a constant value for all the cases treated here of 117 W/cm2. The evaluation of the performance of these elements is made using COMSOL Multiphysics. This Finite Element Method package determines the heat dissipation in the structure produced by an incoming electromagnetic wavefront in the infrared. This heat distribution produces a temperature pattern that allows to obtain the temperature difference at the location of the junctions in the resonant structures. The electromagnetic and thermal constants are those of the materials involved in the simulation and extracted from reference .
Figure 2 shows a collection of temperature patterns for several states of polarization. We can see how the temperature increases selectively along the structure when varying the polarization state of the radiation. Actually, the distribution of temperature follows a profile determined by the geometry and material parameters. Figure 3 shows the temperature profile when moving along the structure. For the case of the dipole configuration this profile is considered along the lead line. For the spiral, the chosen path represented in Fig. 3 turns with the spiral and moves from one side to the other of the structure. In this figure we have marked the location of the junctions as vertical lines. It is important to note here that the Seebeck signal does not depend on the variation along the metal structures. It is only related to the difference in temperature at the location of the junctions between metals. The calculation has been made for several states of polarization having different values of the Stokes parameters S1 and S3. The dipoles are illuminated with a linear polarization light that moves along the equator of the Poincaré sphere (plane S3 = 0). The light incident on the spirals is a linear combination of dextro and levo circular polarization. In this case, the polarization states are moving along a meridian of the Poincaré sphere where S2 = 0. All these states of polarization have the same value of total irradiance.
The antenna arrangement proposed in this contribution produces a signal that is dependent on the polarization state of the incident light. Figure 4 shows the Seebeck voltages obtained from the dipole configuration and the spirals when varying the value of the S1 and S3 parameters. For the spirals we have also checked the contribution of linear polarized light to the signal. Also, for the dipoles we have evaluated the contribution of elliptical polarization to the signal. We have obtained that spirals respond to linear polarization following a linear dependence wiht S2. On the other hand, we have not found any significant contribution of the S3 parameter when illuminating the two orthogonal dipole arrangement. In Fig. 4 we have plotted error bars that represent some uncertainties in the practical realization of the elements. For the case of the dipole arrangement we have taken into account the temperature distribution on the surface of the junction. Because of the different thermal constants of the materials used, and the geometry of the antenna, the temperature varies across the junction. In Fig. 5 we have represented the temperature map on the junction surface. The number within each graph is the mean and standard deviation of the temperature distribution on the surface. This standard deviation is taken as the error in each one of the junctions. The error bars plotted in the S3 plot are obtained when moving the location of the junctions by a distance related with the uncertainty in the manufacturing. This uncertainty is fixed as 50 nm. In this case the temperature changes about 9 mK.
Therefore, after evaluating the behavior of the dipoles and spirals arrangements we may write the dependence of the signal obtained for each arrangement as it follows:Fig. 4. For the case treated here we find that α11 = α22 = 4.51 ± 0.03μV, α32 = 1.0536 ± 0.0009μV, α33 = 4.5963 ± 0.0003μV, δ1 = δ2 = 0.002 ± 0.02μV, and δ3 = 0.0040 ± 0.0006μV.
3. Detection of the Stokes parameters
In the previous section we have analyzed how the signals obtained from Seebeck nanoantenna arrays are linearly proportional to the Stokes parameters. In this section we are interested in having the value of the Stokes parameters in terms of the signals given by the arrays. To do this we have prepared polarization states that are illuminating the antenna arrangement.
The results of this evaluation have also been also represented in Fig. 6 on the Poincaré sphere. All selected beams are in the same octant of the sphere for better representation. The location of the original polarization states are represented as blue dots on the sphere. The red dots represent the points defined by S⃗ant and obtained from the signal delivered by the arrangement, V⃗ant, and obtained using Eq. (9). These red dots predict the state of polarization. The values given for the Stokes parameter by the Seebeck nanoantenna arrangement are not on the surface of the sphere. This location has been corrected by normalizing it using the modulus of S⃗ant. The normalization has made it possible to represent the ellipse of polarization for each case. Figure 6 also shows the cases treated here representing in blue the original ellipse, described by S⃗, and plotting in dashed red line the ellipses given by the normalized Stokes parameter obtained from S⃗ant. In this figure we have also plotted in yellow the polarizaton states used to find the linear fitting coefficients described in Eqs. (5) – (7).
From a practical point of view, we also propose an antenna-based polarimetric pixel. In Fig. 7 we show a possible realization of the pixel showing the small footprint of the element. In this arrangement we have included two thermocouples connected in series for the detection of S1, S2, and S3. The signal for the S0 parameter can also be obtained from a thermocouple where the hot junction is located at the feed point of an antenna having no selective detection of the polarization state.
The Seebeck effect has been used to provide a transduction mechanism for the signal coupled to an array of two optical antennas working in the infrared. Since the Seebeck voltage is proportional to the difference in temperature at the location of the junctions, and this difference in temperature is related to the heat produced by the optical irradiance coupled to the antenna structures, then the signal can be used to provide a detection mechanism proportional to the Stokes parameters of a light beam.
Two dipoles oriented perpendicularly to each other are responsible for the detection of the S1 and S2 parameters. Two archimedian spirals having opposite helicities are combined to produce a signal proportional to S3. In both cases, the hot and cold junctions are located at the feed point of the antennas. Therefore, each arm of the antenna are made of different materials. The results given here have been obtained using a Finite Element Method that combines the electromagnetic and thermal response of these structures. We have also evaluated some uncertainties linked with practical realizations of the structure. The crosstalk between Stokes parameters has been analyzed and characterized. The system has been tested using a collection of elliptical states of polarizations showing good agreement between the actual Stokes parameters of the input beams and the values of the Stokes parameters given by the proposed resonant structures. Finally we have outlined a practical realization of a pixel containing four subpixels, one for each Stokes parameter. This polarimetric pixel based on Seebeck nanoantennas has a very small area allowing a high spatial resolution for polarimetric imaging, where the Stokes parameters are obtained independently.
This work has been partially supported by project ENE2009-14340 from the Ministerio de Ciencia e Innovación of Spain, by project “Centro Mexicano de Innovación en Energía Solar” from “Fondo Sectorial CONACYT-Secretaría de Energía-Sustentabilidad Energética”, and by grant CV-45809 from Consejo Nacional de Ciencia y Tecnología of México.
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