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We present a novel design method and sensing scheme for an electro-optic field probe using multi-stratified layers of electro-optic wafers. A serial stack of cascaded layers is found to be capable of enhancing the performance of interferometric electro-optic light modulation that utilizes the slopes of interference fringe patterns and field-induced electro-optic phase retardations within wafers. The absolute sensitivity of the probe is also characterized with a micro-TEM cell that generates electric fields distributions with accurate, calculable strength for use in probe calibration. The sensitivity of a multi-layered probe-per unit electro-optic wafer volume - was enhanced by 6 dB compared to that of a single-layer one.
©2010 Optical Society of America
1. Introduction
The electro-optic (EO) sensing technique, due to its electrically transparent nature, has served as a promising and useful method where minimally invasive measurements of electric fields are necessary [1]. EO sensing, as well as photoconductive switching/sampling [2,3], has been widely adopted for detection of fast electrical transients having bandwidths that exceed the measurement capability of electronic methods. Besides their application to time-domain transient measurements that employ pump-probe-style interactions, the EO sensors are also suitable for detecting the vector components of continuous-wave electric-field radiation without requiring a pulsed laser source [4,5]. However, one of the biggest challenges that has prevented the EO-sensing method from becoming a more attractive scheme is its distinctly lower sensitivity compared to techniques employing metal-based field sensors, which are commonly used in antennas or radiated-field measurements.
Conventional single- or double-pass EO configurations exhibit limited sensitivity due to the inherently minute nature of the Pockels effect on the refractive index of the sensor medium. To address this issue, numerous research efforts have attempted to increase the electro-optic interaction length within EO sensors. One successful scheme is to build a resonator out of the EO crystal. A balanced resonator (Fabry-Perot type) is commonly used to enhance the EO phase retardation [4–6], and such resonant retardation can be enhanced even more significantly through the use of an unbalanced structure (i.e., of Gires-Tournois type) [7]. Another resonant scheme, which avoids the need for coatings on the sensor, is the utilization of highly reflective surfaces on the device under test (DUT) for coupling the modulated light back onto the original optical path [8–11]. However, such interface-coupling configurations cannot be realized without a reflective interface. Moreover, since the coupling efficiency relies strongly on the external conditions associated with the DUT, this method would not be suitable for maintaining the delicate alignment and resonance condition when a probe-scanning procedure is necessary.
In this paper, we present a novel design approach for realizing such a resonance-based, electro-optic-sensing scheme, without relying on coatings or highly reflective surfaces. Utilizing more than one blank EO wafer, the reflectance resonance characteristics can be improved, and thus the EO phase retardation [12] and the slope of the function responsible for phase-to-amplitude modulation-conversion (which are also associated with the multi-stratified embodiment) are also enhanced.
2. Spectral response of a multilayer EO-sensor system
Prior to presenting the EO analysis of a multilayer system, the steady-state transfer functions for both reflection and transmission, based on feedback theory, are reviewed [13]. For a single layer case (N = 1), the overall reflectance is associated with reflections at the front boundary, r 2, as well as at the back boundary, r 1. A similar relationship is also valid for the double layer case (N = 2), in which the overall reflection at the foremost boundary, R (2), is obtained from r 3 and R (1) (the overall reflection from the second interface). The numbering convention is clarified in Fig. 1 , where the input is shown arriving from the left.
Fig. 1 Optical N-layer system with arbitrary refractive indices and thicknesses for the constituent layers. The light incidence is from the left. The layers are numbered starting from the right with layers 1 and 2, an arbitrary intermediate layer k is in the middle, and the last layer (N) is on the left.
This recursive relation can be extended to the multiple, N-layer case as shown in Fig. 1. The overall reflection at the k + 1th boundary (≡ R ( k )), where k is an arbitrary internal layer, is generalized with r k and R ( k- 1) . As a result, the overall reflection at the N th layer is expressed as Eq. (1)
whereandAt the N + 1th boundary, the reflected beam intensity is Ir ( N ) = |R ( N )|2, and thus It (1) = |T (1)|2 is the remaining transmitted beam out-of layer 1 for a lossless medium. (The conventional definition tk = 2nk/(nk-nk- 1) is the Fresnel electric-field transmission coefficient. Here, tk is a transmission field coefficient to satisfy the Ir ( N ) + It (1) = 1 relation.) The transmission also has recursive characteristics associated with any previous equivalent transmission. However, the transmission at each layer also has to include the previous equivalent reflectance (i.e., feedback) terms. This is because transmissions share the identical feedback terms described using reflection.
The generalized transmission out-of an N-layer system is written as
whereThe superscripts on R and T again indicate the layer numbers (Fig. 1).3. Modeling the EO response of a multilayer system
A single dielectric layer is basically a Fabry-Perot etalon, and thus it has unique transmission characteristics. The transmittance (or reflectance) of the etalon will be modulated as the refractive index of an EO medium is perturbed by an applied electric field. Such EO modulation effects can be enhanced by adding other layers of appropriate thicknesses.
Figure 2 is a schematic of a fiber-mounted EO probe that consists of three stratified layers. The incoming optical beam is transmitted through a fiber core (ncore ~1.5), after which it passes layer 3 (LiTaO3), layer 2 (an optical adhesive with n2 ~1.5), and layer 1 (identical to layer 3). Layers 1 and 3 are x-cut LiTaO3 wafers, and thus both of their optic-axes are perpendicular to the travel of the optical beam.
Fig. 2 Structure of a multi-layered electro-optic probe (Layers 1 and 3 are thin LiTaO3 that yield phase modulations δ1 and δ3).
Reformulating Eq. (2) into the standard feedback format, the transmission from such a three-layer system can be expressed as Eq. (3).
where the first-order feedback terms arethe second-order feedback terms areand the third-order feedback terms areThe known birefringence and EO parameters for LiTaO3 at 1558 nm are ne = 2.1224, no = 2.1186, r33 = 27.4 pm/V, and r13 = 6.92 pm/V [14]. The simulated reflectance (Ir (3) = |R(3)|2 = 1-|T(1)|2 = 1-It (1) for lossless media) of this probe for 50-μm-thick EO wafers and a 10-μm-thick adhesive gap is shown in Fig. 3 . The polarization of the laser was assumed to be linear along the x-direction (extra-ordinary axis), and the refractive indices of the fiber and adhesive were set at 1.5.
For efficient EO amplitude modulation, the slope of the reflected-intensity fringes needs to be steeper over a broad spectral region. This condition can be achieved by reducing the gap between wafers or by using a gap-medium of lower index. The EO crystal LiTaO3 has two birefringence terms, given as Δno,e (E,polarization) = (ne-no) + (ne3r33-no3r13)E/2 [15]. These terms are the natural and electrically-induced birefringences, respectively. The Δno,e term can therefore be controlled with E fields and polarizations. As the light is linearly polarized along the e-axis (or o-axis), the phase retardation by the crystal is δe(E,λ) = 4πneh/λ + 2πne3r33hE/λ (or δo(E,λ) = 4πnoh/λ −2πno3r13hE/λ), because the light sees only one refractive index throughout the crystal. As the polarization of the laser and electric fields are applied along the optic-axis, the refractive index of the LiTaO3 changes by (ne3r33)E/2. This index modulation is so tiny that the sensitivity of the EO amplitude modulation is proportional to ∂It (3)(E,λ)/∂n(E). Since It (1)(E,λ) + Ir (3)(E,λ) = 1-loss(λ), the sensitivity becomes ∂It (1) / ∂n = -∂Ir (3) /∂n, where n is an electro-optical variable (here, n = n1 = n3 = ne). The general expression of EO sensitivity for an N-layer system that contains EO refractive index nEO can be described as ∂It (1)(E,λ)/∂nEO(E). Its full analytical expression (which is quite lengthy) or numerical solution also can be obtained with commercial mathematical tools. It should be noted that this approach also inherently deals with resonance-based phase enhancement terms as described in refs. 8, 9 and 10.
The reflectance and corresponding EO sensitivity within the wavelength region appropriate to optical-communication sources are computed and shown in Fig. 4 . Figure 4(a) shows the EO sensitivity (reflectance change over refractive-index modulation) compared to the slopes of the reflectance with the simulation parameters as in Fig. 3. However, as we remove the gap between the EO wafers (that is h1 = 100 μm, h2 = h3 = 0), the fringe slope is reduced, as observed with the correspondingly reduced EO sensitivity in Fig. 4(b).
Fig. 4 Simulated reflectance fringes and corresponding EO strength for the probe in Fig. 2. (a) h1 = h3 = 50 μm, h2 = 10 μm, n1 = n3 = ne (b) h1 = 100 μm, h2 = 0 μm, n1 = ne. (The right vertical axis is the reflectance change over 0.01% of refractive-index modulation).
4. Experimental evaluation of a multi-layer EO probe
A fiber-mounted electro-optic probe was assembled based on the proposed configuration in Fig. 2. The thickness of each LiTaO3wafer was 50 ± 5 μm, and a narrow gap between the two wafers was filled with UV-curing cement. The wafer stack was pressed together in order to minimize the gap between the EO wafers. The reflectance of the probe, measured using a widely tunable external-cavity diode laser as the optical source, is shown in Fig. 5 . The envelope of the fringe peaks implies that the gap is ~2 μm thick, and thus the range of sharp fringes is reasonably broad. In addition, the minor fringes between the main ones originate from the slight difference in the two wafer’s thicknesses. The thickness of each wafer has controlled the mode spacing of each fringe.
The experimental setup for exploring the amplitude modulation of the electro-optically-induced birefringence is given in Fig. 6 .
Fig. 6 Experimental schematic for an all-fiber-based EO-probe calibration system. (The gray and black lines are optical fibers and electrical connections, respectively).
The sensitivity of the EO probe, as well as the absolute values of measured electric fields, can be readily characterized using a standard field-generating device such as a micro transverse-electromagnetic (μ-TEM) cell [16,17]. The setup is basically identical to the one used in our previous report [18] except a tunable laser and a gain controllable photodetector are adopted. A fiber-pigtailed external cavity diode laser, manual polarization controller, optical circulator, and fiber-connectorized photodiode were spliced together with the probe to form a continuous, low loss, enclosed optical path. The optic-axis of the sensor is placed parallel to the electric fields to maximize the sensing fields. The laser power is ~4 mW and the electrical input power at 100 kHz was variable up to + 40 dBm so that the EO sensor’s linearity and dynamic range could be explored up to a reasonably high power.
The EO signal diminishes by about half as the multi-layered LiTaO3 probe is replaced with a single, 100-μm-thick EO wafer, as seen in the comparison between Figs. 7(a) and 7(c). Thus, a multi-layered structure is found to be significantly more efficient compared to a single-element sensor with the same EO-medium volume.
Fig. 7 Reflectance fringes (dashed lines) and corresponding EO strength (solid lines) for the probe in Fig. 2. (a) h1~h3~50 μm, h2~2 μm, n1 = n3 = ne case; (b) same as (a) except n1 = n3 = no; (c) same as (a) but h1 ~100 μm only.
The absolute sensitivity of the probe and its dynamic range can be explored utilizing a standard TEM cell through a calibration procedure. The peak signal in Fig. 7(a) corresponds to a photodetector output-power reading on the lock-in amplifier (see Fig. 6) of −65.2 dBm, and this value linearly decreases as the electrical feeding power is reduced from + 40 dBm down to ~ + 2 dBm, until it reaches ~-104 dBm, which is the upper limit of the experiment’s noise floor.
To utilize the interferometric fringes as EO amplitude-modulation slopes, it should be noted that the polarization of the beam ought to be identified. As polarization is linearly aligned along one of either the ne or no axes, the slope contrast becomes maximized by having a single index component. The relatively small, static, natural birefringence (ne-no) gives comparable modulation slope for either (ne or no) birefringent refractive index, with some spectral offset. The reflectance of the respective ne and no alignment cases is shown in Figs. 7(a)-7(b). Figure 7(a) (or Fig. 7(b)) can be identified as representing the situation when the polarization is along the ne (or no) axis by exploring the respective EO sensitivities. That is, the EO sensitivity is proportional to the field-induced birefringence, and the amplitude-modulation slope is primarily determined by the refractive index of the crystal. While the modulation slope for either (ne or no) birefringence index is comparable, the field-induced dynamic birefringence (ne3r33-no3r13)E/2 varies up to approximately four times mainly due to the EO coefficient difference (i.e., r33 >> r13). Hence, it is crucial to ensure the appropriate polarization is realized through the use of a polarization controller.
Fig. 8 Measured EO signal strength (black) and calculated electric field strength (gray) in the μ-TEM cell versus feeding power. (This relates the measured signals and the calculated fields). The interconnecting lines are guides to the eye.
The measured signal versus the variable electrical driving power can be mapped into the absolute electrical field strength in V/m because the field in the TEM cell is readily calculable. The minimum detectable field strength is found to be ~8 V/m at + 2 dBm input power. Such sensitivity is suitable for diagnosing the reactive near-field of transmission antennas or radiators because the typical field strength exceeds the noise equivalent level by 1-2 orders of magnitude. Thus 20-40 dB of dynamic sensing range can be attained. If further enhancement is desired, adding a third wafer or more would be an appropriate solution. However, stacks with a large number of wafers will, in practice, degrade the coupling efficiency back to the original fiber core. Efficient coupling can be realized by employing a fiber facet with a thermally expanded core or a quarter-pitch GRIN lens, as these would couple less-diverged and retro-reflected beams from even more EO layers.
Finally, this multi-layer technique could be extended to a magneto-optic (MO) or combined EO/MO sensor [19] design to measure magnetic fields independently with enhanced sensitivity. Consequently, a map of the Poynting vector could be experimentally determined after obtaining all the relevant E- and H-field components [19,20].
5. Conclusions
We have demonstrated a cost-effective electro-optic probe utilizing the interference fringes resulting from the multiple reflections within a multi-layer sensor structure. General modeling of the transmission/reflection of a light beam and its electro-optical amplitude modulation function due to field-induced phase retardations associated with multiple EO wafers are presented. The probe performance was evaluated by comparing different polarizations and sensor layouts. The absolute sensitivity of the probe was also determined using a field calculable micro-TEM cell, so the measured fields could be calibrated into absolute field strength in V/m. This design methodology is suitable for enhancing EO modulation depth by utilizing multiple interferometric interactions of EO layers without building dielectric-coated resonators.
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