Electric field enhanced second-harmonic generation

EFISHG is a third-order nonlinear optical process in which one of the three input frequencies is a DC electric field.^[Reference Boyd^11] This externally applied field affects the macroscopic order of the system, namely by orienting the polar chromophores and creating a bulk asymmetry, required for a second-order optical response. The nonlinear optical polarizability (P _i^2ω) of such a system can be written

(1)$${P_i^{2\omega} = \chi _{ijkl}^{(3)} ( - 2\omega ;\,\omega, \,\omega, \,0)E_j^\omega E_k^\omega E_l^0},$$

in which E _j^ω and E _k^ω are the incident optical fields of frequency ω, E _l⁰ is the externally applied DC electric field, and $\chi _{ijkl}^{(3)} \left( - 2\omega ;\,\omega, \,\omega, \,0\right)$ is the third-order susceptibility of the bulk system for EFISHG. In the case of PR polymers, upon the application of the external field, the dipole of the chromophore causes the molecules to rotate and partially align to the field.^[Reference Dalton, Harper and Robinson^12] The bulk nonlinear polarizability in this case can be rewritten in a more convenient manner with an electric-field-dependent second-order polarizability replacing the third-order polarizability as

(2)$$\eqalign{& P_i^{2\omega} = \chi _{ijk}^{(2)} ( - 2\omega ;\,\omega, \,\omega )({E^0})\; E_j^\omega E_k^\omega; \cr & \quad \chi _{ijk}^{(2)} ( - 2\omega ;\,\omega, \,\omega )({E^0}) = \; \chi _{ijkl}^{(3)} ( - 2\omega ;\,\omega, \,\omega, \,0)E_l^0.}$$

Typical chromophores exhibit cylindrical symmetry with hyperpolarizability confined to the long axis of symmetry along which the dipole is oriented. This results in the tensor elements of the polarizability only being significant in the directions perpendicular or parallel to the poling field. As the purpose of the current study was to analyze the relative field dependence of the orientation of the chromophores subject to a spatially varying bias, only the general form of the second-order polarizability is necessary for this analysis.

The optical intensity of the generated second-harmonic [I(2ω)] is related to χ ⁽²⁾, E ⁰ and the incident optical power [I(ω)] through

(3)$$I(2\omega ) \propto \; {({\chi ^{(2)}})^2}{I^2}(\omega ) \propto \; {({E^0})^2}{I^2}(\omega ).$$

This quadratic dependence of the generated second-harmonic optical intensity on the degree of molecular orientation translates into a similar quadratic relationship to the applied electric field.

EFISHG through multiphoton microscopy

The magnitude of the second-order susceptibility for this type of poled polymer system is relatively small, requiring high fundamental frequency intensity to produce a significant signal at the second-harmonic frequency. The system employed in these measurements was a near-IR laser-scanning microscope, described in detail by Kieu et al.^[Reference Kieu, Mehravar, Gowda, Norwood and Peyghambarian^13] The fundamental illumination was provided by a 1560 nm, 190 fs pulse width, 75 MHz repetition rate fiber laser with 50 mW average power. A linear polarizer inserted into the collimated fiber laser output linearly polarizes the excitation source and results in 25 mW average power at the measurement plane. The orientation of the linear polarization was aligned with the direction of chromophore orientation using a half wave plate prior to the scan mirror. The pulses have a Gaussian shape in the time domain with an estimated peak power of 1.54 kW and energy per pulse of 0.333 nJ.^[Reference Kieu and Mansuripur^14] The microscope objective in this system was a NewFocus 0.5 NA aspheric lens, resulting in a focused beam waist of 0.609 μm. Thus, the peak power intensity was able to reach 1.06 × 10¹² W/cm².

In this type of focusing system, the objective lens imparts polarization changes to the beam profile with respect to the incident wavefront. For investigation of poled polymer systems using EFISHG, this could lead to an axial polarization component to the observed signal. For the case of an aplanatic, well-corrected system, such as the aspheric objective used in this microscope, this effect can be quantified using a geometry-based ray-bending analysis.^[Reference Mansuripur^15] For a 0.5 NA objective and incident x polarization, the magnitude of the electric-field vector components at the focal plane, |E _x|², |E _y|², and |E _z|² normalized to a total intensity of unity are 0.98, 7.15 × 10⁻⁵, and 0.018, respectively. Additionally, taking into account the I(2ω)∝I ²(ω) relation, the y- and z-polarization contributions to these measurements are negligible.

The interdigitated electrodes were 30 nm thick ITO on glass and were lithographically fabricated with 10 μm electrode width, 100 μm inter-electrode spacing, and 5 mm finger length. The polymer composite was prepared by melt processing between the electrode and a blank glass cover slide with an active thickness of 100 μm. The second-harmonic signal (780 nm) was collected with 0.5 μm in-plane microscope resolution and in 2 μm vertical steps through the bulk of the PR polymer. Each measurement plane was parallel to the electrode plane. Figure 2 shows a log scale cross-section through the resulting volume of second-harmonic signal in false color [log₁₀ (counts)] with an applied bias of 4000 V. The central 100 μm repeating structure corresponds to the region between two electrode fingers of opposite polarity with the electrodes located at the zero point on the vertical graph axis.

Figure 2.

Cross-section of the intensity [log₁₀ (counts)] of the second-harmonic signal measured through the thickness of the polymer in the case of (a) hole-conducting PATPD, 2000 V applied bias and (b) non-conductive polystyrene, 4000 V applied bias. The grounded electrode is the right electrode imaged, located ~50 μm.

Preliminary EFISHG investigations were undertaken using the standard hole conducting polymer base PATPD, with results shown in Fig. 2(a). The asymmetric form of the SHG signal is due to a decrease in electric-field magnitude near the bias electrode. This indicates the presence of photoconductivity in the device and the location near the bias electrode signifies that the mobile species are holes, as would be expected in a hole-conducting polymeric system such as this. Typically the bulk effect of photoconductivity is relatively small but in the presence of intense optical fields, this effect obscures the direct electric-field relationship to the intensity of second-harmonic signal. Subsequent characterizations were performed on a reduced conductivity system to decrease the number of assumptions that had to be made in the analysis. The polymer composite used in the final EFISHG characterizations [see sample measurement in Fig. 2(b)] was composed of the non-conductive polymer polystyrene, ECZ and 7-DCST in the weight percent ratio of 55:20:25. The polystyrene composite was comparable with that of the standard PATPD-based formulation in glass transition temperature (measured by differential scanning calorimetry) and chromophore orientation speed and magnitude (characterized by transmission ellipsometry), ensuring that the EFISHG results would be applicable to the PATPD PR uniformity analysis.

Measurements were taken over a range of applied electric fields from 0 to 4000 V and were confined to the central lengthwise region of the electrodes, far from the ends. Analysis along the length of the electrodes verifies that the orientational response is uniform in that direction in this measurement area. The intensity of the second-harmonic signal taken in the plane of the electrode (0,0) is shown in Fig. 3. Based on the quadratic relation between the SHG signal and applied field, all of the data points would be expected to lie on the same E ⁰² curve. The results, however, indicate the existence of an effective orientational barrier between 2500 and 3000 V, below which the applied field was not high enough to fully overcome the thermal reorientation of the chromophores. The SHG intensity versus applied electric field was fit to I(2ω) = C × (E ⁰)² for the low- and high-field cases with the resulting trends shown in Fig. 3. Above 3000 V applied bias, the standard chromophore orientation response to the electric field in the PR response models was valid. As indicated by the results at low bias, an orientational threshold voltage exists for the system under test and a modification to the following model was necessary for the regions of the device with lower field magnitudes.

Figure 3.

Intensity of the second-harmonic signal as a function of applied electric field. Low range fit is for applied electric fields <2500 V, while the high range fit is for the >2500 V range.

Diffraction uniformity model

A map of the electric field was calculated for the experimental coplanar electrode geometry for use in the PR model to predict diffraction uniformity. By assuming an infinite extent of the electrodes along their length, this 3D geometry can be reduced to a two-dimensional (2D) analysis, producing the applied electric fields and orientation in the PR material with minimal computation time. This assumption is valid due to the configuration of our electrodes, with the length 50 times larger than the electrode width and spacing and the illumination occurring well away from the ends of the electrodes. The Laplace equation, ∇²V = 0, describes the electrostatic potential in a non-conductive media according to Dirichlet boundary conditions which set the applied bias at the electrodes, and Newmann conditions which constrain the form of the response along the remaining boundaries.

The response of the polymer system to an external bias depends on the overall magnitude of the electric field [Fig. 4(a)] in some cases; in others it is the projection of the electric-field vector onto the grating vector that is the dominant factor. Figure 4(b) shows the calculated electric-field projection onto a horizontal grating vector resulting from an untilted geometry. Figure 4(c) displays the comparison of the theoretical horizontal projection of the applied electric field and the square-root of the EFISHG values [I(2ω)∝(E ⁰)²] in the plane of the electrode (z = 0) at 4000 V applied bias. Since the EFISHG measurements are qualitative in nature for this application, the measurement was normalized to the central field value for the model. It can be seen that there is a good degree of agreement between the electric-field model and the measurement which means that upon taking into account the low-field orientation correction factor, this electric-field model can be used for predicting PR response to coplanar electrodes. The decreased peak intensity of the SHG signal at the edges of the range seen in Fig. 4(c) is due to the slightly curved focal plane of the scanning beam across the extent of the image plane. The central electrode region spanning from −55 to +55 μm in Fig. 4 is the section over which the generated signal that was collected originates from the same 2 μm vertical measurement step. Outside this range, the focal location of the fundamental beam is higher, reducing the peak intensity with which the second harmonic was generated. Owing to this, the central electrode region is the one that is used for measurement analysis, such as in Fig. 3.

Figure 4.

(a) Electric-field magnitude and (b) horizontal vector component applied by coplanar electrodes at 4000 V bias. Electrode width 10 μm, spacing between electrodes 100 μm, magnitude units in V/μm; (c) comparison of theoretical horizontal electric-field component at the electrode surface (z = 0) with the value obtained from EFISHG measurements at 4000 V applied bias.

Dynamic models of photorefractivity in polymers have long been used to direct the formulation of new and more efficient PR devices. Ostroverkhova modified the Schildkraut and Buettner model describing the relationship between material parameters and electric fields and their contribution to the PR response.^[Reference Ostroverkhova and Singer^16–Reference Schildkraut and Buettner^18] The response depends on a variety of material parameters such as sensitizer densities and trap levels, but in this case the focus is on the effect of the electric-field-dependent parameters. These are the photogeneration cross-section (s), hole drift mobility (μ), and trapping (γ) and recombination (γ _T) rates with field dependencies on either the magnitude of the applied field or the projection of the field onto the grating vector. The calculation parameters for this analysis were set to match the subsequent test cases, with angle of 30° between the two writing beams, tilt angles ranging from 0° to 20° with respect to the sample normal (effectively changing the orientation of the grating vector with respect to the sample normal), a recording wavelength of 532 nm, and illumination of 30 mW/cm². Using the electric-field model to evaluate these parameters across the electrodes, and incorporating the low-field adjustments from EFISHG measurements, the resulting space-charge field was used to calculate the refractive index modulation of the form

(4)$$\Delta n = {\rm constant} \times {E^0} \times \left\vert {{E_{{\rm sc}}}} \right\vert \times \cos \,\varphi,$$

where φ is the angle between the grating vector and the electric-field vector. The constant term is related to material parameters and is not field dependent. The orientational corrections to the electric field from Fig. 3 decrease the low applied field contribution to the index modulation.

The index modulation calculated here results in the diffraction grating that is recorded in response to the interference of the writing beams in the material, so the decrease in electric field with increasing distance from the electrodes [Figs. 4(a) and 4(b)] and subsequent refractive index modulation that is an order of magnitude higher near the edges of the electrodes than in the remaining bulk of the material affects the diffraction of the reading beam. Computational propagation of the reconstructed beam through the grating to analyze diffraction uniformity across the beam was performed using the Crank–Nicholson method.^[Reference Press, Teukolsky, Vetterling and Flannery^19] Implementation of periodic boundary conditions took advantage of the repeating nature of the electrode structure, so only three repeating electrode structures were used along with a device thickness of 100 μm. Solution verification was performed by calculating the diffraction grating due to a uniform electric field under the same writing conditions and propagating both planar and Gaussian wavefront optical fields through it. Both incident field results showed the expected 100% uniformity of the diffracted beam. These tests additionally validated the use of an infinite extent plane wave in the following analysis to remove the Gaussian envelope from the uniformity analysis.

Figure 5(a) shows the results of the spatial distribution of the diffracted beam after propagation through the PR polymer across one period of the electrode structure. The effect of the spatially inhomogeneous electric fields is to concentrate diffraction of the reconstructed beam near the electrode edges, which affects the resulting beam uniformity in a high spatial frequency manner. Tilting the writing beams with respect to the normal to the sample plane increases the number of these high index regions that the reconstruction beam encounters. For a range of tilt angles from 0° to 20°, the high-frequency components peak power to root-mean-squared (RMS) ratio decreases by 15%. Also called the crest factor in signal processing, the peak to RMS ratio is a measure of how extreme the peaks are in a signal, and indicates an increase in the spatial homogeneity of the reconstructed beam with increasing tilt angle.

Figure 5.

Propagated reconstructed beam at 0°, 10°, and 20° degree tilt angle geometries of the recording beams in (a) electrode and (b) full beam scales.

To analyze the low spatial frequency effects of diffraction from the underlying electrode structure, an 800 μm diameter Gaussian beam was propagated through basic split-step beam propagation and absorbing boundary conditions with a 40 mm analysis space. The grating parameters and thickness remained the same with respect to the previous model. The change in the analysis method used here was due to the increased robustness of this propagator at larger sampling steps (to counter the larger analysis space in computation time), which, while obscuring some of the fine detail represented in Fig. 5(a), reproduced the lower spatial frequency results accurately. Figure 5(b) shows the propagation results of the full width of the reading beam in the output plane (total diffracted angle 30°). The central portion is the primary diffracted beam from the high-frequency grating and is Bragg matched to that grating. The presence of side lobes is due to diffraction from the underlying low spatial frequency refractive index modulation from the periodicity of the applied electric field. The angle of diffraction from this grating period is only ~0.09° and both of the side lobes deviate slightly from the Bragg angle of the high-frequency grating, reducing the efficiency of their diffraction. With increasing tilt angle, the diffracted order that is closer to the sample normal than the reconstructed Bragg angle (left peak) is less efficiently diffracted by the base structure due to its increasing Bragg mismatch.

Model validation through non-degenerate four-wave mixing

Experimental verification of the applicability of these adjusted models in device design was executed. Non-degenerate four-wave mixing measurements were performed in which a grating was formed by two TE polarized 532 nm mutually coherent beams with an inter-beam angle of 30°. The orientation of the coplanar electrodes was also such that the electrodes extended in the TE direction, ensuring that the generated charges traveled across the fringes to become trapped instead of along them. A 633 nm reading beam counter-propagating at the appropriate angle accounting for the wavelength mismatch diffracted from the grating and was monitored by a CMOS detector for 0°, 10°, and 20° tilt angles. Figure 6(a) shows the original and diffracted beam profiles taken in a cross-section perpendicular to the plane of diffraction. As expected, there was no significant change in the profile of the beam since this is the direction along the fringes and perpendicular to the underlying electrode index modulation. Figure 6(b) is a cross-section in the plane of the electrode index modulation and multiple orders flanking the primary diffracted beam are present. Note that the original beam intensity was scaled to the range of the central diffracted lobes. Variability between the intensity of the central lobes for the different tilt angles is due to minor alignment deviations between the configurations but normalization of these results would have unduly emphasized the fine structure of some of the measurements. The decrease in the intensity of the left order follows that of the theoretical model [Fig. 5(b)]. The fine structure inhomogeneity from the micron-scale diffraction analysis is seen in the side lobes orders but is insignificant with respect to the magnitude of the full diffracted beam. This indicates that the non-uniformity due to the spatial electric-field modulation is a more significant effect on the secondary orders.

Figure 6.

Cross-sections of the measured incident and diffracted beam profiles taken (a) out of the plane of diffraction and (b) in the plane of diffraction for various tilt angles.