Homodyne detection of second-harmonic generation as a probe of electric fields

We demonstrate and analyze a new method for probing electric field strengths using optical second-harmonic generation. The technique, based on a homodyne detection scheme, employs interference between the field-induced second-harmonic radiation from the sample and strong second-harmonic radiation from a reference. The scheme provides a linear relationship between the measured secondharmonic signal strength and the amplitude of the electric field being probed, thus providing easy calibration of the amplitude of the electric field and direct information on its sign. Experimental results are presented for direct and homodyne detection of in-plane fields in silicon structures. A discussion of the expected signal-to-noise characteristics is presented and the results are compared to experimental findings. Homodyne detection of electric fields with strengths on the order of 100 V/cm can be achieved with reasonable integration times. PACS: 42.65.-k; 42.65.Ky; 42.70.Nq Optical second-harmonic generation (SHG) has been widely recognized as a surface-sensitive probe in centrosymmetric materials [1, 2]. As is well known, this sensitivity arises from the fact that the SHG process is dipole-forbidden in a centrosymmetric medium. The breaking of the inversion symmetry at a surface or interface greatly alters SHG from the sample. The same principle is operative when an electric field E0 is applied to a centrosymmetric material. The electric field E0, as a polar vector, acts to lift the inversion symmetry of the material [2]. The efficiency for SHG is thus strongly influenced and measurement of SHG provides a sensitive probe of electric fields present within the optical probing volume. The effect of an applied electric field is thus quite different from that of an applied magnetic field. The latter can alter existing SHG contributions, but, as an axial vector, does not cause the inversion symmetry of the bulk to be lifted [2]. The high sensitivity of SHG for probing of the electric fields was demonstrated early in the history of nonlinear optics. Such an electric field-induced SHG (EFISH) process was first reported by Terhune et al. [3] for a calcite sample and extended soon thereafter to semiconductors and metals by Lee et al. [4]. More recently, systematic studies of the influence of electric fields on SHG have been pursued for centrosymmetric media [5–7], as well as for certain non-centrosymmetric materials with high symmetry [8]. The EFISH process in aqueous environments has also been intensively investigated in several interesting regimes [4, 9, 10]. It has been shown to be both of fundamental interest and a useful tool for probing chemical processes at interfaces. An especially attractive feature of the EFISH process for probing electric fields lies in the possibility for measurements with extremely high time resolution. Time resolution down to the femtosecond regime can be achieved by sampling the material system with ultrafast pulses from a modelocked laser. This approach has been applied successfully to study the influence of chargecarriers on the dynamics of internal electric fields in insulators [10] and semiconductors [11, 12], and to probe microwave [13] and ultrafast transient electric fields directly in time domain [14, 15]. While the background-free character of the EFISH process in a centrosymmetric medium constitutes the principal attractive feature of the method, it also imposes certain complications and limitations. In particular, if we do indeed observe negligible SHG in the absence of the applied field E0, then we expect the induced second-harmonic (SH) polarization to vary linearly with E0. In this case, the radiated SH field will also be linear in E0. The measured quantity is, however, the SH intensity I2ω, which will then scale quadratically with the electric field E0 being probed. Such a quadratic relation implies both the need for careful calibration and, more importantly, the loss of information on the sign of the electric field. While the sign of E0 can be recovered from a measurement of the phase of the radiated SH field, it is clearly desirable to have an experimental methodology free of these complications. Such a method is the homodyne detection scheme presented in this paper. The homodyne detection scheme is a well-established method in which a weak signal of interest is combined with