1. Introduction

Recently knowledge of electro-optic (EO) coefficient of photonic materials is very important in many application areas, especially in areas of high-speed optical modulator and switch application. There have been many types of the electro-optic coefficient measurement methods, such as Mach-Zehnder interferometer method with a laser source and reference medium [1,2], an ellipsoid method of a thin sample with reflective coating or plate on its backside [3–6], an interference spectrum measurement method caused by multiple internal reflection within sample surfaces of a Fabry-Perot interferometer type [7,8], a relative phase change measurement method between two orthogonally polarized beams passing through birefringent samples [9–11], a finite fringe interferogram method using a bulk-optics based Mach-Zehnder interferometer [12], and a direct EO modulation performance test method with an EO modulator made of sample material [13]. These methods have certain drawbacks because some of them measure the EO coefficient only at single wavelength, require a specially prepared geometry of the samples, or have a measurement accuracy limited by experimentally measurable parameters. Dispersion effect of the EO coefficient has been measured with a Mach-Zehnder interferometer method by Mendez, et al. [14] and with the multiple internal reflection method within the sample by Yonekura, et al. [8]. They used several discrete laser sources to measure the dispersion effect at discrete wavelengths. Especially the latter method required knowledge of the precise internal reflectivity to achieve an accurate EO coefficient as well as preferred high internal reflectivity. Knowledge of the dispersion effect of the EO coefficient is very important to EO modulators or switches suitable for wide wavelength range coverage. There has been no report for measurement of the dispersion effect of EO coefficient over a continuous wavelength range.

In this paper we report, for the first time to our knowledge, a new simple method to measure the continuous dispersion spectrum of the linear EO coefficient using a white-light interferometry. This method is based on measurement of the phase changes of the white light interference patterns between with and without an electric voltage applied to the sample. Since the phase change is related to the EO coefficient of the sample, the low coherence interferometric method provides an accurate measurement of the EO coefficient over the whole spectral range of the low coherence light source. This method can also allow a simultaneous measurement of unclamped EO coefficients of birefringent materials along their ordinary and extraordinary axes.

2. Experimental setup and theory

A schematic diagram of the experimental setup used for measuring the relative phase of the interference spectrum is shown in Fig. 1 . The setup is based on a fiber-type Mach-Zehnder interferometer composed of two 50/50 fiber couplers with a broadband light source at its input side and an optical spectrum analyzer (OSA) at the output side. Each arm of the interferometer has a polarization controller and a pair of gradient index (GRIN) lenses for collimated beam transmission and reception. One arm of the interferometer, called “reference arm”, has an adjustable free-space spacer to change the spacing between the GRIN lenses, and the other arm, called “sample arm”, has a nonlinear crystal sample with a pair of electrodes formed over its top and bottom sides in a perpendicular direction to the beam path to apply an electrical voltage across them. The broadband light source used in our experiment was a semiconductor optical amplifier (SOA) with a central peak wavelength at 1505 nm and a full width at half maximum (FWHM) of about 60 nm. We used a conventional z-cut congruent LN crystal (MTI corp.) with dimension of 10 mm × 10 mm × 0.5 mm, and the electrode was formed by Pt coating over an area of 7.8 mm × 7.8 mm at the bottom and top sides so that the electric field direction is located along the crystallographic z-direction [15]. Measurement errors for the crystal thickness and electrode length were less than 0.002 mm and 0.01 mm, respectively. An unpolarized white light beam was directed to propagate along the crystallographic y-direction so that the electric field was applied normal to the beam propagation direction. Two polarization controllers composed of single-mode fiber spooled pads, each of which was placed in the reference and sample arms, respectively, were used to rotate the polarization profiles of the input beams for the maximum visibility of an output interference pattern. The reference arm was also adjusted to a fixed length so that the optical path length (OPL) along the reference arm was located between those of the ordinary and extraordinary waves in the sample arm in order to observe the combined interference patterns of the two polarized waves. Experimental measurements were first carried out with no electric voltage applied to the sample, and a phase-dependent interference spectrum caused by the difference of OPL between reference wave and ordinary/extraordinary waves in the sample arm was measured. Then, a similar interference spectrum with an electric voltage applied to the sample was measured.

 

Fig. 1 Experimental setup for measurement of the EO coefficient.

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The output beam intensity of the interferometer can be derived as

The EO coefficient is related to shift of the phase difference between the two cases without and with an electric voltage applied.

3. Measurements and results

Figure 2 shows a measured interferometer output spectrum and numerically analyzed data in a process to find the relative phase shift between the two cases without and with an applied electric voltage. Figure 2(a) shows the measured interference spectrum of the Mach-Zehnder interferometer without an electric voltage applied across the sample in the experimental setup in Fig. 1. This optical spectrum contains two patterns caused by interference between each of the ordinary and extraordinary waves inside the birefringent sample and the wave through the reference arm. The measured interference spectrum was first subtracted by the spectral intensity emerging out through one of the sample and reference arms, when the other arm was blocked, which provided a resultant spectrum corresponding to last term of Eq. (1). Then, the wavelength-domain spectrum is converted into a frequency-domain spectrum by using the relationship of$f=c/\lambda$. A numerical Fourier-transformation is performed over the frequency-domain spectrum, and then two different peaks in time-domain, each corresponding to an oscillation period of the interference signal for the ordinary and extraordinary waves, respectively, appear as shown in Fig. 2(b). Selection of each of the peaks with a bandpass filtering process and application of an inverse Fourier transform process to the selected one provide separation of the two interference spectra. Each of the separated interference spectra was then normalized by using the spectral intensity measured from each of the separate arms according to the last term of Eq. (1) in order to obtain a normalized interference spectrum corresponding to the pure cosine term. Figures 2(c) and (d) show the normalized and separated interference spectra for the ordinary and extraordinary waves passing through the sample under no applied voltage, respectively, after reconverting their frequency domain back into the wavelength-domain. The previous normalization process can be skipped and replaced instead with a numerical process, in which the original interference spectrum can be separated first into two interference spectra for the ordinary and extraordinary waves by applying the numerical Fourier-transformation and bandpass filtering processes and then each of the separated interference spectra is divided by its envelop spectrum calculated from a Hilbert transformation on it at spectral power level. Most of the noises contained in the interferometer output due to environmental instabilities can be removed during the Fourier transform and filtering processes. The separated interference spectra can also be obtained directly by using a polarized input beam aligned along each of the birefringent axes instead of adopting the above processes, but more careful alignment process is needed which makes the measurement difficult. Even in this case the Fourier transform and filtering processes are needed to eliminate the noises caused by environmental instabilities and to obtain more accurate results. Since the phase difference between two adjacent peaks is 2π, the phase difference spectra are obtained by calculating the phase difference value for mth peak with respect to a particular peak at a fixed reference wavelength to be 2mπ, and plotted in Figs. 2(e) and (f) with open square dots for the ordinary and extraordinary waves, respectively.

 

Fig. 2 (a) Measured interferometer output spectrum, and (b) an oscillation period curve of the interference signals calculated for the ordinary and extraordinary waves from the Fourier transformation of the spectrum in (a), (c) and (d) numerically separated interference spectra of the ordinary and extraordinary waves of a congruent LN crystal, respectively, under no applied voltage and (e) and (f) calculated relative phases vs. wavelength curves of the ordinary and extraordinary waves, respectively, for the cases without and with applied voltage.

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The above process can be done to obtain the phase difference spectra for the cases without and with an appropriate electric voltage applied to the sample. Then, two plots with closed circular dots in Figs. 2(e) and (f) are obtained for the ordinary and extraordinary waves, respectively, when an electric voltage is applied. Then, the wavelength dependent EO coefficients of the sample can be calculated by using the phase difference values and its known refractive index value n₀(λ) at each wavelength in Eq. (4). The measured EO coefficients versus wavelengths are plotted in Fig. 3 . Since a direct current (DC) voltage is used in the experiment, the measured phase shift contains the converse piezoelectric effect of the sample, and thus the EO values obtained here correspond to the unclamped EO coefficients [15].

 

Fig. 3 Measured unclamped EO coefficients vs. wavelength

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The spectral dispersion of the unclamped EO coefficient of the congruent LN crystal shown in Fig. 3 was obtained by applying voltages from 500 V to 800 V with increment of an 100 V interval over a wavelength range from 1450 nm to 1610 nm. Even though we could get the interference spectrum over a wavelength range from 1430 nm to 1630 nm, its intensity at the both edges of the SOA spectrum was very low, and parts of the wavelength range near the edges were ignored because of significant errors caused by the low intensities. The measurement was repeated 5 times at every applied voltage, and thus the total number of the EO coefficient measurement was 20 times. The averaged values and standard deviations of the EO coefficients were plotted as the center line and error bars, respectively, in Fig. 3. The values corresponding to the error bar might be caused by environmental instability of the fiber-type interferometer, which was not fully eliminated during the Fourier transform and filtering processes, and/or by fitting errors occurred during the numerical processing in relative phase calculation. The refractive indices of the congruent LN crystal along the ordinary and extraordinary axes used during the calculation of the EO coefficient were obtained from the dispersion equation in Ref. 17.

The measured phase difference at 1510 nm wavelength increases with the applied voltage for each of the extraordinary and ordinary waves are shown in Figs. 4 (a) and (b) , respectively. The results indicate a verified linearity of the measured EO coefficient. The EO coefficients, r ₃₃ ^T and r ₁₃ ^T of the congruent LN crystal calculated from the linear curve fitting to the plots in its crystallographic extraordinary and ordinary axes were 28.5 ± 0.15 pm/V and 8.5 ± 0.13 pm/V, respectively. These values are comparable to the effective EO coefficients of a congruent LN crystal at 1510 nm wavelength reported in Ref. 8, which included an inverse piezoelectric (IPE) effect on the conventional unclamped EO coefficients. According to Ref. 8 the IPE effect affects the EO coefficients by amount of about 0.6% and 1.7% for the r ₃₃ ^E and r ₁₃ ^E coefficients, respectively.

 

Fig. 4 Phase difference change vs. applied voltage for the (a) extraordinary and (b) ordinary waves in the congruent LN crystal

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4. Conclusions

We have proposed a new method for measurement of the EO coefficient based on a white light interferometry. The continuous dispersion curves of the linear EO coefficients along the crystallographic extraordinary and ordinary axes of a congruent LN crystal have been successfully measured over a wavelength range from 1450 nm to 1610 nm. This method can be extended to measure the continuous dispersion curves of the EO coefficients over a wide wavelength range with a broad white light source.