A general mathematical model based on Mueller-matrix calculation is presented to describe the optical behavior of a dual-crystal electro-optic modulator. The two crystals inside the modulator are oriented at ± 45° with respect to the horizontal, thereby cancelling natural birefringence and temperature-induced birefringence. We describe the behavior of the modulator as a function of the ellipticity of the crystals, the rotation angles of the crystals and the applied voltage. By fitting the measured data with a Mueller-matrix model that uses values for the ellipticity and orientation angles of the crystals, the simulated data and the experimental measurements could be matched. This Mueller-matrix includes physical properties of the thermally compensated electro optic modulator, and the matrix can be used in simulations where these device-specific properties are important, for instance in the modeling of a polarization-sensitive optical coherence tomography system.
© 2016 Optical Society of America
Electro-optic modulators (EOM) are typically used to control the amplitude, polarization and phase of light. Different types of crystals, such as lithium niobate (LiNbO3), gallium arsenide (GaAs) and potassium dihydrogen phosphate (KDP) are used in EOMs [1,2]. Control of the amplitude, polarization and phase is obtained by changing the refractive index or birefringence of a crystal with an appropriately modulated signal in the form of an external voltage [1,2], typically at GHz frequency [3,4]. When a voltage is applied to the crystal, the electro-optic effect causes a change in the refractive indices along the crystal’s ordinary and extraordinary axes. EOMs have been used in applications in different scientific fields. In quantum cryptography, modulators are used to reduce the excess of noise in the signals [5–10]. Modulators are used to characterize the polarization-dependent loss (PDL) of the optical components in fiber-communications systems [11,12]. In biomedical imaging, the EOM is used to replace a quarter wave plate to increase the speed of the measurements of the Stokes vectors of the light returning from the human retina [13,14]. Similarly, different polarization states can be generated at high speed to analyze the birefringence of biological tissues in polarization-sensitive optical coherence tomography (PS-OCT) [15,16].
In comparison to previous work on EOMs, Song et al. provided a theoretical description of a dual-crystal EOM with crystals oriented at 45° with each other, treated as one single linear retarder . Mackey et al. also gave a theoretical description of a dual-crystal EOM treated as two ideal linear retarders oriented at +/− 45°, assuming a 180° phase change between the driving voltages of the two crystals . Furthermore, the theoretical description of technologies that have been referred to so far only use the modulators as an optical component that can quickly generate various linear polarization states, without considering its physical intrinsic characteristics.
In this manuscript, a mathematical model is presented using Mueller calculation to describe the Mueller matrix of a dual-crystal electro-optic modulator. While Jones calculus may be more appropriate for the straightforward experiment that we have used in this manuscript, our model can also be used in a system where partially polarized light returns from biological tissue through the modulator. The Mueller-matrix approach allows us to describe the behavior of the modulator in terms of the modulated signal applied to the crystals, ellipticity and small misalignments in the rotation angles of the crystals. The modulator consists of two magnesium doped lithium niobate (MgO:LiNbO3) crystals arranged in series and oriented at with respect to the horizontal. The magnesium doping of the crystal helps to prevent damage to the crystal, particularly at lower (visible) wavelengths . We analyze the ellipticity and orientation angles of the crystals that are involved in the generation of signal amplitude.
In the first section of this paper, we describe our mathematical model. For clarity, we divide this first part into two subsections. In the first subsection, we describe the electro-optical configuration of a uniaxial crystal and we show the mathematical representation of birefringence that is induced by a voltage supplied to the two crystals, while the natural birefringence and temperature-induced birefringence are cancelled. In the second subsection, we applied the Mueller-matrix analysis to describe the behavior of the modulator as a function of its birefringence. Then we depict an experimental setup that permits verification of our theoretical model. The results obtained experimentally and results generated with the theoretical model are compared in the discussion, and our findings are summarized in a conclusion.
2. Analytical description
2.1 Electro-optics effect of a uniaxial crystal modulator
When a voltage is applied to a modulator crystal in a transverse mode operation as shown in Fig. 1, the refractive indices along its ordinary and extraordinary axes change due to the linear electro-optic effect or Pockels effect [19,20].
The propagation inside the crystal can be described by the index ellipsoid . In the case of a uniaxial crystal, the index ellipsoid is where are oriented along the ordinary refractive index , and along the extraordinary refractive index . The linear change in the coefficients because of an electric field that is applied is described by
If the voltage is applied along the axis and the propagation of the light is in the y direction, the electro-optic tensor for lithium niobate can be described by [19–21]Eq. (2) in Eq. (1) , the resulting coefficients becomeEqs. (3)-(5), the refractive indices ,and becomeEqs. (6)-(8) as the crystal remains uniaxial with the same principal axes. Due to the change of the refractive indices along the principal axes, the light modulation is dependent on the polarization state. The birefringence is defined as nz – nx and by subtracting Eq. (6) from Eq. (8), it follows thatEq. (10) the first term is the natural birefringence associated with the length of the crystal and the second term is associated with the applied voltage.
Similarly, and taking the various orientations of the crystal with respect to the direction of the light beam into account, the birefringence of a second crystal which is arranged optically in series with respect to the first crystal can be determined . We are considering two identical crystals with the same length and their principal axes rotated by 90°. The voltages applied to the crystals are the same. Keeping the different orientations of the two crystals in mind, the static birefringence and the voltage-induced birefringence components can be defined for the two crystals as
2.2 Mueller matrix of an EOM modulator with static birefringence compensation
For our model, we assume the following test setup, Fig. 2 where the EOM will be analyzed between a polarizer at angleand analyzer at angle . The EOM is modeled as a dual crystal modulator oriented at angle and the fast axes of the crystals are crossed. Both crystals will be run with the same input voltage and the light power that passes through the EOM is collected by a photodetector.
Using the Mueller-matrix approach [23–25], a linear polarizer at desired fast axis orientation can be represented by24] can be modeled by14,17] only their linear retardance behavior was modeled, while our approach also accounts for the circularity that is carried by the ellipticity parameter . In the ideal case of non-ellipticity, = 0, Eq. (14) represents a common linear retarder and by = 1, Eq. (14) represents a circular retarder. The sign of indicates the direction of vibration as either left () or right () clockwise direction. The azimuthal angle is a measure of the physical orientation of the crystal and is the induced retardance by the voltage between parallel plates of the crystal at the fast axis orientation. is the voltage at which the induced retardance reaches the half wave value. This value depends on the crystal composition and the physical parameters such as the length and frontal area of the crystal . As explained in Fig. 2, the fast axis orientation of the second crystal is rotated by 90° with respect to the first crystal, to cancel changes due to temperature and natural birefringence. The Mueller matrix of a dual crystal EOM modulator is then defined asEquation (19) is a more realistic model that considers the elliptical retardance (w) and two orientation parameters related to a common (Δθ1) and a relative (Δθ2) orientation difference between crystals. These two parameters are related to a small misalignment in rotation between the crystals, which is likely to happen in a practical situation. and represent the static retardance and the induced retardance induced by the crystal, respectively. For the case of an ideal EOM with perfectly aligned crystals, we set the ellipticity and Δθ1 = Δθ2 = 0. These are the parameters for a linear retarder without static retardance,, while the voltage-induced retardance is doubled (in comparison to a single crystal). After expanding Eq. (19) and considering an orientation of and small angle approach for w, Δθ1 and Δθ2, the Mueller-matrix representation becomesEqs. (21)-(24) to reduce the size of Eq. (20)Equation (20) now represents a Mueller matrix of a thermally compensated EOM with a voltage induced retardance of , where is the voltage at which a retardance equal to half a wave is induced. The circular retardance parameter w is related to the ellipticity condition. For the analysis of the physical behavior of the modulator, the device can be placed between an ideal polarizer and an ideal rotating analyzer. By placing the first polarizer in a vertical position and the analyzer at angle , the output Stokes vector can be represented byFig. 3(a) shows the intensity that was detected at six orientations of the analyzer when an ideal EOM is used (Δθ1 = 0, Δθ2 = 0, w = 0) and Fig. 3(b) shows the results obtained with an elliptical retarder in the model (Δθ1 = −0.5°, Δθ2 = −0.5°, w = 0.001). The most obvious difference between the two plots in Fig. 3 is the gap that occurs between the + 45° and −45° measurements in the non-ideal situation.
3. Experimental results
We will refer again to Fig. 2 for the experimental optical setup that was used for measurement of the intensity of the light through the different components. First, we used light from a broadband light source, a superluminescent diode (Superlum HP-SLD-371) with a center wavelength of 840 nm, a full width at half maximum (FWHW) bandwidth Δλ = 50 nm and a power of P = 10 mW. This is a typical light source for polarization-sensitive optical coherence tomography, which can rely on an EOM to induce various polarization states. The light from the source was coupled with a single-mode optical fiber to a bench (Thorlabs), with a collimator to provide a collimated beam with a beam size of approximately 0.2 mm. Then light passed through an optical isolator (Thorlabs), which also acted as a polarizer with a fast axis orientation at 90°. In order to ensure that the light entering the electro-optic modulator was polarized, the first polarizer (which is part of an isolator, IO-3D-780-VLP (Thorlabs)) was analyzed with the Axometric system. We obtained the spectral variation of the linear diattenuation/polarizance in a range from 700 nm to 800 nm. The linear diattenuation was 0.9970 ± 0.0006 with an orientation of −46.00° ± 0.03°and the linear polarizance was 0.986 ± 0.002 with an orientation of −88.90° ± 0.10°, demonstrating that the first polarizer in the isolator indeed works as a polarizer with 99% efficiency at an output linear polarization state of 90°. The electro-optical modulator (New Focus 4012) which consists of two MgO:LiNbO3 crystals oriented at and with respect to the horizontal was positioned directly behind the isolator. The optical alignment of the components can influence the measurements. The crystals in the temperature-compensated modulator are aligned by the manufacturer and factory sealed, which means that we do not have the opportunity to optimize their alignment. We could however optimize the position of the modulator itself. The modulator was placed on an optical fiber bench coupler (Thorlabs), which has two fiber collimators that can be fine tuned to send a collimated beam through the isolator and modulator. We aligned these components by maximizing the intensity at the output of the bench with a photodetector. Similar measurements with the Axoscan on the modulator demonstrated that the modulator performs best (based on the induced retardance values, fitted w parameter and the magnitude of the detected angles) when transmission losses are minimized. An analyzer (linear polarizer, Thorlabs LPNIR050-MP2) was inserted behind the modulator and used with different orientations of 0°, 30°, + 45°, −45°, 90° and 120° with respect to the horizontal. Finally, a photodetector (New Focus model 2107) was used to detect the intensity of the light as a function of time. Its output was recorded with an oscilloscope (Tektronix TDS3034C) and a function generator (NF WF1974) was used for generation of the modulation signal. A ramp function with a maximum voltage of 4.8 V and an offset of 1.3 V was used as a modulation signal. This signal was amplified with a New Focus high-voltage amplifier (NF 3211) and then sent to the New Focus 4012 dual-crystal modulator. Figure 4 shows the detected intensity for different orientations of the analyzer, recorded with the oscilloscope as a function of the input voltage. A full modulation of the detected signal as shown in Fig. 4 requires careful selection of the offset and amplitude .
To fit the data with our model, we needed values for the ellipticity w and orientation parameters Δθ1 and Δθ2. While the latter two were impossible to measure, as the modulator is carefully sealed, we were able to measure w independently with the Axoscan (Axometrics Inc.) at different input voltages. In an Axoscan system that was calibrated as recommended by Axometrics, the average value of ellipticity was w = 0.016 ± 0.058. In these measurements, we took the standard deviation as an error. Since the EOM has an aperture that is considerably smaller than the Axoscan’s beam, we also calibrated the Axoscan with the EOM in situ, to improve the signal to noise ratio. This resulted in w = −0.014 ± 0.057.
Using the values from Fig. 4, the induced retardance can be plotted as a function of the detected intensity, as shown in Fig. 5. The common linear retarder model predicts that the intensities at degrees will show no modulation and crossing points between 30° and 120° will be matched with measurements obtained at 90° and 0°. A fitting process on the data, using Eq. (10) helps to retrieve the modulator parameters such as w, Vπ and the orientation angles Δθ1 and Δθ2. Figure 5 shows the results obtained experimentally with the following EOM parameters: , w = −0.003 and orientation parameters of Δθ1 = −0.3° and Δθ2 = −0.6° related to a common and relative difference parameter of both crystals, respectively. The w = −0.003 value for the ellipticity is well within the measured range of w when the Axoscan was calibrated with the EOM in the system. Simulations with different values for w, ranging between −0.001 and −0.005 show very similar results, for instance with a slightly better fit for the 30° orientation measurement, and a slightly worse fit for another orientation angle.
As measurements may depend on the bandwidth of the used light source, the same analysis was made using a narrow bandwidth light source (Δλ = 1 nm (FWHM) at λc = 785 nm, (Thorlabs, model S1FC780)). Moreover, using this light source we also studied the temperature dependence of the modulator, as its design promises a stable performance that is independent of the temperature. Figure 6(a) shows the detected normalized intensity dependence against input voltage and Fig. 6(b) shows the temperature-dependent variation of the elliptical parameter w. For these conditions an elliptical parameter of w = −0.003 and relative orientation of Δθ1 = −0.03° between crystals and common orientation angle of Δθ2 = −0.79° were measured. With an input voltage ranging from −0.36 V to 4.44 V, the results were similar to the previous measurements with the broadband light source, giving a similar value for the ellipticity with small variations in the orientation parameters. The temperature-dependent variation is very low, and demonstrates that this EOM is indeed thermally compensated over a temperature range from 24° C to 29° C having a mean value of w = −0.0032.
4. Discussions and conclusions
In the present study, we have outlined the physical behavior of a dual crystal electro-optic modulator with Mueller-matrix analysis. A general equation is presented as a function of the crystal’s static retardance, induced retardance and orientation angle of the analyzer. We then added parameters, so that the ellipticity and the orientations of the crystals are taken into account. It is very well possible that other parameters play a role as well. The voltage-induced birefringence could vary as a function of the input wavelength. To check this, we compared two light sources, one with a 50 nm bandwidth (FWHM), the other with a bandwidth of 1 nm (FWHM) and found similar results. Moreover, the two crystals may have different lengths, or have been cut at angles that are different from our assumptions. By using a rotating analyzer and knowing the input induced voltage against the detected intensity as a function of time, the elliptical retarder parameter and the small error orientation between the crystals can be retrieved by means of appropriate fitting. Figure 3 shows that when the EOM modulator does not have ellipticity and perfectly oriented crystals, it has a common crossing points in intensity between all six orientations at . Furthermore, at an orientation of the analyzer of no modulation is encountered.
Experimentally, we found that crossing points between analyzer angle of 0°, 90°, 30° and 120° occur at different voltages. In addition, the intensity at an analyzing angle of ± 45° does not remain constant. We looked at the ellipticity and the orientations of the crystals for an explanation of this behavior and found that the EOM behavior could be explained if the two crystals act as two elliptical retarders. Measurements with the Axoscan confirmed that the EOM indeed has a (very low) circular retardance component. It is important to remark that the Δθ1, Δθ2 and w parameters are the same for the fits at different analyzer orientations, Fig. 5. The ellipticity parameter w is small, while the EOM orientation of 44.7° is very close to the 45° that is provided by the manufacturer. Since the EOM device was sealed by the manufacturer, it was not possible to verify the angles of the crystals. According to the fitting, both crystals have a relative orientation of 89.4°, which is close to 90° that is required for the static birefringence compensation. Again, it may be possible that other physical parameters are involved, but a model that does include the ellipticity and angle orientation can be fitted accurately to the current experimental data.
Our proposal can be useful to avoid discrepancies between theory and measurements, or to better interpret measurements. For example, Song et al. and Mackey et al. [14,17] used a Mueller-matrix approach that included the modulated signal applied to the modulator, the parameter Vπ and the orientation angle of the crystal as provided by the manufacturer. However, they did not consider a parameter that takes a slight misalignment between the crystals into account.
In conclusion, by adding parameters for the ellipticity w and the rotation of the crystal different from the values provided by the manufacturer, the results from an experiment in which the EOM is placed between polarizers can be better matched with simulated Mueller-matrix results.
Bio-Imaging and Sensing Project at Utsunomiya University, Japan (David Serrano); Consejo Nacional de Ciencia y Tecnologia, Mexico. (CONACYT; Joel Cervantes).
The authors thank Prof. Mitsuo Takeda (Center for Optical Research and Education at Utsunomiya University) for his comments and revisions.
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