Mueller-matrix modeling and characterization of a dual-crystal electro-optic modulator

Joel Cervantes-L; David I. Serrano-Garcia; Yukitoshi Otani; Barry Cense

doi:10.1364/OE.24.024213

1. Introduction

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 [14]. 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 [17]. 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 $\pm 45 º$ with respect to the horizontal. The magnesium doping of the crystal helps to prevent damage to the crystal, particularly at lower (visible) wavelengths [18]. 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].

Fig. 1 Configuration of a transverse modulator crystal. The electric field is applied along the $z$ direction and the light propagation is along the y direction. The length of the crystal is $L$ and $d$ is the crystal thickness. Electrodes are placed on the top and the bottom of the crystal.

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The propagation inside the crystal can be described by the index ellipsoid [21]. In the case of a uniaxial crystal, the index ellipsoid is $x^{2} / n_{x}^{2} + y^{2} / n_{y}^{2} + z^{2} / n_{z}^{2} = 1$ where $(n_{x}, n_{y})$ are oriented along the ordinary refractive index $n_{0}$ , and $n_{z}$ along the extraordinary refractive index $n_{e}$ . The linear change in the coefficients because of an electric field that is applied is described by

Δ {(\frac{1}{n^{2}})}_{i} = \sum_{j = 1}^{3} r_{i j} E_{j},

where

r_{i j}

is the electro-optic tensor,

E_{j}

is the electric field and j corresponds to the principal axes x, y and z (i stands for

1 = x, 2 = y, 3 = z, 4 = y z, 5 = x z, and 6=xy

).

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]

r_{i j} = (\begin{matrix} 0 & - r_{22} & r_{13} \\ 0 & r_{22} & r_{13} \\ 0 & 0 & r_{33} \\ 0 & r_{51} & 0 \\ r_{51} & 0 & 0 \\ - r_{22} & 0 & 0 \end{matrix}) .

Inserting Eq. (2) in Eq. (1) [21], the resulting coefficients become

Δ {(\frac{1}{n^{2}})}_{1} = r_{13} E_{z}

Δ {(\frac{1}{n^{2}})}_{2} = r_{13} E_{z}

Δ {(\frac{1}{n^{2}})}_{3} = r_{33} E_{z}

By using Eqs. (3)-(5), the refractive indices

n_{x}

,

n_{y}

and

n_{z}

become

n_{x} = n_{0} - \frac{1}{2} n_{0}^{3} r_{13} E_{z},

n_{y} = n_{0} - \frac{1}{2} n_{0}^{3} r_{13} E_{z},

n_{z} = n_{e} - \frac{1}{2} n_{e}^{3} r_{33} E_{z} .

When an electric field

E_{z}

is applied along the z-axis of the lithium niobate crystal, its refractive indices are modified in accordance with Eqs. (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 n_z – n_x and by subtracting Eq. (6) from Eq. (8), it follows that

n_{z} - n_{x} = (n_{e} - n_{o}) - \frac{1}{2} (n_{e}^{3} r_{33} - n_{o}^{3} r_{13}) E_{z} .

This then results in a birefringence

Γ = \frac{2 π}{λ} (n_{e} - n_{o}) L - \frac{π}{λ} (n_{e}^{3} r_{33} - n_{o}^{3} r_{13}) \frac{L}{d} V,

where

L

is the crystal length,

d

is the crystal thickness and

λ

is the wavelength in vacuum. In Eq. (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 [22]. 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

Γ_{1} = \frac{2 π}{λ} (n_{e} - n_{o}) L + \frac{π}{λ} (n_{e}^{3} r_{33} - n_{o}^{3} r_{13}) \frac{L}{d} V = δ_{s} + δ_{i},

Γ_{2} = \frac{2 π}{λ} (n_{e} - n_{o}) L - \frac{π}{λ} (n_{e}^{3} r_{33} - n_{o}^{3} r_{13}) \frac{L}{d} V = δ_{s} - δ_{i} .

where the subscripts 1 and 2 are for the first and second crystal, respectively.

δ_{s}

and

δ_{i}

represent the static retardance and the induced retardance in the crystal. Therefore, the birefringence of the two crystals combined into a single optical unit that is thermally uniform is defined as

Δ Γ = Γ_{1} - Γ_{2}

. The subtraction results in a cancellation of both the natural birefringence and temperature-dependent birefringence, while the voltage-induced birefringence is doubled when the fast axis of the second crystal is rotated by

90 º

with respect to the first crystal’s fast axis. These results are important in the next section, where we present our mathematical model to describe the physical behavior of a dual-crystal electro-optic modulator.

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 angle $θ_{p}$ and analyzer at angle $θ_{A}$ . The EOM is modeled as a dual crystal modulator oriented at angle $θ$ 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.

Fig. 2 Schematic representation of the beam propagation through the different components. Light from a light source with a wavelength λ is coupled into an optical fiber and collimated. Then the light passes through a linear polarizer oriented at an angle $θ_{p}$ with respect to the axis of polarization. After the polarizer, the light passes through the EOM which consists of two MgO:LiNbO3 crystals. The first crystal is oriented at an angle $θ$ and the second one is rotated by 90° with respect to the first crystal’s fast axis, $θ + 90 °$ . The voltage applied to the crystals is the same. The arrows that are drawn onto the crystals represent the orientation of the electric field. After the modulator, the light travels towards an analyzer, which is a linear polarizer with an angle $θ_{A}$ with respect to the axis of polarization. Finally, the light power is collected by a photodetector.

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Using the Mueller-matrix approach [23–25], a linear polarizer $P (θ)$ at desired fast axis orientation $θ$ can be represented by

P (θ) = \frac{1}{2} (\begin{matrix} \begin{matrix} \begin{matrix} 1 \\ \cos 2 θ \end{matrix} \\ \begin{matrix} \sin 2 θ \\ 0 \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} \begin{matrix} \cos 2 θ \\ \cos^{2} 2 θ \end{matrix} \\ \begin{matrix} \sin 2 θ \cos 2 θ \\ 0 \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} \begin{matrix} \sin 2 θ \\ \sin 2 θ \cos 2 θ \end{matrix} \\ \begin{matrix} \sin^{2} 2 θ \\ 0 \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} 0 \\ 0 \end{matrix} \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix})

and an elliptical retarder

R (θ, δ, w)

with ellipticity,

γ = \tan | w |

, azimuth angle

θ

and retardance

δ

[24] can be modeled by

R (θ, δ, w) = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & D^{2} - E^{2} - F^{2} + G^{2} & 2 (D E + F G) & - 2 (D F + E G) \\ 0 & 2 (D E - F G) & - D^{2} + E^{2} - F^{2} + G^{2} & 2 (D G - E F) \\ 0 & - 2 (D F - E G) & - 2 (D G + E F) & - D^{2} - E^{2} + F^{2} + G^{2} \end{matrix}),

where the next parameters are defined for notational purposes

D = \cos (2 w) \cos (2 θ) \sin (δ / 2),

E = \cos (2 w) \sin (2 θ) \sin (δ / 2),

F = \sin (2 w) \sin (δ / 2),

G = \cos (δ / 2) .

Each crystal in the EOM will act as an elliptical retarder with combined properties of retardance and ellipticity due to the induced birefringence. Although EOMs have been mathematically modeled before [14,17] only their linear retardance behavior was modeled, while our approach also accounts for the circularity that is carried by the ellipticity parameter

γ = \tan | w |

. In the ideal case of non-ellipticity,

γ

= 0, Eq. (14) represents a common linear retarder and by

γ

= 1, Eq. (14) represents a circular retarder. The sign of

w

indicates the direction of vibration as either left (

-

) or right (

+

) clockwise direction. The azimuthal angle

θ

is a measure of the physical orientation of the crystal and

δ = V π / V_{π}

is the induced retardance by the voltage

V

between parallel plates of the crystal at the fast axis orientation.

V_{π}

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 [2]. 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 as

M_{E O M} = R (θ + Δ_{θ 1} - 90 º, δ_{s} + δ_{i}, - w) \cdot R (θ + Δ_{θ 1} + Δ_{θ 2}, δ_{s} - δ_{i}, w) .

Equation (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.

δ_{s}

and

δ_{i}

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

w = 0

and Δθ₁ = Δθ₂ = 0. These are the parameters for a linear retarder without static retardance,

δ_{s}

, while the voltage-induced retardance is doubled (in comparison to a single crystal). After expanding Eq. (19) and considering an orientation of

θ = 45 °

and small angle approach for w, Δθ₁ and Δθ₂, the Mueller-matrix representation becomes

M_{E O M} (w, Δ_{θ_{1}}, Δ_{θ_{2}}, δ_{s}, δ_{i}) = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & - \cos 2 δ_{i} & M_{2, 3} & \sin 2 δ_{i} \\ 0 & M_{3, 2} & 1 & R_{3, 4} \\ 0 & - 2 \cos δ_{i} \sin δ_{i} & M_{4, 3} & - \cos 2 δ_{i} \end{matrix}),

where the next parameters

M_{3, 2}, M_{2, 3}, M_{4, 3}

and

M_{3, 4}

are defined according to Eqs. (21)-(24) to reduce the size of Eq. (20)

M_{3, 2} = - 4 Δ_{θ 1} \cos^{2} δ_{i} - 2 w [2 \cos (δ_{s} + δ_{i}) - \sin 2 δ_{s}] - 2 Δ_{θ 2} [\sin (δ_{s} + δ_{i}) + 1]

M_{2, 3} = - 4 Δ_{θ 1} \cos^{2} δ_{i} - 2 Δ_{θ 2} [\cos 2 δ_{i} + \sin (δ_{s} - δ_{i})] - 2 w [- 2 \cos (δ_{s} - δ_{i}) - \sin 2 δ_{s}]

M_{4, 3} = - 4 w \sin^{2} δ_{s} + Δ_{θ 2} [- 2 \cos (δ_{s} - δ_{i}) - 2 \sin 2 δ_{i}] - 2 Δ_{θ 1} \sin 2 δ_{i}

M_{3, 4} = - 4 w \sin^{2} δ_{s} - 2 Δ_{θ 2} \cos (δ_{s} + δ_{i}) + 2 Δ_{θ 1} \sin 2 δ_{i}

Equation (20) now represents a Mueller matrix of a thermally compensated EOM with a voltage induced retardance of

δ_{i} = V π / V_{π}

, where

V_{π}

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

θ_{A}

, the output Stokes vector can be represented by

S_{o u t} [w, Δ_{θ_{1}}, Δ_{θ_{2}}, δ_{s}, δ_{i}] = A [θ_{A}] \cdot M_{E O M} [w, Δ_{θ_{1}}, Δ_{θ_{2}}, δ_{s}, δ_{i}] \cdot P [90 º] \cdot S_{i n} .

Taking non-polarized light as an input by

S_{i n} = {[1 0 0 0]}^{T}

, the detected intensity can be obtained by taking the first element of the output Stokes vector, with an induced retardance

δ_{i} = V π / V_{π}

as

\begin{array}{l} I (θ_{A}, Δ_{θ_{1}}, Δ_{θ_{2}}, δ_{s}, δ, w) = \frac{1}{4} [1 + \cos 2 θ_{A} \cos 2 δ_{i} + 2 \sin 2 θ_{A} (Δ_{θ 1} + Δ_{θ 2} + Δ_{θ 1} \cos 2 δ_{i} + \\ 4 w \cos (δ_{i} + δ_{s}) - 2 w \sin 2 δ_{s} + 4 Δ_{θ 1} \sin (δ_{i} + δ_{s}))] . \end{array}

The intensity is then detected as a function of Δθ_1, Δθ₂, w, the induced voltage V and the analyzer orientation θ_A, intrinsic parameters of the EOM. By selecting the correct voltage settings and analyzer orientations, the intrinsic parameters of the crystal can be retrieved. As a matter of comparison, Fig. 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.

Fig. 3 Normalized amplitude obtained by means of Eq. (26) at different orientation angles of the analyzer. a) Amplitude plot for perfectly aligned crystals without ellipticity and rotation error. b) Amplitude plot for the ideal situation, with w = 0.001, Δθ₁ = −0.5°, Δθ₂ = −0.5° and θ_A = 0°, 30°, + 45°, −45°, 90°, 120°. Due to these non-ideal parameters, the plots do not cross at a normalized amplitude of 0.5. The most obvious difference for these non-ideal parameters with the ideal situation is the gap that develops between the + 45° and −45° measurements.

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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 $+ 45 º$ and $- 45 º$ 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 [4].

Fig. 4 Intensity detected with a photodetector when the EOM modulator was placed between a polarizer at 90° and an analyzer with orientations of 0°, 30°, +/−45°, 90° and 120°, respectively. The black plot represents the input voltage, a triangle function, sent to the amplifier.

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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 $\pm 45 º$ 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, V_π and the orientation angles Δθ₁ and Δθ₂. Figure 5 shows the results obtained experimentally with the following EOM parameters: $V_{π} = 4.8 V$ , 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.

Fig. 5 Graph of the recorded photodiode voltage when the analyzer was set at 0°, 30°, + 45°, −45°, 90° and 120° respectively, for voltages ranging from −3.8 V to 1.2 V, normalized with respect to the highest detected signal. The highest signal is obtained with the analyzer at 0° and 90°, while the analyzer at 45° provides almost no modulation in the detected signal. The data was fit with our model, a V_π of 4.8V, w = −0.003 and orientation parameters of Δ_θ1 = −0.3° and Δ_θ2 = −0.6° related to a common and relative difference parameter of both crystals.

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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 λ_c = 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.

Fig. 6 Measurements on the electro-optic modulator using a narrow bandwidth laser (Δλ = 1 nm (FWHM) at λ_c = 785 nm). Figure 6(a) shows the recorded photodiode voltage when the analyzer was set at 0°, 30°, + 45°, −45°, 90° and 120° respectively, for input voltages ranging from −0.36 V to 4.44 V, normalized with respect to the highest detected signal. The elliptical parameter after fitting was equal to w = −0.003 and the orientation parameters were Δθ₁ = −0.03° and Δθ₂ = −0.79°, the common and relative difference parameter of both crystals, respectively. Figure 6(b) shows the temperature-dependent variation of the elliptical parameter w, which is very small over this temperature range.

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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 $V = V_{π} / 2$ . Furthermore, at an orientation of the analyzer of $\pm 45 º$ 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 V_π 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.

Funding

Bio-Imaging and Sensing Project at Utsunomiya University, Japan (David Serrano); Consejo Nacional de Ciencia y Tecnologia, Mexico. (CONACYT; Joel Cervantes).

Acknowledgments

The authors thank Prof. Mitsuo Takeda (Center for Optical Research and Education at Utsunomiya University) for his comments and revisions.

References and links

1. R. Hunsperger, “Integrated optical detectors,” in Integrated Optics: Theory and Technology, R. Hunsperger, ed. (Springer Series in Optical Sciences, 1984).

2. T. A. Maldonado, “Electro-optic modulators,” in Handbook of Optics, M. Bass, ed. (McGraw Hill, 1995).

3. Newport, “Practical Uses and Applications of Electro-Optic Modulators,” www.newport.com/practical-uses-and-applications-of-electro-optic-modulators.

4. Thorlabs, “Free-Space EO Modulator Lab Facts,” www.thorlabs.com/images/TabImages/Free-Space_EO_Modulator_Lab_Facts.pdf.

5. D. Huang, P. Huang, D. Lin, and G. Zeng, “Long-distance continuous-variable quantum key distribution by controlling excess noise,” Sci. Rep. 6, 19201 (2016). [CrossRef] [PubMed]

6. J. Lodewyck, M. Bloch, R. García-Patrón, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N. J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76(4), 042305 (2007). [CrossRef]

7. L. S. Madsen, V. C. Usenko, M. Lassen, R. Filip, and U. L. Andersen, “Continuous variable quantum key distribution with modulated entangled states,” Nat. Commun. 3, 1083 (2012). [CrossRef] [PubMed]

8. V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009). [CrossRef]

9. P. Jouguet, S. Kunz-Jacques, and E. Diamanti, “Preventing calibration attacks on the local oscillator in continuous-variable quantum key distribution,” Phys. Rev. A. 87, 062313 (2013).

10. R. Filip, “Continuous-variable quantum key distribution with noisy coherent states,” Phys. Rev. A 77(2), 022310 (2008). [CrossRef]

11. T. Ferreira da Silva, C. S. Nobre, and G. P. Temporão, “Polarization-dependent loss characterization method based on optical frequency beat,” Appl. Opt. 55(8), 1838–1843 (2016). [CrossRef] [PubMed]

12. M. R. Fernández-Ruiz, J. Huh, and J. Azaña, “Time-domain Vander-Lugt filters for in-fiber complex (amplitude and phase) optical pulse shaping,” Opt. Lett. 41(9), 2121–2124 (2016). [CrossRef] [PubMed]

13. H. Song, Y. Zhao, X. Qi, Y. T. Chui, and S. A. Burns, “Stokes vector analysis of adaptive optics images of the retina,” Opt. Lett. 33(2), 137–139 (2008). [CrossRef] [PubMed]

14. H. Song, X. Qi, W. Zou, Z. Zhong, and S. A. Burns, “Dual electro-optical modulator polarimeter based on adaptive optics scanning laser ophthalmoscope,” Opt. Express 18(21), 21892–21904 (2010). [CrossRef] [PubMed]

15. C. E. Saxer, J. F. de Boer, B. H. Park, Y. Zhao, Z. Chen, and J. S. Nelson, “High-speed fiber based polarization-sensitive optical coherence tomography of in vivo human skin,” Opt. Lett. 25(18), 1355–1357 (2000). [CrossRef] [PubMed]

16. B. Cense, T. C. Chen, B. H. Park, M. C. Pierce, and J. F. de Boer, “Invivo depth-resolved birefringence measurements of the human retinal nerve fiber layer by polarization-sensitive optical coherence tomography,” Opt. Lett. 27(18), 1610–1612 (2002). [CrossRef] [PubMed]

17. J. R. Mackey, K. K. Das, S. L. Anna, and G. H. McKinley, “A compact dual-crystal modulated birefringence-measurement system for microgravity applications,” Meas. Sci. Technol. 10(10), 946–955 (1999). [CrossRef]

18. R. K. Choubey, P. Sen, P. K. Sen, R. Bhatt, S. Kar, V. Shukla, and K. S. Bartwal, “Optical properties of MgO doped LiNbO3 single crystals,” Opt. Mater. 28(5), 467–472 (2006). [CrossRef]

19. A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (John Wiley Sons Inc, 1984).

20. P. R. Hobson, “Lasers and Electro-Optics: Fundamentals and Engineering (2nd Edition), by Christopher C. Davis,” Contemp. Phys. 1–1 (2014).

21. T. T. Ng, A Fast Polarization Independent Phase Shifter Of Light, (PhD thesis, National University of Singapore, 2007).

22. W. Sima, T. Liu, Q. Yang, R. Han, and S. Sun, “Temperature characteristics of Pockels electro-optic voltage sensor with double crystal compensation,” AIP Adv. 6(5), 055109 (2016). [CrossRef]

23. R. A. Chipman, “Polarimetry,” in Handbook of Optics II, M. Bass, ed. (McGraw Hill, 1995), pp 22.1–22.37.

24. D. Kliger and J. Lewis, Polarized Light in Optics and Spectroscopy (Academic, 2012).

25. D. H. Goldstein, Polarized Light (CRC, 2010).

Mueller-matrix modeling and characterization of a dual-crystal electro-optic modulator

Abstract

1. Introduction

2. Analytical description

2.1 Electro-optics effect of a uniaxial crystal modulator

2.2 Mueller matrix of an EOM modulator with static birefringence compensation

3. Experimental results

4. Discussions and conclusions

Funding

Acknowledgments

References and links

Cited By

Figures (6)

Equations (26)

Optics Express