Control of residual amplitude modulation in Lithium Niobate phase modulators

John F. Diehl; Christopher E. Sunderman; Joseph M. Singley; Vincent J. Urick; Keith J. Williams

doi:10.1364/OE.25.032985

1. Introduction

Residual amplitude modulation (RAM) is the unwanted intensity modulation present in a phase modulated system. RAM is defined as the ratio of the generated intensity and phase modulated power, but for simplicity, in this paper we will leave the discussion to the amplitude of the intensity modulation by itself. It limits the performance of photonic sensors and systems employing a phase modulated architecture, such as those found in fiber-based gyroscopes, spectroscopy, and gravity wave sensors [1–3]. For the field of microwave photonics, RAM is of concern for optical beam forming networks utilizing photonic phase shifters. Amplitude fluctuations caused during a phase adjustment could result in a degradation of side lobe cancellation, potentially leading to a break in isolation for a receive/transmit system.

Previous works focusing on Lithium Niobate (LiNbO₃) phase modulators have investigated the major causes of RAM, including etalon, photorefractive, and polarization effects [4–7]. As the phase modulators improved in design and performance, the etalon and photorefractive effects were significantly reduced for state of the art devices. High quality anti-reflective coatings and optimized fiber coupling angles reduce the etalon effects present in previous component designs. The reduction of photorefractive effects was accomplished by introducing a Mg dopant to the LiNbO₃ [8]. Work has been done limiting the effect of polarization error by utilizing servos [6] or temperature tuning of the device [9].

This work models the RAM as a byproduct of polarization misalignment, taking into account the impact of chromatic dispersion present in the optical fiber following the phase modulator, which becomes important as the modulation frequencies extend beyond the Ka-Band. It uses a simple set of transfer matrices, useful for generating simple models of the output field of electrooptic modulators [10]. This simplification is justified by assuming the orthogonal polarizations become spatially separated in the crystal, due to the birefringent properties of the LiNbO₃. It is important to note that while this assumption offers a useful model for the RAM, it does not result in a precise expression for the optical field inside the device. That is not the intent of this work. This model presents a closed form expression for the expected RAM as a function of the effective polarization misalignments on the input and output faces of the modulator. It then gives insight into a relatively simple and immediate control mechanism for minimizing RAM for high sensitivity applications. A method that the model predicts is similar to the temperature tuning method proposed in [9], yet with an immediate reaction time.

2. Theory

The model proposed in this work utilizes simple transfer matrices, similar to those in [10]. We start with an input electric field with linear polarization given by

E_{i n} = [\begin{matrix} 1 \\ 0 \end{matrix}] E_{0} e^{i ω_{0} t}

where

E_{0}

is the real amplitude of the input field and

ω_{0}

is the angular frequency of the laser carrier. Due to manufacturing errors and misalignment of the polarization maintaining fiber, there will be a small rotation of the polarization away from the modulator’s preferred axis. As the light passes through the birefringent medium, the field rotates and eventually becomes spatially separated, resulting in the bulk of the input field travelling on the preferred axis and a small percentage travelling on the orthogonal axis. This angle of error is given by

θ_{1}

in Eq. (2). This allows us to treat the starting Jones vector like a simple transfer matrix. The two beams pass through the modulator, gaining a varying phase shift, driven by the input electrical signal. Due to the nature of the device, the preferred axis is more efficient in converting the input electrical signal to an optical phase shift. If we assume the input electrical signal consists of a DC component in addition to a sinusoidal AC component of angular frequency Ω, the transfer matrix looks similar to an asymmetric Mach-Zehnder intensity modulator. Next, we consider that the two polarizations see different effective delays and losses as they travel through the modulator. The difference in travel time is modeled using the

T (τ)

operator used in [10] for the MZI. The loss of each polarization is given as a scalar constant

α_{1}

or

α_{2}

. Finally, the two polarizations are spatially recombined and launched into the output optical fiber. This second angle of error is given by

θ_{2}

. The resulting chain of transfer matrices, when assembled, is

\begin{array}{l} [\begin{matrix} E_{o u t, o} \\ E_{o u t, e} \end{matrix}] = \frac{1}{2} [\begin{matrix} \cos (θ_{2}) & i \sin (θ_{2}) \\ i \sin (θ_{2}) & \cos (θ_{2}) \end{matrix}] [\begin{matrix} α_{1} T (τ) & 0 \\ 0 & α_{2} \end{matrix}] \\ \times [\begin{matrix} e^{i φ_{d c, 1}} e^{i φ_{r f, 1} \sin (Ω t)} & 0 \\ 0 & e^{i φ_{d c, 2}} e^{i φ_{r f, 2} \sin (Ω t)} \end{matrix}] [\begin{matrix} \cos (θ_{1}) & i \sin (θ_{1}) \\ i \sin (θ_{1}) & \cos (θ_{1}) \end{matrix}] [\begin{matrix} E_{0} e^{i ω_{0} t} \\ 0 \end{matrix}] \end{array}

where

E_{o u t, o}

and

E_{o u t, e}

are the output fields in the ordinary and extraordinary orientations, respectively, and

T (τ)

is an operator that shifts the function in time, such that

T (τ) f (t) = f (t - τ)

. This time shift is only significant at optical frequencies, meaning it can be ignored when acting on the RF modulation

(Ω t) .

As mentioned, the phase shifts seen by the two polarizations differ in magnitude. A few key assumptions are made to reach this step. The first is that the two polarizations are spatially separated during the interaction length of the modulator. In practice, this is not true, as there will be overlap. This assumption greatly simplifies the model, and does an excellent job in presenting the mechanism for RAM, though it does not give an accurate description of the optical field inside the device. The DC components are given as

ϕ_{d c, 1}

and

ϕ_{d c, 2}

for the two components, which is of the form

φ_{d c, i} = \frac{V_{d c}}{V_{π, d c, i}} π

where

V_{d c}

is the DC voltage of the input electrical signal, and

V_{π, d c, i}

is the dc voltage required to shift the optical carrier by pi radians for that field (this value is what differs for the two polarizations). The alternating component of the input electrical signal is given as

ϕ_{r f, i} sin (Ω t)

, where

φ_{r f, i} = \frac{V_{r f}}{V_{π, r f, i}} π

Like the DC case,

V_{r f}

is the peak voltage of the AC signal, and

V_{π, r f, i}

is the voltage required to see a

π

phase shift on the optical carrier (this term is frequency dependent). Multiplying the transfer matrices gives the output electric fields as

\begin{array}{l} E_{o u t, o} = \frac{1}{2} α_{1} E_{0} e^{i φ_{d c, 1}} e^{i ω_{0} t} \\ \times [\cos (θ_{1}) \cos (θ_{2}) e^{- i ω_{0} τ} e^{i φ_{r f, 1} \sin (Ω t)} - \bar{α} \sin (θ_{1}) \sin (θ_{2}) e^{i \bar{φ_{d c}}} e^{i φ_{r f, 2} \sin (Ω t)}] + c . c . \end{array}

For the ordinary field, and

\begin{array}{l} E_{o u t, e} = \frac{i}{2} α_{1} E_{0} e^{i φ_{d c, 1}} e^{i ω_{0} t} \\ \times [\cos (θ_{1}) \sin (θ_{2}) e^{- i ω_{0} τ} e^{i φ_{r f, 1} \sin (Ω t)} + \bar{α} \sin (θ_{1}) \cos (θ_{2}) e^{i \bar{φ_{d c}}} e^{i φ_{r f, 2} \sin (Ω t)}] + c . c . \end{array}

for the extraordinary field. Here, we have substituted

\bar{ϕ_{d c}} = ϕ_{d c, 2} - ϕ_{d c, 1}

and c.c. means the complex conjugate of what precedes it. We have also factored out the modulators optical loss (which should be very close to

α_{1}

) and left

\bar{α}

to represent the difference in loss between the two polarizations. After the orthogonal fields exit the phase modulator, they will typically pass through a few meters of fiber. In most instances, the effects of chromatic dispersion can be ignored for fiber lengths as short as this, but in the case of RAM, it must be included. We do this in the same manner as [10], using the identity

e^{i φ_{r f, i} \sin (Ω t)} = \sum_{n = - \infty}^{\infty} J_{n} (φ_{r f, i}) e^{i n Ω t},

(where

J_{n}

is the nth order Bessel function of the first kind), and then adding the contribution of the propagation constant required due to the propagation down the remaining optical fiber (of length L) following the modulator

β (ω) = β_{0} + β_{1} (ω - ω_{0}) + \frac{1}{2} β_{2} {(ω - ω_{0})}^{2} + \dots

where

ω

is the angular frequency of the particular field (the angular frequency of the carrier

ω_{0}

plus or minus an integer multiple of the modulation angular frequency Ω). The terms

β_{0}

and

β_{1}

are unimportant when calculating the detected photocurrent for a link with one carrier frequency, and can be left out. Terms higher than

β_{2}

can be ignored, as they are small. Expanding Eqs. (5) and (6) and inserting the propagation constant results in the orthogonal optical fields at the photodetector. They are of the form

\begin{array}{l} E_{o u t, o} = c . c . + \frac{1}{2} α_{1} E_{0} e^{i φ_{d c, 1}} e^{i ω_{0} t} \\ \times [\cos (θ_{1}) \cos (θ_{2}) e^{- i ω_{0} τ} \sum_{n = - \infty}^{\infty} J_{n} (φ_{r f, 1}) e^{i n Ω t} e^{- \frac{i}{2} β_{2} {(n Ω)}^{2} L} \\ - \bar{α} \sin (θ_{1}) \sin (θ_{2}) e^{i \bar{φ_{d c}}} \sum_{n = - \infty}^{\infty} J_{n} (φ_{r f, 2}) e^{i n Ω t} e^{- \frac{i}{2} β_{2} {(n Ω)}^{2} L}] \end{array}

for the ordinary field, and

\begin{array}{l} E_{o u t, e} = c . c . + \frac{i}{2} α_{1} E_{0} e^{i φ_{d c, 1}} e^{i ω_{0} t} \\ \times [\cos (θ_{1}) \sin (θ_{2}) e^{- i ω_{0} τ} \sum_{n = - \infty}^{\infty} J_{n} (φ_{r f, 1}) e^{i n Ω t} e^{- \frac{i}{2} β_{2} {(n Ω)}^{2} L} \\ + \bar{α} \sin (θ_{1}) \cos (θ_{2}) e^{i \bar{φ_{d c}}} \sum_{n = - \infty}^{\infty} J_{n} (φ_{r f, 2}) e^{i n Ω t} e^{- \frac{i}{2} β_{2} {(n Ω)}^{2} L}] \end{array}

for the extraordinary field. The c.c. was moved to the front of the equation for formatting reasons only. It can be seen that Eqs. (9) and (10) show an optical field with an infinite number of sidebands spaced at the frequency of the input AC drive signal. The next step is to calculate the photocurrent output of the detector. This photocurrent is proportional to

E^{*} E

, which gives a DC (unmodulated) component, a component at the modulation frequency Ω, and an infinite number of harmonics at nΩ. We will ignore all terms of

| n | > 1

, as they either do not contribute to the fundamental frequency or their size is negligible compared to the preceding terms. Another assumption made is the input RF signal is small in amplitude compared to

V_{π}

, which makes

J_{0} (ϕ_{r f, i}) \to 1

and

J_{1} (ϕ_{r f, i}) \to ϕ_{r f, i} / 2

. This is done purely for the simplicity of the following expressions. For input signals with amplitudes approaching

V_{π}

, the following steps would be taken without making this assumption. Making these assumptions, and calculating the photocurrent contribution from each of the orthogonally polarized optical fields gives independent photocurrents of the form

\begin{array}{l} I_{Ω, o} \propto \frac{1}{2} α_{1}^{2} E_{0}^{2} \sin (Ω t) [\cos^{2} (θ_{1}) \cos^{2} (θ_{2}) φ_{r f, 1} \sin (\frac{1}{2} β_{2} Ω^{2} L) \\ + {\bar{α}}^{2} \sin^{2} (θ_{1}) \sin^{2} (θ_{2}) φ_{r f, 2} \sin (\frac{1}{2} β_{2} Ω^{2} L) \\ - \bar{α} \sin (θ_{1}) \cos (θ_{1}) \sin (θ_{2}) \cos (θ_{2}) \\ \times [(φ_{r f, 1} + φ_{r f, 2}) \cos (ω_{0} τ + \bar{φ_{d c}}) \sin (\frac{1}{2} β_{2} Ω^{2} L) \\ + (φ_{r f, 1} - φ_{r f, 2}) \sin (ω_{0} τ + \bar{φ_{d c}}) \cos (\frac{1}{2} β_{2} Ω^{2} L)]] \end{array}

for the ordinary field, and

\begin{array}{l} I_{Ω, e} \propto - \frac{1}{2} α_{1}^{2} E_{0}^{2} \sin (Ω t) [\cos^{2} (θ_{1}) \sin^{2} (θ_{2}) φ_{r f, 1} \sin (\frac{1}{2} β_{2} Ω^{2} L) \\ + {\bar{α}}^{2} \sin^{2} (θ_{1}) \cos^{2} (θ_{2}) φ_{r f, 2} \sin (\frac{1}{2} β_{2} Ω^{2} L) \\ + \bar{α} \sin (θ_{1}) \cos (θ_{1}) \sin (θ_{2}) \cos (θ_{2}) \\ \times [(φ_{r f, 1} + φ_{r f, 2}) \cos (ω_{0} τ + \bar{φ_{d c}}) \sin (\frac{1}{2} β_{2} Ω^{2} L) \\ + (φ_{r f, 1} - φ_{r f, 2}) \sin (ω_{0} τ + \bar{φ_{d c}}) \cos (\frac{1}{2} β_{2} Ω^{2} L)]] \end{array}

for the extraordinary field. The proportionality of Eqs. (11) and (12) are given instead of an equivalence, because the exact photocurrent will depend on the photodetector used. These values can be written as an equivalence with an addition of a scalar, photodiode dependent quantity so long as the same photodetector is used for each field. Of course, these currents Eqs. (11) and (12) are immediately summed by the photodetector (with a small phase correction due to the polarization dependent path-length of the modulator). They are given separately here, due to the nature of the experimental setup described in the following section of this paper. Since the measurement we will take is a gain measurement, we must use Eqs. (11) and (12) to calculate the gain of a photonic link. This is given as

G = \frac{P_{o u t}}{P_{i n}} = \frac{I^{2} Z_{o u t} Z_{i n}}{V_{r f}^{2}}

where I is the photocurrent amplitude of the output signal,

Z_{o u t}

and

Z_{i n}

represent the impedance of the photodetector and modulator, respectively. Equation (13), inserting Eq. (11), Eq. (12), or a sum of the two for I (averaging over time) will give the theoretical fit for the data to follow.

A key point to notice in Eqs. (11) and (12) is the lumping of the terms $ω_{0} τ + \bar{ϕ_{d c}}$ as an effective bias control. This means that changes on the input DC signal have the same impact to the RAM as changes in $ω_{0} τ$ . So instead of making changes to the laser frequency $ω_{0}$ or the effective path-length difference between the two polarizations (by tuning the temperature), one can simply adjust a control voltage to minimize RAM. To show this temperature equivalence, as well as support the assumption that etalon effects are not significantly impacting these new modulators, we will solve for the term $ω_{0} τ$ (assuming there is zero electrical bias). If we take only the ordinary field’s photocurrent, and insert for I, the resulting equation can be rewritten as

\begin{array}{l} φ_{r f, 1} \sin (ω_{0} τ + \frac{1}{2} β_{2} Ω^{2} L) - φ_{r f, 2} \sin (ω_{0} τ - \frac{1}{2} β_{2} Ω^{2} L) \\ = (\sqrt{\frac{G V_{r f}^{2}}{Z_{o u t} Z_{i n}}} - \frac{1}{2} α_{1}^{2} E_{0}^{2} (\cos^{2} (θ_{1}) \cos^{2} (θ_{2}) φ_{r f, 1} \sin (\frac{1}{2} β_{2} Ω^{2} L) \\ + {\bar{α}}^{2} \sin^{2} (θ_{1}) \sin^{2} (θ_{2}) φ_{r f, 2} \sin (\frac{1}{2} β_{2} Ω^{2} L))) \\ \div (- \frac{1}{2} α_{1}^{2} E_{0}^{2} \bar{α} \sin (θ_{1}) \cos (θ_{1}) \sin (θ_{2}) \cos (θ_{2})) \end{array}

We can immediately neglect the dispersion related terms on the left hand side of this equation, as they are much smaller than $ω_{0} τ$ . We can also substitute Eqs. (3) and (4) in for $ϕ_{r f, i}$ , resulting in a closed form expression for $ω_{0} τ$ , written as

\begin{array}{l} \sin (ω_{0} τ) = \frac{V_{π, r f, 1} V_{π, r f, 2}}{π (V_{π, r f, 2} - V_{π, r f, 1})} (\sqrt{\frac{G}{Z_{o u t} Z_{i n}}} - \frac{π α_{1}^{2} E_{0}^{2}}{2} \\ \times (\frac{\cos^{2} (θ_{1}) \cos^{2} (θ_{2})}{V_{π, r f, 1}} \sin (\frac{1}{2} β_{2} Ω^{2} L) + {\bar{α}}^{2} \frac{\sin^{2} (θ_{1}) \sin^{2} (θ_{2})}{V_{π, r f, 2}} \sin (\frac{1}{2} β_{2} Ω^{2} L)) \\ \div (- \frac{1}{2} α_{1}^{2} E_{0}^{2} \bar{α} \sin (θ_{1}) \cos (θ_{1}) \sin (θ_{2}) \cos (θ_{2})) \end{array}

We can now give a prediction for the temperature dependence of $ω_{0} τ$ using the following equation:

ω_{0} τ = \frac{ω_{0}}{c} (\frac{\partial Δ n}{\partial T} L_{\mod} + Δ n \frac{\partial L_{\mod}}{\partial T}) \partial T

where

L_{m o d}

is the interaction length of the modulator,

Δ n

is the difference in the index of refraction between the orthogonal polarizations, c is the speed of light in vacuum, T is temperature,

\partial Δ n / \partial T

is the change in the index difference versus temperature, and

\partial L_{m o d} / \partial T

is the coefficient of thermal expansion for LiNbO_3. When the values given in Table 1 are inserted, this leaves the effective time delay due to temperature as scalar times the change in temperature. This is valid under small changes in temperature (to where the measured values of these parameters are still accurate). Inserting Eq. (16) into Eq. (15) allows us to convert a change in gain, to an estimated change in temperature for the device. The importance of this will become clear in the following section.

Table 1. Table of parameters used in model.

View Table

This section has given a closed form expression for the unwanted amplitude modulation given an input signal for both polarizations. It has also given a form for predicting the temperature dependence of this unwanted amplitude. As stated before, a DC input voltage is an equivalent to a change in temperature, and will be used to control the RAM seen in experiment.

3. Experiment

The first experiment to be conducted is a simple gain measurement taken on a vector network analyzer (VNA). The experimental setup, shown in Fig. 1, consists of an optical source (EM4: 1550nm 100mW CW laser) coupled to a LiNbO_3. phase modulator (EOSpace: 40GHz PM) with a polarization maintaining optical fiber. The output of the modulator is coupled to another length of polarization maintaining fiber (~1m), which, for some of the measurements below, is connected to an inline linear polarizer. Next is a variable optical attenuator to set the power to be within spec of the following photodiode (U2T: 75 GHz waveguide device). The modulator’s electrical input is preceded by a bias tee (Agilent: 40 GHz). The system is driven by a 50 GHz VNA (Agilent: PNA Series).

Fig. 1 Experimental Setup for RAM measurements. VNA calibrated with RF cables and bias tee inline to isolate the photonic link for measurement accuracy.

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To test the model presented previously, three measurements will be required. One with the linear polarizer set to pass the ordinary mode, one with the linear polarizer set to pass the extraordinary mode, and one without the linear polarizer. For each of these setups, the DC bias voltage is set to a random voltage and the appropriate term in the model is used as a fitting parameter (every other term remains constant, and $ω_{0} τ$ was set to zero for simplicity). Table 1 shows the values for the variables present in the model. Figure 2(a) shows the measured gain due to the ordinary field for two random bias voltages, and the corresponding theory. Figure 2(b) shows the results for the contribution due to the extraordinary mode only. These data were isolated by using an in-line polarizer to strip the unwanted polarization before detection. Finally, shown in Fig. 3, are the measurement results for both contributions summed in current at the photodetector. For this particular set of data, a small phase correction must be applied to the model due to the polarization maintaining fiber’s birefringence. The polarization dependent delay was measured by analyzing the phase of the output signal for the two independent polarizations. The correction is small (1.5 picoseconds), but important to get an accurate comparison.

Fig. 2 Measurement results for the a) ordinary mode, and b) extraordinary mode at two random bias voltages. Measured data in black, theory in red dashes. The measurement noise floor prevented the system from resolving the null shown in the model.

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Fig. 3 Measurement results for both contributions at two random bias voltages. Measured data in black, theory in red dashes. Lower noise floor is achievable due to higher total gain (summed photocurrent).

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From here we can discuss the temperature dependence of the model presented in the previous section. Given that the modulator is a sealed device with open space, metal contacts, and the crystal itself, a detailed thermal model is unnecessarily complicated. Instead, we will simply analyze the temperature dependence by monitoring the order of magnitude of calculated temperature change. By placing a small resistive heater against the modulator, and warming the device with ½ Watt of power, we can expect the temperature change to be relatively small across the crystal.

The measurement is taken at a single frequency over a five-minute window while the heater is running, or the device is cooling down after running for some time. The amplitude of the unwanted intensity modulation is monitored over this time, and from that data, the temperature is extracted via the model presented in the previous section. Figure 4 shows the results of this measurement for the case of a heating device, and a device cooling down after heating. The data is normalized to show the predicted change in temperature over time.

Fig. 4 Predicted temperature change during a heating (red) and cooling (blue dashed) cycle. These data were extracted from measured gain data using Eqs. (13), (15), and (16).

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The theory supplied in the previous section, when applied given the measured data predicts a temperature change on the order of one-degree C. In the case of etalon effects, the temperature change would have to expand the crystal enough to impact its length on the RF signal’s scale. This would require a temperature change on the order of 10,000 degrees, which of course is not realistic with a ½ Watt heater. The purpose of this experiment is two-fold. First, it tests the model for a realistic temperature dependence. Second, it shows that bulk reflections do not account for the RAM we are measuring with this particular device.

The final measurement to take is one showing the extent at which one can reduce the RAM of a phase modulated link by adjusting the DC bias voltage. As mentioned earlier in this work, a change in voltage should simulate a change in temperature, although with a near-instantaneous reaction time. The measurement will be taken at 1 GHz to remove any contribution seen by chromatic dispersion (chromatic dispersion converts the phase modulated signal into an intensity modulated signal, as described in [10]. Its contribution cannot be negated through bias control). The results of this final experiment are shown in Fig. 5, where the bias voltage was swept from −1.5 V to + 1.5 V. The measurement is a time sensitive measurement; as small temperature drifts can significantly impact the level of RAM. To reduce the effects of temperature drift, the entire measurement took place over approximately 15 seconds. Even with this precaution, it can be seen that as the measurement progressed, the slight temperature shifts present in the environment skewed the data slightly. These data show a RAM suppression of >40 dB for this particular device, which offers an excellent means of in situ control with an engineered feedback system.

Fig. 5 Measured RAM level as a function of a sweeping bias control voltage (black), fit to model (red dashed).

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A key limitation of this control mechanism is in its application. For systems driving the phase modulator with a DC (or near-DC) voltage as a form of control, adding an additional bias voltage is not possible. For systems which drive the modulator with an oscillating signal of a frequency high enough to employ a bias tee on the input, this mechanism will offer a mechanism for minimizing the RAM observed during operation.

4. Conclusion

This paper has provided a closed form expression for the unwanted amplitude modulation present in phase modulated links employing a LiNbO₃ device. It assumed that there was some polarization misalignment present on both ends of the modulator and that the resulting orthogonal components of the optical field were separated in space as they travelled through the interaction length of the crystal. These assumptions result in an effective asymmetric Mach-Zehnder modulator with poor efficiency. The temperature dependence of the RAM is measured and used to check the model’s predicted behavior as it relates to changes in temperature, further testing the theory described. Finally, the model predicts that a DC voltage can be used to minimize the RAM present in these systems, and data is provided, confirming this assertion.

The experimental data agreed well with the proposed theory. Some error would be expected given the assumptions made. The fact that the orthogonal fields most likely had some spatial overlap would result in some deviations that may be heavily device dependent. The small signal approximations in the final steps of the theory could result in deviations from the theoretical prediction as the Bessel functions begin to roll over. However, these errors are small enough, given the operating conditions of this setup, to demonstrate the validity of the control mechanism described.

The take away point to be made is that the use of a bias tee before the modulator may allow for the minimization of RAM in phase modulated links. A feedback mechanism could be engineered to keep this voltage set, minimizing the RAM, and keeping the system operating at an ideal sensitivity.

References and links

1. D. A. Pogorelaya, M. A. Smolovik, V. E. Strigalev, A. S. Aleynik, and G. Deyneka, “An investigation of the influence of residual amplitude modulation in phase electro-optic modulator on the signal of fiber-optic gyroscope,” J. Phys. Conf. Ser. 735, 012040 (2016). [CrossRef]

2. E. Jaatinen, D. J. Hopper, and J. Back, “Residual amplitude modulation mechanisms in modulation transfer spectroscopy that uses electro-optic modulators,” Meas. Sci. Technol. 20(2), 025302 (2009). [CrossRef]

3. K. Kokeyama, K. Izumi, W. Z. Korth, N. Smith-Lefebvre, K. Aria, and R. X. Adhikari, “Residual amplitude modulation in interferometric gravitational wave detectors,” arXiv:1309.4522v1 (2013).

4. E. A. Whittaker, M. Gehrtz, and G. C. Bjorklund, “Residual amplitude modulation in laser electro-optic phase modulation,” J. Opt. Soc. Am. B 2(8), 1320–1326 (1985). [CrossRef]

5. J. Sathian and E. Jaatinen, “Intensity dependent residual amplitude modulation in electro-optic phase modulators,” Appl. Opt. 51(16), 3684–3691 (2012). [CrossRef] [PubMed]

6. N. C. Wong and J. L. Hall, “Servo control of amplitude modulation in frequency-modulation spectroscopy: demonstration of shot-noise-limited detection,” J. Opt. Soc. Am. B 2(9), 1527–1533 (1985). [CrossRef]

7. J. Sathian and E. Jaatinen, “Polarization dependent photorefractive amplitude modulation production in MgO:LiNbO3 phase modulators,” in IQEC/CLEO Pac. Rim 2011 (2011).

8. D. A. Bryan, R. Gerson, and H. E. Tomaschke, “Increased optical damage resistance in lithium Niobate,” Appl. Phys. Lett. 44(9), 847–849 (1984). [CrossRef]

9. L. Li, F. Liu, C. Wang, and L. Chen, “Measurement and control of residual amplitude modulation in optical phase modulation,” Rev. Sci. Instrum. 83(4), 043111 (2012). [CrossRef] [PubMed]

10. J. F. Diehl and V. J. Urick, “Chromatic Dispersion Induced Second-Order Distortion in Long-Haul Photonic Links,” J. Lightwave Tech. 34(20), 4646–4651 (2016). [CrossRef]

11. V. J. Urick, F. Bucholtz, J. D. McKinney, P. S. Devgan, A. L. Campillo, J. L. Dexter, and K. J. Williams, “Long-Haul Analog Photonics,” J. Lightwave Tech 29(8), 1182–1205 (2011). [CrossRef]

Symbol	Quantity	Value
$\frac{1}{2} α_{1}^{2} E_{0}^{2}$	DC Photocurrent^a	0.005 A
$θ_{1}$	Input Alignment Error	0.157 radians
$θ_{2}$	Output Alignment Error	0.157 radians
$β_{2}$	Dispersion for SMF-28	21.5 $p s^{2} / n m Δ k m$ for SM-28 64 $p s^{2} / n m Δ k m$ for PM fiber
$V_{π, r f, 1}$	Vpi of phase modulator in ordinary mode.	Data Provided by Manufacturer
$V_{π, r f, 2}$	Vpi of phase modulator in extraordinary mode	Data Provided by Manufacturer
$\partial Δ n / \partial T$	Index difference dependence on Temperature^b	$33.7 \times 10^{- 6} ° C^{- 1}$
$\partial L / \partial T$	Coefficient of Thermal Expansion^b	$9.1 \times 10^{- 6}$ $m ° C^{- 1}$
$L_{m o d}$	Interaction Length of Modulator	5 cm (Estimated)
$Δ n$	Difference in Indices of Refraction for two modes	0.085

Control of residual amplitude modulation in Lithium Niobate phase modulators

Abstract

1. Introduction

2. Theory

3. Experiment

4. Conclusion

References and links

Cited By

Figures (5)

Tables (1)

Equations (16)

Optics Express