DE102024201192A1 - Device and method for generating a bias voltage for an electro-optical modulator - Google Patents

Abstract

Embodiments provide a device [e.g. circuit, such as control circuit] for generating a [e.g. time-variable] bias voltage V _DC for an electro-optical modulator, wherein the device is configured to obtain an output power signal P̂ _out dependent on an optical output power of the electro-optical modulator, wherein the device is designed to generate the bias voltage V _DC as a function of an amplitude ratio between the fundamental wave and the first harmonic of the output power signal P̂ _out and as a function of a phase angle of a pilot signal applied to the electro-optical modulator.

Description

Embodiments of the present invention relate to an apparatus and method for generating a bias voltage for an electro-optical modulator. Some embodiments relate to an apparatus and method for controlling the bias voltage for the precise locking of the minimum operating points in Mach-Zehnder modulators.

In quantum computing, light is one of the most important tools for controlling both the position and quantum state of atoms used for data processing. In practical systems, laser light is used to manipulate the quantum states of qubits in the desired manner. In addition to physical effects such as decoherence and quantum noise, the precision of qubit manipulation has a significant impact on the achievable error rate in quantum computing. Therefore, the laser transmission chain is subject to strict requirements regarding frequency accuracy, spectral bandwidth, output power stability, and so on.

One of the key components, alongside the laser, is the optical modulator, which must modulate or switch a constant-power laser light to generate light pulses or pulse trains with a desired shape. Electro-optical (EOM) modulators, and in particular the Mach-Zehnder modulator (MZM), are preferred for this purpose. However, there is neither a simple linear relationship between the modulator's control signal and the modulator's output, nor can they be assumed to have time-invariant properties.

One of the most challenging technical problems when using EOMs is controlling the control voltage. EOMs generally drift due to refractive index changes caused by temperature fluctuations, aging, or other pyroelectric, photorefractive, or photoconductive effects. This shifts the transfer function and places the modulation signal at a different operating point, significantly impairing the modulation quality.

The present invention is therefore based on the object of creating a concept which makes it possible to reduce or even compensate for a drift shift of the operating point of maximum attenuation of an electro-optical modulator.

This problem is solved by the independent patent claims.

Advantageous further developments can be found in the dependent patent claims.

Embodiments provide a device [e.g. circuit, such as control circuit] for generating a [e.g. time-variable] bias voltage V _DC for an electro-optical modulator, wherein the device is configured to obtain an output power signal P̂ _out dependent on an optical output power of the electro-optical modulator, wherein the device is designed to generate the bias voltage V _DC as a function of an amplitude ratio between the fundamental wave and the first harmonic of the output power signal P̂ _out and as a function of a phase angle of a pilot signal applied to the electro-optical modulator.

Embodiments make it possible to compensate for the drift phenomenon by generating a suitable DC bias voltage, e.g., which reduces or even compensates for the drift shift by means of feedback (e.g., with the aid of a feedback (e.g., feedback bias control loop) and with the aid of a pilot signal (e.g., pilot tone), thus controlling the drifting modulator bias point.

In embodiments, the device is configured to generate the pilot signal V _pilot for the electro-optical modulator.

In embodiments, the device is configured to apply the bias voltage V _DC and/or the pilot signal V _pilot to at least one control input of the electro-optical modulator.

For example, the device can be configured to apply a combination/superposition of bias voltage V _DC and pilot signal V _pilot to a control input of the electro-optical modulator. Alternatively, the device can also be configured to apply the bias voltage V _DC to a first control input and the pilot signal V _pilot to a second control input of the electro-optical modulator.

In embodiments, the device is configured to superimpose or combine the bias voltage V _DC with the pilot signal V _pilot to obtain a control voltage V _C for the electro-optical modulator.

In embodiments, the device is configured to apply control voltage V _C to a control input of the electro-optical modulator.

In embodiments, the device is configured to apply the bias voltage V _DC to a first control input of the electro-optical modulator and to apply the pilot signal V _pilot to a second control input of the electro-optical modulator.

In embodiments, the output power signal P̂ _out describes an estimated optical output power of the electro-optical modulator.

In embodiments, the pilot signal V _pilot is a time-limited, sinusoidal signal.

In embodiments, the electro-optical modulator is a Mach-Zehnder modulator.

In embodiments, the device comprises a photodetector configured to detect at least a portion of an optical output power P̂ _out of the electro-optical modulator to obtain the output power signal.

In embodiments, the device is configured to estimate a differential voltage ΔV as a function of the amplitude ratio between the fundamental wave and the first harmonic of the output power signal P̂ _out and as a function of the phase angle of the pilot signal applied to the electro-optical modulator, wherein the differential voltage ΔV describes a difference between the current bias voltage V _DC = V̂ _min and a target bias voltage V _min at which the optical output power P̂ _out of the electro-optical modulator has a minimum value.

In embodiments, the device is configured to track [e.g., correct] the bias voltage V _DC as a function of the estimated difference voltage ΔV toward the desired bias voltage V _min .

In embodiments, the device is configured to determine the differential voltage ΔV based on the following equation: $$\Delta V=\frac{{\mathcal{F}}_{1f2f}^{+}\left({\omega }_d\right)}{{\mathcal{F}}_{1f2f}^{+}\left(2{\omega }_d\right)}\cdot \frac{V_d}{4}\cdot e^{-i\left({\phi }_d-\frac{\pi }{2}\right)}$$ where ${\mathcal{F}}_{1f2f}^{+}\left({\omega }_d\right)$ is a Fourier transform of the fundamental wave of the output power signal, where ${\mathcal{F}}_{1f2f}^{+}\left(2{\omega }_d\right)$ is a Fourier transform of the first harmonic of the output power signal,
where V _d is an amplitude of the pilot signal, and
where φ _d is the phase angle of the pilot signal.

In embodiments, the output power signal P̂ _out is discretely sampled, wherein the device is configured to estimate the differential voltage ΔV based on the following equation: $$\Delta {\widehat{V}}_c=\frac{DFT\left\{{\widehat{P}}_{out}\right\}\left[z_{1{\omega }_d}\right]}{DFT\left\{{\widehat{P}}_{out}\right\}\left[z_{2{\omega }_d}\right]}\cdot \frac{V_d}{4}\cdot e^{-i\left({\phi }_d-\frac{\pi }{2}\right)}$$ where ΔV̂ _c is an estimate of the differential voltage ΔV,
where DFT{P̂ _out }[z _{1ω d} ] is a discrete Fourier transform of the fundamental wave of the discretely sampled output power signal,
where DFT{P̂ _out }[z _{2ω d} ] is a discrete Fourier transform of the first harmonic of the discretely sampled output power signal,
where V _d is an amplitude of the pilot signal, and
where φ _d is the phase angle of the pilot signal.

In embodiments, the device is configured to estimate the phase angle depending on a signal propagation time difference between the output power signal and the pilot signal.

In embodiments, the pilot signal V _pilot is sinusoidal, wherein the output power signal is a sampled output power signal P̂ _out [k] with K samples, where K is an integer multiple N of periods of the pilot signal V _pilot .

In embodiments, the device is configured to shift the output power signal or a sampled version of the output power signal by a propagation delay Δτ of the pilot signal [e.g. propagation delay of the pilot signal between a control input of the electro-optical modulator to which the pilot signal is applied and the output power signal dependent on the optical output power of the electro-optical modulator] [e.g. in the negative time direction] in order to compensate for a propagation-related phase difference Δφ of the phase angle φ _d .

In embodiments, the device is configured to determine the differential voltage ΔV based on the following equation: $$\Delta \widehat{V}=-\frac{imag\left(DFT\left\{{\widehat{P}}_{out}\right\}\left[z_{1{\omega }_d}\right]\right)}{real\left(DFT\left\{{\widehat{P}}_{out}\right\}\left[z_{2{\omega }_d}\right]\right)}\cdot \frac{V_d}{4}$$ where ΔV̂ is an estimate of the differential voltage ΔV,
where imagDFT{P̂ _out }[z _{1ω d} ] an imaginary part of a discrete Fourier transform of the fundamental wave z _{1ω d} of the discretely sampled output power signal P̂ _out ,
where imagDFT{P̂ _out }[z _{2ω d} ] a real part of a discrete Fourier transform of the first harmonic z _{2ω d} of the discretely sampled output power signal P̂ _out ,
where V _d is an amplitude of the pilot signal.

Further embodiments provide an optical arrangement comprising an electro-optical modulator and a device for generating a [e.g. time-variable] bias voltage V _DC for the electro-optical modulator.

• 1 ein schematisches Blockschaltbild eines Mach-Zehnder-Modulators,
• 2 in einem Diagramm eine schematische Ansicht einer Übertragungsfunktion eines idealen Mach-Zehnder-Modulators aufgetragen über die Kontrollspannung,
• 3 in einem Diagramm eine schematische Ansicht einer driftbedingten Verschiebung einer Übertragungsfunktion eines nicht-idealen (Ungleichheit-Faktor f_ib ≠ 1/2) Mach-Zehnder-Modulators aufgetragen über die Kontrollspannung,
• 4 in einem Diagramm einen Verlauf des Verhältnisses zwischen Schwingungen erster und zweiter Grundfrequenz in Abhängigkeit aufgetragen über einen Bias-Phasendriftwinkel,
• 5 ein schematisches Blockschaltbild einer Vorrichtung zur Erzeugung einer Vorspannung für einen elektro-optischen Modulator, gemäß einem Ausführungsbeispiel der vorliegenden Erfindung,
• 6 eine schematische Ansicht einer optischen Anordnung mit einer Lichtquelle, einem elektro-optischen Modulator, einem optischen Strahlteiler, einer Messvorrichtung für optische Leistung und einer Abtastvorrichtung,
• 7 in einem Diagramm eine schematische Ansicht vom Zusammenhang zwischen gesuchter Vorspannung, bekannter Vorspannung und geschätzter Differenzspannung in der Übertragungsfunktion,
• 8 in Diagrammen einen zeitlichen Zusammenhang zwischen Pilotsignal und geschätzter Ausgangsleistung, und
• 9 in Diagrammen einen zeitlichen Zusammenhang zwischen Pilotsignal und geschätzter Ausgangsleistung nach Ausgleich der Laufzeitverzögerung.

Embodiments of the present invention are described in more detail with reference to the accompanying figures. They show:

• 1 a schematic block diagram of a Mach-Zehnder modulator,
• 2 in a diagram a schematic view of a transfer function of an ideal Mach-Zehnder modulator plotted against the control voltage,
• 3 in a diagram a schematic view of a drift-induced shift of a transfer function of a non-ideal (inequality factor f _ib ≠ 1/2) Mach-Zehnder modulator plotted against the control voltage,
• 4 in a diagram a curve of the relationship between oscillations of the first and second fundamental frequency plotted against a bias phase drift angle,
• 5 a schematic block diagram of a device for generating a bias voltage for an electro-optical modulator, according to an embodiment of the present invention,
• 6 a schematic view of an optical arrangement with a light source, an electro-optical modulator, an optical beam splitter, an optical power measuring device and a scanning device,
• 7 in a diagram a schematic view of the relationship between the desired bias voltage, the known bias voltage and the estimated differential voltage in the transfer function,
• 8 in diagrams a temporal relationship between pilot signal and estimated output power, and
• 9 in diagrams a temporal relationship between the pilot signal and the estimated output power after compensation of the propagation delay.

In the following description of the embodiments of the present invention, identical or equivalent elements in the figures are provided with the same reference numerals so that their description is interchangeable.

In the following description of the embodiments, several details are set forth to provide a more complete explanation of embodiments of the present invention. However, it will be apparent to one skilled in the art that embodiments of the present invention may be practiced without these specific details. In other instances, well-known structures and devices are shown in block diagram form rather than in detail in order not to obscure embodiments of the present invention. Furthermore, features of the various embodiments described below may be combined with one another unless expressly stated otherwise.

Before embodiments of the device for generating a bias voltage for an electro-optical modulator are described with reference to 5 to 9 will be explained in detail, the underlying electro-optical modulator and the problem of the drift shift of the operating point of maximum attenuation will first be described in more detail.

One of the best-known electro-optical modulators, the Mach-Zehnder modulator (MZM), is an optical component used to modulate the intensity of laser light. A phase shift between two optical paths is converted into an amplitude change through interference.

1 shows a schematic block diagram of such a Mach-Zehnder modulator 10. This comprises two optical paths (transmission arms) 12 and 14 as well as electrodes 16, 17 and 18, via which opposing electric fields can be applied to the two optical paths 12 and 14.

As in 1 As shown, the incident light is ideally split equally between the two (transmission) arms, to which opposing electric fields are applied. This creates a phase difference between the light waves traveling in the two arms, which interfere when combined at the output. In an ideal MZM, the resulting power of the output signal ranges from zero (0) to the maximum input power, depending on the strength of the electric field. The two waves interfere destructively when the phase shift is a multiple of π and constructively for multiples of 2π.

The basic EOM transfer function between the applied control voltage V _c (t) and the optical output power P _out , including the possible optical misbalance represented by the inequality factor f _ib when splitting into the two arms, is given by: $$P_{out}\left(t\right)=P_{in}\cdot \left(\frac{1}{2}+f_{ib}\cdot cos\left(\frac{\pi }{V_{\pi }}\cdot \left(V_0-V_c\left(t\right)\right)\right)\right).$$

The value ${\phi }_0=V_0\frac{\pi }{V_{\pi }}$ denotes the initial phase shift that would be present without the application of an electric field. V _π is the half-wave voltage that must be applied to the RF electrode (high-frequency electrode) to bring the optical power from the maximum value to the minimum value (or vice versa), and V _c is the control voltage that generates an electric field between the two electrodes. P _in denotes the optical input power, which is ideally constant in many practical applications.

The transfer function T = P _ouc /P _in resulting from Eq. (1) is in 2 for a given value of V _0. In detail, 2 A diagram showing a schematic view of a transfer function T of an ideal MZM with f _ib = 1/2 plotted against the control voltage V _c . The optical throughput is plotted along the ordinate and the electrical bias voltage along the abscissa. The MZM can be operated as an approximately linear intensity modulator if the optical path difference is set such that V ₀ =±V _π /2 (positions Quad±) and the operation takes place in the almost linear region of the half-power point at T = 0.5. Alternatively, the optical path difference can be set such that that V ₀ is a multiple of V _π . In this case, as shown in the example in 2 shown T(V _c = V ₀ +2 · (n - 1) · V _π ) = 0 (Min) and T(V _c = V ₀ +2 · n · V _π ) = 1 (Max), so that the modulator switches the light off and on (“on-off keying”).

One of the most difficult technical problems when using EOMs is maintaining the correct control voltage V _c . EOMs drift due to changes in the refractive index caused by temperature fluctuations, aging, or other pyroelectric, photorefractive, or photoconductive effects. Furthermore, applying a control voltage results in a significant charge carrier shift on the electrodes. This shifts, as 3 As shown, the transfer function T = P _out /Pi _n in the horizontal direction and the modulation signal is placed at a different operating point, which significantly affects the quality of the modulation. In detail, 3 A diagram showing a drift-induced shift of a transfer function T of an MZM (f _ib ≠ 1/2) plotted against the control voltage V _c . The optical throughput is plotted along the ordinate and the electrical bias voltage along the abscissa. In Eq. (1), this corresponds to a drift of V ₀ , meaning that V ₀ can no longer be considered constant.

The solid curve (curve 1) shows the behavior of the transfer function before the drift shift, whereas the dashed curve (curve 2) shows the behavior of the transfer function under the influence of a drift shift of V ₀ by V _drift . To compensate for this drift phenomenon and thus ensure the long-term stability of the transmission chain, two electrical voltages are applied to the 1 The following voltages are applied to the MZM shown: the high-frequency modulation voltage V _RF , which contains the information about the desired switching or modulation behavior of the EOM, and a bias voltage V _DC , which is applied to the same electrode or to a second bias electrode. This bias voltage compensates for the drift shift V _drift to maintain stable operating conditions and thus controls the modulator bias point (operating point). The use of the subscript "DC" in V _DC has become established in the specialist literature; however, in the strict sense, it is not a constant (direct) voltage, but rather a voltage that changes relatively slowly compared to V _RF .

There are different approaches to eliminate the drift problem, either through a better modulator design or by using auxiliary circuits that fix the bias point. However, since a commercially viable solution using the former approach has not yet been developed [1, 2], the use of feedback bias control loops has become a state-of-the-art method. Two different categories can be distinguished here: techniques that use a pilot tone and pilot-tone-free methods [3, 4]. The latter use the optical input/output power or their ratio as the feedback signal to be monitored. The design of the bias controller is relatively straightforward [3, 4]. However, the feedback signal depends strongly on the optical power fluctuation of the MZ modulator input, which limits the practical application of pilot-tone-free techniques. Pilot-tone-free techniques are used especially for multilevel modulation schemes such as QAM, (D)QPSK, or OFDM and the use of multiple MZMs connected in parallel [3,4]. However, pilot-tone-based schemes predominate in many other EOM applications.

The first pilot tone techniques for bias stability control of the electrical modulator bias voltage were developed in the early 1980s. In the beginning, the modulator was mostly operated as a linear intensity modulator at the half-power points Quad±. In the patent "Automatic bias controller for electro-optic modulator" [5] filed in 1991, a small, low-frequency and mean-free rectangular pilot signal composed of various bipolar individual square-wave signals is applied to the electrical modulation signal input of the MZM. Such a pilot signal is often referred to as a dither signal because it deflects the operating point through comparatively small fluctuations. The optical output signal of the modulator is then acquired, and the sizes of the positive and negative deflection amplitudes of the output signal are compared with the corresponding deflections of the pilot signal originally applied at the input. If the modulator is linearly biased as desired, the deflections are symmetrical in both directions. However, if the MZM has deviated from the half-power points Quad±, a deflection in one direction relative to the other is larger than in the dithered signal, which consequently also indicates the direction of the drift shift V _drift . Using this information, the DC bias V _DC is automatically adjusted to the nearest linear bias point Quad±. If the bias range includes multiple bias points within its operating range, the bias is also automatically reset to the smallest linear point with V ₀ =±V _π /2 by a bias reset.

While in [5] a rectangular pilot signal was used, in [6,7] sinusoidal waveforms are used $$V_{dither}\left(t\right)=V_d\cdot sin\left({\omega }_{d}t+{\phi }_d\right)\text{ mit }{\omega }_d=2\pi f_d,V_d>0$$ with low frequency f _d and as low a drive voltage V _d as possible. The frequency f _d of the dither pilot tone (usually within the range 0.5 kHz to 10 kHz) is usually much lower than the frequency components within the spectrum of the time-varying electrical RF signal. The half-power operating point at T = 0.5 is reached exactly when all even harmonics (with 2f _d , 4f _d , ...) of the harmonic pilot signal disappear and only odd harmonics (with 1f _d , 3f _d , ...) are present in the optical feedback path. This is due to the fact that the transfer function at the quad positions at T = 0.5 shows a point-symmetric curve behavior and a Taylor polynomial series expansion of the trigonometric function from Eq. (1) at this expansion point would only show odd exponents in the functional equation.

The second harmonic at 2f _d , which can be regarded as a measure of the excursion from the half-power point, is extracted from the optical output, followed by synchronous demodulation that generates an error signal that adjusts the DC bias voltage V _DC so that the power of the second harmonic converges to zero (0). Since the power of the second harmonic is significantly lower than that of the original pilot signal at f _d , the SNR results in a significantly lower one compared to detecting the pilot signal. To achieve this SNR, an analog low-pass or band-pass filter is usually used, which extracts the second harmonic with appropriate noise reduction. The SNR can also be improved by using two phase-synchronous pilot frequencies f _d1 and f _d2 , at which the difference component |f _d1 - f _d2 | of the two pilot tones is then extracted and minimized. This is described in more detail in [8].

In an analogous manner, in “switch mode” at the other operating points of the transfer function curve of the modulator, ie at the maximum and minimum points, odd order distortion products (1f _d , 3f _d , ...) of the dither signal can be minimized.

The transfer function exhibits an axisymmetric curve behavior at these bias operating points, and a Taylor polynomial series expansion of the trigonometric function from Equation (1) would exhibit only even exponents at these points, as explained in [9]. Since the application usually knows at which of the four common and therefore practically relevant bias operating points of the transfer function it currently has to operate, a corresponding case distinction is also possible by minimizing the signal amplitude at the fundamental frequency 1f _d or the harmonic at 2f _d , as shown in [10].

All bias stability control techniques considered so far [5-10] are predominantly analog in nature, with the pilot signal permanently superimposed on the time-varying electrical RF signal. The error signal is averaged by filtering, and the drift shift V _drift is mapped to the DC bias voltage V _DC using various methods. These methods can be spreadsheet methods or control algorithms such as a PID controller according to [11].

For some applications in digital communications or microwave photonics, arbitrary bias points along the transfer function curve of the MZ modulator are required instead of the otherwise often common four operating bias points (positively (Quad+) or negatively (Quad-) inclined quadrature points, the minima (also called zeros) and the maxima (also called peaks)) to optimize system performance [12, 13]. This is done in [14, 15, 16, 17], where a complete analytical solution approach was also pursued for the first time. By inserting the dither signal from Eq. (2) into Eq. (1), some trigonometric transformations, and a subsequent 4th-order Taylor series expansion, a relationship R ₁ between the oscillations of the first and second fundamental frequencies is obtained, corresponding to $$R_1=\frac{s_{1f_d}}{s_{2f_d}}=-\text{tan}\left({\phi }_{drift}\right)\frac{\left(a-\frac{a^3}{8}\right)}{\left(\frac{a^2}{4}-\frac{a^4}{48}\right)}\text{ mit a}=\frac{\pi }{V_{\pi }}{V}_d{\phi }_{drift}=\frac{\pi }{V_{\pi }}{V}_{drift}$$

In [14,16], the fundamental and harmonic components are calculated using two bandpass filters, whereas in [15] they are calculated using a fast Fourier transform (FFT). By comparing the resulting harmonic ratios with the values stored in a database (lookup table), the unsigned deviation of the DC control voltage V _DC from its ideal value can be calculated, allowing for appropriate adjustment.

4 shows a diagram showing the relationship _R1 between oscillations of the first and second fundamental frequencies plotted against a bias phase drift angle. The ordinate represents the relationship R1, and the abscissa represents the bias phase drift angle in degrees. In other words, 4 shows the course of Eq. (3) as a function of the unknown phase drift angle φ _drift and a = 0.1. Since V _d << V _π is usually chosen, a < 1 holds and the two expressions in brackets in Eq. (3) always assume a positive value.

As in 4 As shown, the minimum (the zero operating voltage point) V _min is located at φ _drift = 180°. However, since the desired phase drift angle φ _drift is unknown, a pure level calculation of the fundamental and first harmonic is carried out after appropriate transformation of Eq. (3), with subsequent ratio formation [14]. This unsigned ratio is compared with older results in a control system, and the bias control value is calculated by comparing the ratio and the specified ratio [18]. Since no phase information φ _d of the dither pilot tone from Eq. (2) is taken into account in Eq. (3), according to the state of the art, there is no instantaneous memoryless information on the relative position (right or left) of the deviation from the minimum operating point (cf. 7 ) before.

If this additional information were available, then a (e.g. significantly) better convergence in the minimum tracking would result, among other things, in the case that the feedback signal (see e.g. 6 , P _out [k]) is very noisy.

The embodiments described below therefore take into account the phase information (e.g. phase angle) of the pilot signal when providing the bias voltage for the electro-optical modulator.

For example, this can improve minimum tracking.

For example, the convergence of the feedback (e.g., feedback bias control loop) can be improved by taking into account the phase information φ _d of the pilot signal (e.g., dither pilot tone from Eq. (2)).

5 shows a schematic block diagram of a device 100 for generating a bias voltage V _DC for an electro-optical modulator (EOM) 102, according to an embodiment of the present invention. The device 100 is configured to receive an output power signal P̂ _out that is dependent on an optical output power P̂ _out of the electro-optical modulator (EOM) 102 and to generate the bias voltage V _DC for the electro-optical modulator (EOM) 102 as a function of an amplitude ratio between the fundamental wave and the first harmonic of the output power signal P̂ _out and as a function of a phase angle of a pilot signal V _pilot applied to the electro-optical modulator (EOM) 102.

In embodiments, when providing the bias voltage V _DC for the electro-optical modulator (EOM) 102, not only the amplitude ratio between the fundamental wave and the first harmonic of the output power signal P̂ _out is taken into account, but also the phase angle of the pilot signal V _pilot applied to the electro-optical modulator (EOM) 102.

In embodiments, the pilot signal V _pilot for the electro-optical modulator (EOM) 102 can be generated by the device, e.g., by means of a pilot signal generating device of the device 100, or else by another (e.g., external) pilot signal generating device, wherein the device 100 then receives the pilot signal V _pilot and/or the phase angle or information about the phase angle of the pilot signal V _pilot from the other pilot signal generating device.

In embodiments, the device 100 may be configured to apply the bias voltage V _DC and/or the pilot signal V _pilot to at least one control input of the electro-optical modulator.

For example, the device 100 may be configured to apply a combination or superposition of bias voltage V _DC and pilot signal V _pilot to a control input of the electro-optic modulator (EOM) 102. For example, the device 100 may be configured to superimpose or combine the bias voltage V _DC with the pilot signal V _pilot to obtain a control voltage V _C for the electro-optic modulator (EOM) 102. The control voltage V _C may, for example, be applied to a control input of the electro-optic modulator (EOM) 102.

Alternatively, the device 100 may also be configured to apply the bias voltage V _DC to a first control input and the pilot signal V _pilot to a second control input of the electro-optic modulator (EOM) 102.

In embodiments, the output power signal P̂ _out may describe an estimated optical output power of the electro-optic modulator (EOM) 102.

For example, the output power signal P̂ _out can be determined/estimated via a beam splitter 104 and an optical power measuring device 106. For example, the beam splitter 104 can direct a portion of the optical output power P̂ _out of the electro-optic modulator (EOM) 102 to the optical power measuring device 106, wherein the optical power measuring device 106 then estimates the output power P̂ _out of the electro-optic modulator (EOM) 102 based on the portion of the optical output power P̂ _out of the electro-optic modulator (EOM) 102 obtained via the beam splitter 104.

The beam splitter 104 and/or the optical power measuring device 106 may be implemented internally or externally to the device 100.

The optical power measuring device 106 may be, for example, a photodetector.

As this is 5 As indicated, the device 100 can be part of an optical arrangement 110, which comprises the device 100 for generating a bias voltage V _DC for an electro-optic modulator (EOM) 102 and the electro-optic modulator (EOM) 102. The optical arrangement 110 can optionally have a light source (e.g., laser) that provides the optical input power P _in applied to an input of the electro-optic modulator (EOM) 102.

Further embodiments are described in more detail below.

The embodiments described below enable the estimation of the bias voltage required to achieve the lowest possible optical output power, i.e., the maximum possible attenuation of the incoming optical power, for an electro-optical modulator (EOM) based on a defined pilot signal with given phase information φ _d . This voltage is also referred to below as the "black level voltage."

• • die Übertragungsfunktion des EOM zumindest in grober Näherung durch Gl. (1) beschreibbar ist,
• • der EOM sich während des anliegenden Pilotsignals bereits zumindest näherungsweise in einem Arbeitspunkt maximaler Dämpfung der optischen Übertragung befindet,
• • die optische Ausgangsleistung des EOM in ihrem zeitlichen Verlauf während des anliegenden Pilotsignals durch eine geeignete Messvorrichtung in Form eines Rückkopplungssignales geschätzt wird,
• • die Phasenbeziehung zwischen dem Pilotsignal und dem Rückkopplungssignal bekannt ist, z. B. durch Kenntnis der Laufzeitverzögerung des Rückkopplungssignals relativ zum Pilotsignal,
• • die optische Eingangsleistung des EOM in dem für die Auswertung des Rückkopplungssignals herangezogenen Zeitabschnitt näherungsweise konstant ist.

In the following examples it is assumed that

• • the transfer function of the EOM can be described at least roughly by Eq. (1),
• • the EOM is already at least approximately at an operating point of maximum attenuation of the optical transmission during the pilot signal,
• • the optical output power of the EOM is estimated in its temporal course during the applied pilot signal by a suitable measuring device in the form of a feedback signal,
• • the phase relationship between the pilot signal and the feedback signal is known, e.g. by knowing the propagation delay of the feedback signal relative to the pilot signal,
• • the optical input power of the EOM is approximately constant in the time period used for the evaluation of the feedback signal.

A possible example application is the operation of an EOM as an optical switch, as used, for example, in quantum computers, to generate light pulses with a defined course (e.g. with regard to shape, duration, amplitude, energy) from a light source with approximately constant optical power (e.g. laser), under the condition that as little light power as possible is allowed to pass through in the "off" switching state.

Since the phase information φ _d of the dither pilot tone is taken into account in exemplary embodiments, instantaneous, memoryless information is available regarding the relative position (right or left) of the deviation from the minimum operating point. This results in significantly better convergence during minimum tracking, especially in the case of a highly noisy feedback signal.

Embodiments allow for the estimation of the bias voltage V _min required to achieve the minimum possible optical power P _out,min = min(P _out ) at the optical output of an electro-optical modulator (EOM). In embodiments, this estimation is performed using a pilot signal in the form of a comparatively small sinusoidal voltage applied to the control input of the EOM.

The mathematical principles and the derivation of the method are described below.

1. Assumptions made

In the following, it is assumed that the power P _in at the optical EOM input is constant and that an estimated value P̂ _out for the light power P̂ _out at the optical EOM output is available using a suitable measuring device. The measuring device can be implemented, for example, by partially coupling the outgoing light (beam splitter) into a photodetector and evaluating its electrical output signal. The arrangement considered is shown in 6 shown schematically.

In detail, 6 a schematic view of an optical arrangement 110 with a light source (e.g. laser) 101, an EOM 102, an optical beam splitter 104, an optical power measuring device 106, and a scanning device 108. The light source is designed to apply a (constant) optical power P _in to an input of the EOM 102. The EOM 102 is designed to modulate the optical power P _in applied at the input as a function of a control voltage V _c (t) = V _DC + V _RF and to provide an optical output power P _out (t) to an output of the EOM. The optical beam splitter 104 directs a portion of the optical output power P _out (t) of the EOM 102 to the optical power measuring device 106, wherein the optical power measuring device 106 then estimates the output power P _out (t) of the EOM 102 based on the portion of the optical output power P _out (t) of the EOM 102 obtained via the beam splitter 104 to provide an estimated output power signal P̂ _out (t) describing the optical output power P _out (t) of the EOM 102. The sampling device 108 is configured to sample the output power signal P̂ _out (t) to obtain a sampled output power signal P̂ _out [k].

In embodiments, it is assumed that at the time of estimation, a known bias voltage V _DC = V̂ _min is already present at the EOM, which corresponds approximately to the ideal bias voltage V _min , which leads to the minimum achievable optical output power P _out,min (see also 7 ). Thus, at the time of estimation, the EOM is at an operating point with respect to V _DC at which the optical output power is at least in the order of magnitude of the minimum achievable.

During the estimation, no further voltage is applied to the control input of the EOM except the sinusoidal pilot signal and the constant bias voltage.

2. Approximation of the transfer function in the vicinity of its minimum

Starting from the current, known operating point of the EOM with the bias voltage V _DC = V̂ _min , the voltage V _min is sought that leads to the minimum possible optical output power of the EOM according to Equation (1). The previous estimated value V̂ _min deviates from the exact value V _min by the differential voltage ΔV to be estimated, according to the relationship $$\Delta V=V_{min}-{\widehat{V}}_{min}.$$

Knowing the differential voltage ΔV thus allows the calculation of the desired voltage V _min . The following describes the method for estimating ΔV, which directly leads to an estimate of V _min .

The idealized transfer function according to Eq. (1) is periodic and thus, according to the model, exhibits an infinite number of minima. To simplify the following presentation, the minimum at V _min = V ₀ - V _π is referenced without loss of generality, see 7 .

In detail, 7 A diagram showing the relationship between the desired bias voltage V _min , the known bias voltage V̂ _{min ,} and the estimated differential voltage ΔV in the transfer function. The ordinate represents the optical throughput (transfer function from control voltage to optical output power) and the abscissa represents the control voltage V _c .

If the EOM is located at the operating point V _DC = V̂ _min as assumed above and superimposes the sinusoidal pilot signal voltage V _RF (t) = V _pilot (t) = -V _d · sin(ω _d t + φ _d ) with the angular frequency ω _d = 2πf _g and the initial phase φ _d , the total control voltage is $$V_c\left(t\right)={\widehat{V}}_{min}-V_d\cdot sin\left({\omega }_{d}t+{\phi }_d\right)=V_0-V_{\pi }-\Delta V-V_d\cdot sin\left({\omega }_{d}t+{\phi }_d\right).$$

The initial phase φ _d describes the phase position of the sinusoidal pilot signal relative to a definable time t = 0. It consists of a freely selectable phase φ ₀ and a runtime-dependent phase difference Δφ, which will be discussed in more detail below: $${\phi }_d={\phi }_0+\Delta \phi$$

If V _c (t) from Eq. (5) is inserted into Eq. (1), the optical power P _out is: $$P_{out}=P_{in}\cdot \left(\frac{1}{2}+f_{ib}\cdot cos\left(\pi +\frac{\pi }{V_{\pi }}\cdot \left(\Delta V+V_d\cdot sin\left({\omega }_{d}t+{\phi }_d\right)\right)\right)\right).$$

Near its minimum at π, ie at its point of expansion x ≈ π and accordingly for |ΔV|/V _π << 1 and V _d /V _π << 1, the trigonometric function cos(x) can be described with high accuracy by the quadratic Taylor series according to $$\text{cos}\left(x\right)\approx -1+\frac{1}{2}{\left(x-\pi \right)}^2$$ If this approximation is applied to Eq. (7), the following expression P _out,T2 results as an approximation for P _out : $$P_{out}\approx P_{out,T_2}=P_{in}\cdot \left(\frac{1}{2}+f_{ib}\cdot \left(-1+\frac{1}{2}{\left(\frac{\pi }{V_{\pi }}\right)}^2{\left(\Delta V+V_d\cdot sin\left({\omega }_{d}t+{\phi }_s\right)\right)}^2\right)\right).$$

By applying the binomial formula to the quadratic term (ΔV + V _d · sin(ω _d t + φ _d )) ² we obtain, after transformations: $$\begin{array}{l}{\left(\Delta V+V_d\cdot sin\left({\omega }_{d}t+{\phi }_d\right)\right)}^2=\\ =\frac{1}{2}{\left(V_d\right)}^2+{\left(\Delta V\right)}^2+2\Delta V\cdot V_d\cdot sin\left({\omega }_{d}t+{\phi }_d\right)+\frac{1}{2}{\left(V_d\right)}^2\cdot sin\left(2{\omega }_{d}t+2{\phi }_d-\frac{\pi }{2}\right)\end{array}$$

3. Frequency components of the optical output power with phase information

Considering the spectral components of the output signal P _out,T2 from Eqs. (9) and (10), three components emerge: P _out,T2 (ω = 0), P _out,T2 (ω = ω _d ), and P _out,T2 (ω = 2ω _d ). The real-valued scaling factors S _*,T2 , which represent the magnitudes of the three spectral components, as well as the corresponding phases φ _*,T2 are: • DC component at ω = 0: $S_{DC,T_2}=P_{in}\cdot \left(\frac{1}{2}-f_{ib}+\frac{f_{ib}}{2}{\left(\frac{\pi }{V_{\pi }}\right)}^2\left(\frac{1}{2}{\left(V_d\right)}^2+{\left(\Delta V\right)}^2\right)\right)$ • 1f _d fundamental wave at ω = ω _d : $S_{1f_d,T_2}=P_{in}\cdot \frac{f_{ib}}{2}{\left(\frac{\pi }{V_{\pi }}\right)}^2\cdot 2\cdot \Delta V\cdot V_d$ with the phase offset φ _1fd,T2 = φ _d • 2f _d harmonic at ω = 2ω _d : $S_{\text{2}{f}_d,T_2}=P_{in}\cdot \frac{f_{ib}}{2}{\left(\frac{\pi }{V_{\pi }}\right)}^2\cdot \frac{1}{2}{\left(V_d\right)}^2$ with the phase offset ${\phi }_{2f_d,T_2}=2{\phi }_d-\frac{\pi }{2}$

Both the DC and the 1f _d fundamental wave components (S _DC,T2 , S _1fd,T2 ) depend on the variable ΔV to be estimated, while the 2f _d harmonic component S _{2f d ,T2} is independent of ΔV.

Thus, there are, in principle, several ways to estimate the desired ΔV. To be as independent as possible from physically determined quantities such as P _in , f _ib , the 1f _d and 2f _d components are related to each other in the exemplary embodiments for determining ΔV, taking into account amplitude and phase information as well as the phase angle φ _d . In contrast to conventional methods, the phase information is also taken into account in the exemplary embodiments.

4. Estimation of the error deviation ΔV taking into account the fundamental and first harmonic:

The Fourier transform $\mathcal{F}\left\{P_{out,T_2}\left(t\right)\right\}$ the 1f _d - and 2f _d -components of P _out,T2 are considered. Since the DC component is irrelevant for the following considerations, it is not considered for the sake of clarity. Because the output power P̂ _out ≈ P _out,T2 is a purely real signal, negative frequencies are also not considered here. This is achieved by the Fourier operator ${\mathcal{F}}_{1f2f}^{+}$ expressed.

For the Fourier transform of the 1f _d - and 2f _d -components of P _out,T2, the following results for positive values of ω: $$\begin{array}{l}\mathcal{F}\left\{P_{out,T_2}\left(t\right)\right\}\left(\omega >0\right)={\mathcal{F}}_{1f2f}^{+}\left\{S_{1f_d,T_2}\cdot sin\left({\omega }_{d}t+{\phi }_d\right)+S_{2f_d,T_2}\cdot sin\left(2{\omega }_{d}t+2{\phi }_d-\frac{\pi }{2}\right)\right\}\left(\omega \right)=\\ =S_{1f_d,T_2}\cdot i\cdot \sqrt{\frac{\pi }{2}\cdot }{e}^{-i{\phi }_d}\cdot \delta \left(\omega -{\omega }_d\right)+S_{2f_d,T_2}\cdot i\cdot \sqrt{\frac{\pi }{2}\cdot }\left(e^{-i2{\phi }_d}\right)\cdot \delta \left(\omega -2{\omega }_d\right)\end{array}$$

If the values of the above Fourier transforms at ω = ω _d and ω = 2ω _d are related to each other, the result after appropriate reductions is: $$\frac{{\mathcal{F}}_{1f2f}^{+}\left({\omega }_d\right)}{{\mathcal{F}}_{1f2f}^{+}\left(2{\omega }_d\right)}=\frac{S_{1f_d,T_2}\cdot i\cdot \sqrt{\frac{\pi }{2}}\cdot e^{-i{\phi }_d}}{S_{2f_d,T_2}\cdot \sqrt{\frac{\pi }{2}}\cdot \left(-e^{-i2{\phi }_d}\right)}=\frac{S_{1f_d,T_2}\cdot e^{i\left(-{\phi }_d+\frac{\pi }{2}\right)}}{S_{2f_d,T_2}\cdot e^{i\left(\pi -2{\phi }_d\right)}}=4\frac{\Delta V}{V_d}\cdot e^{i\left({\phi }_d-\frac{\pi }{2}\right)}$$

After transformation, ΔV is: $$\Delta V=\frac{{\mathcal{F}}_{1f2f}^{+}\left({\omega }_d\right)}{{\mathcal{F}}_{1f2f}^{+}\left(2{\omega }_d\right)}\cdot \frac{V_d}{4}\cdot e^{-i\left({\phi }_d-\frac{\pi }{2}\right)}$$

It turns out that the value of ΔV, given a known phase position of the pilot signal φ _d and its amplitude V _d, can be determined from the quotient of the Fourier transform of the output power P̂ _out ≈ P _out,T2 at ω = ω _d and ω = 2ω _d . While conventional methods only allow the estimation of the magnitude of ΔV according to Equation (4), they do not contain any information about the sign, unlike the exemplary embodiments.

Obtaining the estimated values of ΔV with a time-discrete representation of the output power P _out

The above equation (13) is based on the evaluation of the continuous-time Fourier transform of the approximated output power P _out,T2 ≈ P _out . In real systems, signals are usually present in the form of discrete-time samples in multiples of the sampling interval T _s , which are based on a measurement. This results in both a discretization of the time axis and an additive disturbance n(t). The discrete-time sequence P̂ _out $${\widehat{P}}_{out}\left[k\right]\approx P_{out,T_2}\left(t=k\cdot T_s\right)+n\left(t=k\cdot T_s\right)$$ represents approximately the signal P _out,T2 at the times t = k · T _s plus a term n[k] that takes into account an additive disturbance at the same times. For the spectral representation, the Fourier transform of a continuous signal used above is $\mathcal{F}\left\{P_{out,T_2}\right\}$ by the discrete Fourier transform DFT{P _out } of the sequence P̂ _out [k]. The estimated value ΔV̂ _c of ΔV is thus: $$\Delta {\widehat{V}}_c=\frac{DFT\left\{{\widehat{P}}_{out}\right\}\left[z_{1{\omega }_d}\right]}{DFT\left\{{\widehat{P}}_{out}\right\}\left[z_{2{\omega }_d}\right]}\cdot \frac{V_d}{4}\cdot e^{-i\left({\phi }_d-\frac{\pi }{2}\right)},$$ where z _1ωd indicates the element of the Discrete Fourier Transform (DFT) which corresponds in the continuous frequency domain to the angular frequency ω _d and correspondingly the index z _{2ω d} the angular frequency 2 _{ω d} corresponds.

Ideally, a purely real value would result for ΔV̂ _c , since it is the estimate of a real-valued physical quantity (voltage). Due to the above-mentioned additive noise n[k] in P̂ _out , parasitic imaginary components also arise when determining ΔV̂ _c via DFT. A mapping of the generally complex estimated value ΔV̂ _c to a real value with simultaneous reduction of the noise components can be achieved by forming the real part of ΔV̂ _c as a final step, i.e. $$\Delta \widehat{V}=real\left(\Delta {\widehat{V}}_c\right)$$

As defined in Eq. (4), the desired value V _min can be estimated using the estimated value ΔV̂ starting from a known operating point V _DC = V̂ _min , ie V _min ≃ V̂ _min + ΔV̂.

Determination of φ _d

As can be seen from Eq. (15), the estimation of ΔV̂ in terms of magnitude and sign requires knowledge of the phase angle φ _d = φ ₀ + Δφ, see Eq. (6). This describes the effective phase shift of the sinusoidal pilot voltage V _pilot (t) relative to time t = 0. This temporal zero point can, in principle, be chosen arbitrarily, but then applies jointly to the pilot signal V _pilot and to the estimated output power P̂ _out , both in its continuous-time representation P̂ _out (t) and in the discrete-time representation as the sequence P̂ _out [k].

This is in 8 The above example shows a sinusoidal pilot signal with φ ₀ = 0, which was chosen to be identically zero for t < 0 for better illustration and without loss of generality. The time t = 0 was chosen to refer to the beginning of the pilot signal.

In detail, 8 The diagram shows a temporal relationship between the pilot signal V _pilot and the estimated output power P̂ _out . The ordinates describe the amplitudes of the pilot signal V _pilot and the estimated output power P _out , while the abscissas describe the respective times t and t'.

In real causal systems, there is always an unavoidable time delay Δτ > 0 of P̂ _out (t) relative to V _pilot (t) due to signal propagation times, e.g. in measuring devices, amplifiers, etc., see 8 below. The output signal attributable to the beginning of the pilot signal at time t = 0 thus becomes effective in P̂ _out at time t = Δτ or t' = 0. Since the estimation of ΔV̂ is based on the evaluation of P̂ _out , its time axis is decisive. Based on the time axis t' of P̂ _out , the pilot signal V _pilot (t) experiences a phase shift of Δφ = Δτ · ω _d due to the (propagation time) delay Δτ. The effective phase φ _d with respect to the time axis of P̂ _out is thus calculated as $${\phi }_d={\phi }_0+\Delta \tau \cdot {\omega }_d,$$ where φ ₀ can be chosen freely.

Compensation of the propagation delay Δτ

As an alternative to the explicit determination of φ _d, the propagation delay Δτ can be taken into account by shifting the signal P̂ _out (t) or, after sampling, its discrete-time sequence P̂ _out [k] by Δτ in the negative time direction (to the "left"). Due to this shift by -Δτ, P̂ _out and V _pilot (t) are time-aligned to each other, resulting in Δφ = 0. The signals after shifting P̂ _out by -Δτ are in 9 shown.

In detail, 9 The diagram shows a temporal relationship between the pilot signal V _pilot and the estimated output power P̂ _out after compensation for the propagation delay. The ordinates represent the amplitudes of the pilot signal V _pilot and the estimated output power P _out , while the abscissas represent time t.

Thus, the running time Δτ is compensated and φ _d = φ ₀ applies. If φ ₀ = 0 is chosen, then φ _d = 0 results. In this case, equation (11) simplifies to $$\mathcal{F}\left\{P_{out,T_2}\left(t\right)\right\}\left(\left(\omega >0|{\phi }_d=0\right)\right)=S_{1f_d,T_2}\cdot i\cdot \sqrt{\frac{\pi }{2}}\cdot \delta \left(\omega -{\omega }_d\right)-S_{2f_d,T_2}\cdot \sqrt{\frac{\pi }{2}}\cdot \delta \left(\omega -2{\omega }_d\right).$$

The spectral component at ω = ω _d is thus purely imaginary, the component at ω = 2 _{ω d} purely real. Taking this fact into account, ΔV is thus obtained after transformations $$\Delta V=-\frac{imag\left({\mathcal{F}}_{1f2f}^{+}\left({\omega }_d\right)\right)}{imag\left({\mathcal{F}}_{1f2f}^{+}\left(2{\omega }_d\right)\right)}\cdot \frac{V_d}{4}$$ and for ΔV̂ accordingly $$\Delta \widehat{V}=\frac{imag\left(DFT\left\{{\widehat{P}}_{out}\right\}\left[z_1{\omega }_d\right]\right)}{real\left(DFT\left\{{\widehat{P}}_{out}\right\}\left[z_2{\omega }_d\right]\right)}\cdot \frac{V_d}{4}.$$

It turns out that this approach effectively reduces noise interference in the signal P̂ _out , since these typically have both a real and an imaginary spectral component. In the numerator and denominator of Eq. (20), corresponding noise interference at ω _d and 2ω _d is therefore halved again by the separate formation of the real and imaginary parts.

Practical aspects

In practically realizable systems, the sequence P̂ _out [k] is of limited length K. In order to concentrate as much energy as possible in the elements belonging to the corresponding (pos. and neg.) frequency in the discrete Fourier transform of a purely sinusoidal signal, the sequence to be transformed (in this case P̂ _out [k]) should contain an integer multiple of signal periods.

In the case of the present invention, this means that the sequence P̂ _out [k] with K elements contains an integer multiple N of pilot signal periods of length T _d = 1/f _d , ie $$K\cdot T_s=N\cdot T_d\text{ mit }K,N\in \mathbb{N},$$ where the sampling interval according to T _s = 1/f _s results from the sampling frequency of P̂ _out (t). Fulfillment of the condition according to Eq. (20) can be achieved in particular by a suitable choice of the pilot signal frequency relative to the sampling rate and a suitable length of the sequence P̂ _out [k].

5. Summary / further examples

In exemplary embodiments, the estimated value ΔV̂ of the differential voltage to be estimated ΔV = V _min - V̂ _min is not obtained from the pure amplitude ratio of the fundamental wave and the first harmonic of the estimated optical output power P _out , as was previously the case, but the effective phase angle φ _d of the dither pilot signal is also included in the calculation in exemplary embodiments. By taking the magnitude and phase into account in the derivation, a signed estimate of the differential voltage is possible. This achieves significantly better convergence during minimum tracking; memory-based control loops are obsolete, especially in the case where the estimated output power P̂ _out (t) is very noisy.

Estimating ΔV̂ by magnitude and sign requires knowledge of the phase angle φ _d of the dither pilot signal. In exemplary embodiments, the initial phase angle φ d can be controlled by knowing the unavoidable time delay Δτ > 0, i.e., the signal propagation time difference of P̂ _out ( _t ) relative to V _pilot (t), which is measured in advance in an initialization step, and by considering integer multiples of pilot signal periods.

For the special case where the phase angle of the dithered pilot signal is φ _d = 0 and where integer multiples of pilot signal periods are considered in the DFT window, the estimation of ΔV̂ according to Equation (20) can be implemented particularly easily in exemplary embodiments, since only two real-valued spectral components are required. By omitting the other components, which contain only noise components, the noise interference at ω _d and 2ω _d is halved again by the separate formation of the real and imaginary parts, respectively.

Although some aspects have been described in the context of a device, it should be understood that these aspects also represent a description of the corresponding method, so that a block or component of a device can also be understood as a corresponding method step or as a feature of a method step. Analogously, aspects described in the context of or as a method step also represent a description of a corresponding block, detail, or feature of a corresponding device. Some or all of the method steps may be performed by (or using) a hardware apparatus, such as a microprocessor, a programmable computer, or an electronic circuit. In some embodiments, some or more of the key method steps may be performed by such an apparatus.

Depending on specific implementation requirements, embodiments of the invention may be implemented in hardware or software. The implementation may be performed using a digital storage medium, such as a floppy disk, a DVD, a Blu-ray Disc, a CD, a ROM, a PROM, an EPROM, an EEPROM, or a FLASH memory, a hard disk, or other magnetic or optical storage device storing electronically readable control signals that can interact or cooperate with a programmable computer system to perform the respective method. Therefore, the digital storage medium may be computer-readable.

Some embodiments according to the invention thus comprise a data carrier having electronically readable control signals capable of interacting with a programmable computer system such that one of the methods described herein is carried out.

In general, embodiments of the present invention may be implemented as a computer program product having a program code, wherein the program code is effective to perform one of the methods when the computer program product is run on a computer.

The program code can, for example, also be stored on a machine-readable medium.

Other embodiments include the computer program for performing one of the methods described herein, wherein the computer program is stored on a machine-readable carrier.

In other words, an embodiment of the method according to the invention is thus a computer program which has a program code for carrying out one of the methods described herein when the computer program runs on a computer.

A further embodiment of the method according to the invention is thus a data carrier (or a digital storage medium or a computer-readable medium) on which the computer program for performing one of the methods described herein is recorded. The data carrier, the digital storage medium, or the computer-readable medium is typically physical and/or non-perishable or non-transient.

A further embodiment of the method according to the invention is thus a data stream or a sequence of signals that represents the computer program for carrying out one of the methods described herein. The data stream or the sequence of signals can be configured, for example, to be transferred via a data communication connection, for example, via the Internet.

A further embodiment comprises a processing device, for example a computer or a programmable logic device, which is configured or adapted to carry out one of the methods described herein.

A further embodiment comprises a computer on which the computer program for performing one of the methods described herein is installed.

A further embodiment according to the invention comprises a device or system designed to transmit a computer program for performing at least one of the methods described herein to a recipient. The transmission can be electronic or optical, for example. The recipient can be, for example, a computer, a mobile device, a storage device, or a similar device. The device or system can, for example, comprise a file server for transmitting the computer program to the recipient.

In some embodiments, a programmable logic device (e.g., a field-programmable gate array, an FPGA) may be used to perform some or all of the functionalities of the methods described herein. In some embodiments, a field-programmable gate array may interact with a microprocessor to perform any of the methods described herein. In general, in some embodiments, the methods are performed by any hardware device. This may be general-purpose hardware such as a computer processor (CPU) or method-specific hardware such as an ASIC.

The devices described herein may be implemented, for example, using a hardware apparatus, or using a computer, or using a combination of a hardware apparatus and a computer.

The devices described herein, or any components of the devices described herein, may be implemented at least partially in hardware and/or in software (computer program).

The methods described herein may be implemented, for example, using a hardware apparatus, or using a computer, or using a combination of a hardware apparatus and a computer.

The methods described herein, or any components of the methods described herein, may be implemented at least partially by hardware and/or by software.

The above-described embodiments are merely illustrative of the principles of the present invention. It is understood that modifications and variations of the arrangements and details described herein will be apparent to others skilled in the art. Therefore, it is intended that the invention be limited only by the scope of the following claims and not by the specific details presented in the description and explanation of the embodiments herein.

List of abbreviations

EOM

electro-optical modulator

DFT

Discrete Fourier transform

MZM

Mach-Zehnder modulator

SNR

signal-to-noise ratio

Designations

ΔV = Vmin - V̂min:

differential voltage to be estimated

Δτ:

temporal runtime delay

fd:

Frequency of a sinusoidal pilot or dither signal

fib:

Imbalance factor for modeling a non-ideal MZM

Fourier transformed, discrete Fourier transformed

S*,T2, φ*,T2:

real-valued scaling factors of the various spectral components with associated phases

T = Pout/Pin:

Transfer function from control voltage to optical output power (normalized to optical input power)

Td = 1/fd:

Duration of a pilot signal period

inherent phase difference between the two modulator branches, which characterizes the position of the first maximum of the transfer function without the application of an electric field

Vπ:

constant value of a half-wave voltage to be applied to the RF electrode to bring the optical power from the maximum value to the minimum value (or vice versa)

Vdrift:

drift shift caused by refractive index changes

Vdither, Vpilot:

Pilot tone signal

Vc = VRF + VDC:

electrical control voltage applied to the electrodes

VDC:

DC bias voltage applied to bias electrode

VRF:

high-frequency modulation signal applied to the RF electrode

Vmin = V0 - Vπ:

Bias voltage leading to the minimum achievable optical output power P _out,min

V̂min, ΔV̂, ΔV̂c, P̂out:

measured values to be estimated

Vd:

Control voltage of the pilot or dither signal

ωd = 2πfd :

Angular frequency of the pilot or dither signal

φd = φ0 + Δφ:

Phase angle

φ0:

Initial phase

Δφ:

runtime-related phase difference

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QUOTES CONTAINED IN THE DESCRIPTION

This list of documents submitted by the applicant was generated automatically and is included solely for the convenience of the reader. This list is not part of the German patent or utility model application. The DPMA assumes no liability for any errors or omissions.

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Claims (19)

Device for generating a bias voltage V _DC for an electro-optical modulator, wherein the device is configured to obtain an output power signal P̂ _out that is dependent on an optical output power of the electro-optical modulator, wherein the device is designed to generate the bias voltage V _DC as a function of an amplitude ratio between the fundamental wave and the first harmonic of the output power signal P̂ _out and as a function of a phase angle of a pilot signal applied to the electro-optical modulator. Apparatus according to the preceding claim, wherein the apparatus is configured to generate the pilot signal V _pilot for the electro-optical modulator. Device according to one of the preceding claims, wherein the device is configured to apply the bias voltage V _DC and/or the pilot signal V _pilot to at least one control input of the electro-optical modulator. Apparatus according to any one of the preceding claims, wherein the apparatus is configured to superimpose or combine the bias voltage V _DC with the pilot signal V _pilot to obtain a control voltage V _C for the electro-optical modulator. Device according to Claim 4 , wherein the device is configured to apply control voltage V _C to a control input of the electro-optical modulator. Device according to one of the Claims 1 until 3 , wherein the device is configured to apply the bias voltage V _DC to a first control input of the electro-optical modulator and to apply the pilot signal V _pilot to a second control input of the electro-optical modulator. Device according to one of the preceding claims, wherein the output power signal P̂ _out describes an estimated optical output power of the electro-optical modulator. Device according to one of the preceding claims, wherein the pilot signal V _pilot is a time-limited, sinusoidal signal. Device according to one of the preceding claims, wherein the electro-optical modulator is a Mach-Zehnder modulator. Device according to one of the preceding claims, wherein the device comprises a photodetector configured to detect at least a portion of an optical output power P̂ _out of the electro-optical modulator to obtain the output power signal. Device according to one of the preceding claims, wherein the device is configured to estimate a difference voltage ΔV as a function of the amplitude ratio between the fundamental wave and the first harmonic of the output power signal P̂ _out and as a function of the phase angle of the pilot signal applied to the electro-optical modulator, wherein the difference voltage ΔV describes a difference between the current bias voltage V _DC = V̂ _min and a target bias voltage V _min at which the optical output power P̂ _out of the electro-optical modulator has a minimum value. Device according to Claim 11 , wherein the device is configured to adjust the bias voltage V _DC as a function of the estimated difference voltage ΔV towards the desired bias voltage V _min . Device according to Claim 11 or 12 , wherein the device is configured to determine the differential voltage ΔV based on the following equation: $$\Delta V=\frac{{\mathcal{F}}_{1f2f}^{+}\left({\omega }_d\right)}{{\mathcal{F}}_{1f2f}^{+}\left(2{\omega }_d\right)}\cdot \frac{V_d}{4}\cdot e^{-i\left({\phi }_d-\frac{\pi }{2}\right)}$$ where ${\mathcal{F}}_{1f2f}^{+}\left({\omega }_d\right)$ is a Fourier transform of the fundamental wave of the output power signal, where ${\mathcal{F}}_{1f2f}^{+}\left(2{\omega }_d\right)$ is a Fourier transform of the first harmonic of the output power signal, where V _d is an amplitude of the pilot signal, and where φ _d is the phase angle of the pilot signal. Device according to Claim 11 or 12 , wherein the output power signal P̂ _out is discretely sampled, the device being configured to estimate the differential voltage ΔV based on the following equation: $$\Delta {\widehat{V}}_c=-\frac{DFT\left\{{\widehat{P}}_{out}\right\}\left[z_{1{\omega }_d}\right]}{DFT\left\{{\widehat{P}}_{out}\right\}\left[z_{2{\omega }_d}\right]}\cdot \frac{V_d}{4}\cdot e^{-i\left({\phi }_d-\frac{\pi }{2}\right)}$$ where ΔV _c is an estimate of the differential voltage ΔV, where DFT{P̂ _out }[z _{1ω d} ] is a discrete Fourier transform of the fundamental wave of the discretely sampled output power signal, where DFT{P̂ _out }[z _{1ω d} ] is a discrete Fourier transform of the first harmonic of the discretely sampled output power signal, where V _d is an amplitude of the pilot signal, and where φ _d is the phase angle of the pilot signal. Apparatus according to one of the preceding claims, wherein the apparatus is configured to estimate the phase angle in dependence on a signal propagation time difference between the output power signal and the pilot signal. Apparatus according to any one of the preceding claims, wherein the pilot signal V _pilot is sinusoidal, wherein the output power signal is a sampled output power signal P̂ _out [k] having K samples, where K is an integer multiple N of periods of the pilot signal V _pilot . Apparatus according to one of the preceding claims, wherein the apparatus is configured to shift the output power signal or a sampled version of the output power signal by a propagation delay Δτ of the pilot signal in order to compensate for a propagation-related phase difference Δφ of the phase angle φ _d . Device according to the preceding claim, wherein the device is configured to determine the differential voltage ΔV based on the following equation: $$\Delta \widehat{V}=-\frac{imag\left(DFT\left\{{\widehat{P}}_{out}\right\}\left[z_{1{\omega }_d}\right]\right)}{real\left(DFT\left\{{\widehat{P}}_{out}\right\}\left[z_{2{\omega }_d}\right]\right)}\cdot \frac{V_d}{4}$$ where ΔV is an estimate of the differential voltage ΔV, where imagDFT{P̂ _out }[z _{1ω d} ] is an imaginary part of a discrete Fourier transform of the fundamental wave z _1ωd of the discretely sampled output power signal P̂ _out , where imagDFT{P̂ _out }[z _{2ω d} ] a real part of a discrete Fourier transform of the first harmonic z _{2ω d} of the discretely sampled output power signal P̂ _out , where V _d is an amplitude of the pilot signal. An optical arrangement comprising: a light source, and a device according to any one of the preceding claims.