Device and method for generating a bias voltage for an electro-optic modulator

Abstract

Exemplary embodiments provide a device [e.g. a circuit, such as a control circuit] for generating a [e.g. time-variable] bias voltage VDC for an electro-optic modulator, wherein the device is configured to obtain an output power signal (I) dependent on an optical output power of the electro-optic modulator, wherein the device is designed to generate the bias voltage VDC on the basis of an amplitude ratio between the fundamental and the first harmonic of the output power signal (I) and on the basis of a phase angle of a pilot signal applied to the electro-optic modulator.

Description

[0001] 2023P66416 1Device and method for generating a bias voltage for an electro-optical modulator Description Embodiments of the present invention relate to a device and a method for generating a bias voltage for an electro-optical modulator. Some embodiments relate to a device and a method for controlling the bias voltage for the positionally accurate 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 the quantum state of the 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 in terms of frequency accuracy, spectral bandwidth, output power stability, etc. 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. FH250205PCT-2025037744.DOCX. 2023P66416 2The 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 object is achieved 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 VDC for an electro-optical modulator, wherein the device is configured to generate an output power signal ^ dependent on an optical output power of the electro-optical modulator. ^ ^^^ to obtain, wherein the device is designed to determine the bias voltage VDC as a function of an amplitude ratio between the fundamental wave and the first harmonic of the output power signal ^ ^ ^^^and depending on 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 feedback (e.g., a feedback bias control loop) and with the aid of a pilot signal (e.g., a pilot tone), and thus controls the drifting modulator bias point. In embodiments, the device is configured to generate the pilot signal Vpilot for the electro-optical modulator. In embodiments, the device is configured to apply the bias voltage VDC and / or the pilot signal Vpilot 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 VDC and pilot signal Vpilot to a control input of the electro-optical modulator. Alternatively, the device can also be configured to apply the bias voltage V. DC at a first control input and the pilot signal V pilot to a second control input of the electro-optical modulator. FH250205PCT-2025037744.DOCX 2023P66416 3 In embodiments, the device is configured to modulate the bias voltage VDC with the pilot signal V pilot to superimpose or combine to create a control voltage V C for the electro-optical modulator. In embodiments, the device is configured to 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 DCto a first control input of the electro-optical modulator and the pilot signal V pilot to a second control input of the electro-optical modulator. In embodiments, the output power signal ^ ^ ^^^ an estimated optical output power of the electro-optical modulator. In embodiments, the pilot signal Vpilot 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 Pout of the electro-optical modulator to obtain the output power signal. In embodiments, the device is configured to, depending on the amplitude ratio between the fundamental wave and the first harmonic of the output power signal ^ ^ ^^^and to estimate a differential voltage ∆V depending on the phase angle of the pilot signal applied to the electro-optical modulator, wherein the differential voltage ∆V is a difference between the current bias voltage ^ ^^ = ^ ^ ^^^ and a target bias voltage Vmin at which the optical output power Pout of the electro-optical modulator has a minimum value. In embodiments, the device is configured to adjust [e.g., correct] the bias voltage VDC as a function of the estimated differential voltage ∆V toward the target bias voltage Vmin. In embodiments, the device is configured to determine the differential voltage ∆V based on the following equation: FH250205PCT-2025037744.DOCX 2023P66416 4 the fundamental wave of the output power signal,where ^ ^ ^^^^ (2^^) is a Fourier transform of the first harmonic of the output power signal, where Vd is an amplitude of the pilot signal, and where ^ ^ is the phase angle of the pilot signal. In embodiments, the output power signal ^ ^ ^^^ discretely sampled, the device being configured to estimate the differential voltage ∆V based on the following equation: where ∆^ ^ ^ is an estimate of the differential voltage ∆V, where ^^^^^^^^^^^^^^^^ is a discrete Fourier transform of the fundamental of the discretely sampled output power signal, where ^^^^^^^^^^^^^^^^ is a discrete Fourier transform of the first harmonic of the discretely sampled output power signal, where Vd is an amplitude of the pilot signal, and where ^ ^is the phase angle of the pilot signal. In embodiments, the device is configured to estimate the phase angle as a function of a signal propagation time difference between the output power signal and the pilot signal. In embodiments, the pilot signal V pilot sinusoidal, wherein the output power signal is a sampled output power signal ^^^^^[^] 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 sample the output power signal or a sampled version of the output power signal by a propagation delay Δ^ of the pilot signal [e.g., propagation delay of the pilot signal between a control input FH250205PCT-2025037744.DOCX 2023P66416 5of the electro-optical modulator to the pilot signal 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 obtain a runtime-dependent phase difference ∆φ of the phase angle ^ ^ to compensate. In embodiments, the device is configured to determine the differential voltage ∆V based on the following equation: where ∆^^ is an estimate of the differential voltage ∆V, where ^^^^^^^^^^^^^^^^^^^^ is an imaginary part of a discrete Fourier transform of the fundamental wave ^ ^^^ of the discretely sampled output power signal ^^ ^^^ is, where ^^^^^^^^^^^^^^^^^^^ ^ is a real part of a discrete Fourier transform of the first harmonic ^ ^^^ of the discretely sampled output power signal ^^ ^^^where Vd is an amplitude of the pilot signal. Further embodiments provide an optical arrangement with an electro-optical modulator and a device for generating a [e.g. time-variable] bias voltage VDC for the electro-optical modulator. Embodiments of the present invention are described in more detail with reference to the accompanying figures. They show: Fig. 1 a schematic block diagram of a Mach-Zehnder modulator, Fig. 2 a diagram showing a schematic view of a transfer function of an ideal Mach-Zehnder modulator plotted against the control voltage, Fig. 3 a diagram showing a schematic view of a drift-induced shift in a transfer function of a non-ideal (inequality factor ^^^ ≠ 1 / 2) Mach-Zehnder modulator plotted against the control voltage, Fig.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, FH250205PCT-2025037744.DOCX. 2023P66416 6Fig. 5 is 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, Fig. 6 is a schematic view of an optical arrangement with a light source, an electro-optical modulator, an optical beam splitter, a measuring device for optical power and a scanning device, Fig. 7 is a diagram showing a schematic view of the relationship between the desired bias voltage, the known bias voltage and the estimated differential voltage in the transfer function, Fig. 8 is a diagram showing a temporal relationship between the pilot signal and the estimated output power, and Fig. 9 is a diagram showing a temporal relationship between the pilot signal and the estimated output power after compensation for the propagation delay.In the following description of the embodiments of the present invention, like or equivalent elements in the figures are provided with the same reference numerals so that their descriptions are interchangeable. In the following description of the embodiments, several details are set forth to enable a more complete explanation of the 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 different embodiments described below may be combined with one another unless expressly stated otherwise.Before explaining embodiments of the device for generating a bias voltage for an electro-optical modulator in detail with reference to Figs. 5 to 9, the underlying electro-optical modulator and the problem of drift shifting of its operating point of maximum attenuation will first be described in more detail. FH250205PCT-2025037744.DOCX. 2023P66416 7One of the most well-known electro-optical modulators, the Mach-Zehnder modulator (MZM), is an optical component for modulating the intensity of laser light. In this process, a phase shift between two optical paths is converted into an amplitude change through interference. Fig. 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 shown in Fig. 1, 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 output signal power ranges from zero (0) to the maximum input power, depending on the strength of the electric field. The two waves interfere structively 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^^(^) and the optical output power ^^^^, including the power determined by the inequality factor ^. ^^ The possible optically incorrect balance shown when dividing into the two arms is given by: The value ^^ = ^^ ^ ^ ^ indicates the initial phase shift that would occur without the application of an electric field. ^ ^ 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 ^ ^is the control voltage that generates an electric field between the two electrodes.^ ^^ refers to the optical input power, which is ideally constant in many practical applications. FH250205PCT-2025037744.DOCX 2023P66416 8 The transfer function ^ = ^^^^ / ^^^ resulting from Eq. (1) is shown in Fig. 2 for a given value of ^ ^plotted. In detail, Fig. 2 shows a schematic diagram of a transfer function T of an ideal MZM with ^^^ = 1 / 2 plotted against the control voltage ^^. The optical throughput is plotted along the ordinate and the electrical bias along the abscissa. The MZM can be operated as an approximately linear intensity modulator if the optical path difference is set such that ^^=±^^ / 2 (positions Quad±) and operation takes place in the nearly linear region of the half-power point at T=0.5. Alternatively, the optical path difference can be set such that ^ ^ a multiple of ^ ^In this case, as shown in Fig. 2, ^(^^ = ^^+2 ∙ (^ − 1) ∙ ^^) = 0 (Min) and ^(^^ =^^+2 ∙ ^ ∙ ^^) = 1 (Max), so that the modulator switches the light off and on ("on-off gating"). One of the most difficult technical problems when using EOMs is maintaining the correct control voltage ^ ^EOMs drift, on the one hand, 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. As shown in Fig. 3, this shifts the transfer function ^ = ^^^^ / ^^^ in the horizontal direction, and the modulation signal is placed at a different operating point, which significantly impairs the modulation quality. In detail, Fig. 3 shows a schematic diagram of a drift-induced shift in a transfer function T of an MZM (^^^ ≠ 1 / 2) plotted against the control voltage ^^. 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 ^^, meaning ^^ 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 ^. ^ at ^ ^^^^^ To compensate for this drift phenomenon and thus ensure the long-term stability of the transmission chain, two electrical voltages are applied to the MZM shown in Fig. 1: the high-frequency modulation voltage ^ ^^ , which contains the information about the desired switching or modulation behavior of the EOM, and to the same electrode or to a second bias electrode a bias voltage ^ ^^ , which, to maintain stable operating conditions, the drift shift ^ ^^^^^ compensated and thus controls the modulator bias point (operating point). The use of FH250205PCT-2025037744.DOCX 2023P66416 9 of the subscript “DC” in ^ ^^has been established in the technical literature, although it is not a constant (DC) voltage in the true sense of the word, but rather a voltage that is higher than ^ ^^relatively slowly varying voltage. 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 strongly depends 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 particularly used in multilevel modulation schemes such as QAM, (D)QPSK, or OFDM and when multiple MZMs are used in parallel [3,4]. However, pilot-tone-based techniques 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. Initially, 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 determines the direction of the drift shift ^. ^^^^^ The DC bias voltage ^ ^^ With this information, the nearest linear bias point is automatically assigned Quad± FH250205PCT-2025037744.DOCX 2023P66416 10 adjusted. If the bias range includes several bias points within its working range, the bias is also automatically reset to the smallest linear point with ^^=±^^ / 2 by a bias reset. While in [5] a rectangular pilot signal was used, in [6,7] sinusoidal waveforms^^^^^^^(^) = ^^ ∙ ^^^(^^^ + ^^) with ^^ = 2^^^ , ^^ > 0 (2) with low frequency ^ ^ and the lowest possible control voltage ^ ^ used. The frequency ^ ^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 Τ=0.5 is reached exactly when all even harmonics (with 2^ ^ , 4^ ^ , …) of the harmonic pilot signal disappear and only odd harmonics (with 1^ ^ , 3^ ^ , …) are present in the optical feedback path. This is due to the fact that the transfer function at the quad positions at Τ=0.5 exhibits a point-symmetric curve behavior and a Taylor polynomial series expansion of the trigonometric function from Eq. (1) would only show odd exponents in the function equation at this expansion point. The second harmonic at 2^ ^, which is to be regarded as a measure of the deflection from the half-power point, is extracted from the optical output, followed by synchronous demodulation, which generates an error signal representing the DC bias voltage ^ ^^ 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 ^ ^ , results in a significantly lower SNR compared to detecting the pilot signal. To achieve this SNR, an analog low-pass or band-pass filter is typically used, which extracts the second harmonic with appropriate noise reduction. The SNR can also be increased by using two phase-synchronous pilot frequencies ^ ^^ and ^ ^^can be improved, in which the difference component |^^^ − ^^^| of the two pilot tones is 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, distortion products of odd order (1^ ^ , 3^ ^ , ...) of the dither signal can be minimized. FH250205PCT-2025037744.DOCX 2023P66416 11The transfer function exhibits an axisymmetric curve behavior at these bias operating points, and a Taylor polynomial series expansion of the trigonometric function from Eq. (1) would only exhibit 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, the minimization of the signal amplitude at the fundamental frequency 1^ ^ or the harmonic at 2^ ^ A corresponding case distinction, as shown in

[0010] , is also possible. All techniques for bias stability control considered so far [5-10] are predominantly analog in nature, and the pilot signal is permanently superimposed on the time-varying electrical RF signal. The error signal is averaged by filtering, and the drift shift ^ ^^^^^is applied to the DC bias voltage using different methods ^ ^^ These methods can be table methods or control algorithms such as a PID controller according to

[0011] . 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 completely 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 ^ ^between the oscillations of the first and second fundamental frequency, corresponding The calculation of fundamental and harmonic components is performed in [14,16] using two bandpass filters, whereas in

[0015] 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 ^ ^^ from their ideal value, and then readjust accordingly. FH250205PCT-2025037744.DOCX 2023P66416 12Fig. 4 shows a graph 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, Fig. 4 shows the relationship of Equation (3) as a function of the unknown phase drift angle ^^^^^^ and a ^ = 0.1. Since ^^ ≪ ^^ is usually chosen, ^ < 1 applies, and the two bracketed expressions in Equation (3) always assume a positive value. As shown in Fig. 4, the minimum (the zero operating voltage point) ^ ^^^at ^^^^^^ = 180°. However, since the desired phase drift angle ^^^^^^ is unknown, a pure level calculation of the fundamental and first harmonic is carried out after appropriate transformation of Eq. (3), followed by a ratio calculation

[0014] . This unsigned ratio is compared with older results in a control system, and by comparing the ratio and the specified ratio, the bias control value is calculated

[0018] . Since no phase information ^ is therefore contained in Eq. (3) ^of the dither pilot tone from Eq. (2) is taken into account, 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. Fig. 7). If this additional information were available, then a (e.g., significantly) better convergence in the minimum tracking would result, among other things, even in the case that the feedback signal (see, for example, Fig. 6, ^^^^^ [^]) 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, the minimum tracking can be improved.For example, the convergence of the feedback (e.g., feedback bias control loop) can be improved by taking into account the phase information ^^ of the pilot signal (e.g., dither pilot tone from Eq. (2)).Fig.Figure 5 shows a schematic block diagram of an apparatus 100 for generating a bias voltage VDC for an electro-optical modulator (EOM) 102, according to an embodiment of the present invention. The apparatus 100 is configured to generate a bias voltage VDC dependent on an optical output power P. out of the electro-optical modulator (EOM) 102 FH250205PCT-2025037744.DOCX 2023P66416 13 dependent output power signal ^^^^^ and to determine the bias voltage VDC 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 ^ ^ ^^^ and depending on a phase angle of a pilot signal Vpilot applied to the electro-optical modulator (EOM) 102. In embodiments, when providing the bias voltage V DCfor the electro-optical modulator (EOM) 102 not only the amplitude ratio between the fundamental wave and the first harmonic of the output power signal ^ ^ ^^^ taken into account, but also the phase angle of the pilot signal V applied to the electro-optical modulator (EOM) 102 pilot . In embodiments, the pilot signal V pilotfor the electro-optical modulator (EOM) 102 are 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 Vpilot and / or the phase angle or information about the phase angle of the pilot signal Vpilot from the other pilot signal generating device. In embodiments, the device 100 can be configured to apply the bias voltage VDC and / or the pilot signal Vpilot to at least one control input of the electro-optical modulator. For example, the device 100 can be configured to apply a combination or superposition of bias voltage VDC and pilot signal Vpilot to a control input of the electro-optical modulator (EOM) 102.For example, the device 100 may be configured to superimpose or combine the bias voltage VDC with the pilot signal Vpilot to obtain a control voltage VC for the electro-optic modulator (EOM) 102. The control voltage Vc 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 VDC to a first control input and the pilot signal Vpilot to a second control input of the electro-optic modulator (EOM) 102. In embodiments, the output power signal ^ may be ^ ^^^ Describe an estimated optical output power of the electro-optical modulator (EOM) 102. FH250205PCT-2025037744.DOCX 2023P66416 14For example, the output power signal ^^^^^ can be determined / estimated via a beam splitter 104 and an optical power measuring device 106. For example, the beam splitter 104 can measure a portion of the optical output power P out of the electro-optical modulator (EOM) 102 to the optical power measuring device 106, wherein the optical power measuring device 106 then determines the output power Pout of the electro-optical modulator (EOM) 102 based on the portion of the optical output power P obtained via the beam splitter 104 outof the electro-optic modulator (EOM) 102. The beam splitter 104 and / or the optical power measuring device 106 can be implemented internally or externally to the device 100. The optical power measuring device 106 can be a photodetector, for example. As indicated in Fig. 5, the device 100 can be part of an optical arrangement 110 that includes the device 100 for generating a bias voltage VDC 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., a laser) that provides the optical input power Pin present at 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, in an electro-optical modulator (EOM) based on a defined pilot signal with given phase information ^^. This voltage is also referred to below as the "black level voltage." In the following embodiments, it is assumed that ^the transfer function of the EOM can be described at least roughly by Equation (1), ^the EOM is already at least approximately at an operating point of maximum attenuation of the optical transmission during the applied pilot signal, FH250205PCT-2025037744.DOCX. 2023P66416 15^ the optical output power of the EOM is estimated over time during the pilot signal applied 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 evaluating 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), provided that as little light power as possible is allowed to pass through in the "off" switching state. Since in exemplary embodiments, the phase information ^^ of the dither pilot tone is taken into account, instantaneous memoryless information on the relative position (right or left) of the deviation from the minimum operating point is available. This achieves significantly better convergence in minimum tracking, especially in the case of a highly noisy feedback signal. Embodiments allow for the estimation of the bias voltage ^ in an electro-optical modulator (EOM). ^^^, which is necessary to achieve the minimum possible optical power^^^^,^^^ = ^^^(^^^^) at the optical output. In embodiments, this estimation is carried out using a pilot signal in the form of a comparatively small sinusoidal voltage, which is applied to the control input of the EOM. The mathematical principles and the derivation of the method are described below. 1. Assumptions In the following, it is assumed that the power ^^^ at the optical EOM input is constant and an estimated value ^ ^ ^^^ for the light output ^ ^^^ The time profile of the optical EOM output is available using a suitable measuring device. The measuring device can, for example, be partially coupled out of the outgoing light FH250205PCT-2025037744.DOCX 2023P66416 16(beam splitter) into a photodetector and evaluation of its electrical output signal. The arrangement under consideration is shown schematically in Fig. 6. In detail, Fig. 6 shows a schematic view of an optical arrangement 110 with a light source (e.g., laser) 101, an EOM 102, an optical beam splitter 104, a measuring device 106 for optical power, and a scanning device 108. The light source is designed to generate a (constant) optical power P in to an input of the EOM 102. The EOM 102 is designed to modulate the optical power Pin applied to the input as a function of a control voltage Vc(t) = VDC + VRF, and to supply an optical output power P to an output of the EOM. out(t). The optical beam splitter 104 directs a portion of the optical output power Pout(t) of the EOM 102 to the optical power measuring device 106, wherein the optical power measuring device 106 then estimates the output power Pout(t) of the EOM 102 based on the portion of the optical output power Pout(t) of the EOM 102 obtained via the beam splitter 104 to provide an estimated output power signal ^^^^^(^) describing the optical output power Pout(t) of the EOM 102. The sampling device 108 is configured to sample the output power signal ^^^^^(^) to obtain a sampled output power signal ^^^^^[^]. In embodiments, it is assumed that a known bias voltage ^^^ = ^^^^^ is already present at the EOM at the time of the estimation, which approximately corresponds to the ideal bias voltage^ ^^^ which corresponds to the minimum achievable optical output power ^ ^^^,^^^(see also Fig.7). The EOM is thus at the time of estimation with regard to ^ ^^ at an operating point where the optical output power is at least in the order of magnitude of the minimum achievable. During the estimation, no other voltage is applied to the control input of the EOM other than 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 ^ ^^ = ^ ^ ^^^ the voltage ^ ^^^ which, according to Eq. (1), leads to the minimum possible optical output power of the EOM. The previous estimate ^ ^ ^^^ deviates from the exact value ^^^^ by the estimated differential voltage ∆^, according to the relationship ∆^ = ^^^^ − ^^^^^. (4)FH250205PCT-2025037744.DOCX 2023P66416 17The knowledge of the differential voltage ∆^ thus allows the calculation of the desired voltage ^^^^. In the following, the method for estimating ∆^ is derived, which directly provides an estimate of ^ ^^^The idealized transfer function according to Equation (1) is periodic and thus, according to the model, exhibits an infinite number of minima. To simplify the following representation, reference is made to the minimum at ^^^^ = ^^ − ^^, without loss of generality; see Fig. 7. In detail, Fig. 7 shows a diagram schematically illustrating the relationship between the desired bias voltage ^^^^ , the known bias voltage ^^^^^, and the estimated differential voltage ∆^ in the transfer function. The ordinate describes the optical throughput (transfer function from control voltage to optical output power) and the abscissa the control voltage Vc. If the EOM is at the operating point ^^^ = ^^^^^ as assumed above and superimposes the sinusoidal pilot signal voltage ^^^(^) = ^^^^^^(^) = −^^ ∙ ^^^(^^^ + ^^) with the angular frequency ^^ = 2^^^ and the initial phase ^^, the total control voltage is The initial phase ^ ^describes the phase position of the sinusoidal pilot signal relative to a definable time ^ = 0. It consists of a freely selectable phase ^ ^ and a time-of-flight phase difference ∆^, which will be discussed in more detail below:^^ = ^^ + ∆^ (6)If ^^(^) from equation (5) is inserted into equation (1), the optical power ^^^^ is: FH250205PCT-2025037744.DOCX 2023P66416 18 Near its minimum at ^, ie at its point of development ^ ≈ ^ and accordingly for ≪ 1 and ^^⁄ ^^ ≪ 1, the trigonometric function cos(^) can be approximated with high accuracy by the quadratic Taylor series according to cos(^) ≈ −1 +^ ^(^ − ^)^ (8). Applying this approximation to Eq. (7), the following expression ^ ^^^,^^ as an approximation for ^ ^^^ : By applying the binomial formula to the quadratic term ^∆^ + ^^ ∙ ^^^(^ )^^ ^^ + ^^ results after transformations: 3. Frequency components of the optical output power with phase information. If the spectral components of the output signal ^^^^,^^ from equations (9) and (10) are considered, three components result: ^^^^,^^(^ = 0), ^^^^,^^(^ = ^^) and ^^^^,^^(^ = 2^^). The real-valued scaling factors ^ ∗,^^ , which represent the sizes of the three spectral components, as well as the corresponding phases ^ ∗,^^ are:^ DC component at ^ = 0: ^ 1^ Fundamental wave at ^ = ^^: with the phase offset ^^^^,^^ = ^^^ 2^ harmonic at ^ = 2^^ : with the phase offset ^^ ^^^,^^ = 2^^ −^ FH250205PCT-2025037744.DOCX 2023P66416 19Both the DC and the 1^^-fundamental wave component (^^^,^^ , ^^^^,^^) depend on the desired and estimated variable ∆^, while the 2^^-harmonic component^^^^,^^ is independent of ∆^. Thus, there are in principle several ways to estimate the desired ∆^. In order to be as independent as possible from physically determined quantities such as ^ ^^ , ^ ^^ In order to determine ∆^, the 1^^ and 2^^ components are related to each other in exemplary embodiments, taking into account amplitude and phase information as well as the phase angle ^^. In comparison to conventional methods, the phase information is also taken into account in exemplary embodiments. 4. Estimation of the error deviation ∆V taking into account the fundamental and first harmonic: The Fourier transform is the 1^^- and 2^^-components of ^^^^,^^ 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^^^^ ≈ ^^^^,^^ is a purely real signal, negative frequencies are also not considered here. This is achieved by the Fourier operator ^ ^ ^ ^ ^^ expressed.For the Fourier transform of the 1^^- and 2^^-components of ^^^^,^^, for positive values ​​of ^, we obtain: If the values ​​of the above Fourier transform are = ^^ and ^ = 2^^ are related to each other, the result after appropriate reductions is: After transformation, ∆^ is: FH250205PCT-2025037744.DOCX 2023P66416 20 It turns out that the value of ∆^ with known phase position of the pilot signal ^^ and its amplitude ^ ^can be determined from the quotient of the Fourier transform of the output power^^^^ ≈ ^^^^,^^ at ^ = ^^ and ^ = 2^^. While conventional methods only allow the estimation of the magnitude of ∆^ according to Eq. (4), in contrast to exemplary embodiments, they contain no information about the sign. Obtaining the estimated values ​​of ∆^ with a time-discrete representation of the output power ^^^^The above Eq. (13) is based on the evaluation of the continuous-time Fourier transform of the approximated output power ^^^^,^^ ≈ ^^^^. In real systems, signals are usually in the form of time-discrete samples in multiples of the sampling interval ^ ^ based on a measurement. This results in both a discretization of the time axis and an additive perturbation ^(^). The time-discrete sequence ^ ^ ^^^ represents approximately the signal ^^^^,^^ at the times ^ = ^ ∙ ^^ plus a term ^ [ ^] , which takes into account an additive disturbance at the same time points. For the spectral representation, the Fourier transform of a continuous signal ^^^^^^,^^^ used above is replaced by the discrete Fourier transform ^^^^^^^^^^ of the sequence ^^^^^[^]. The estimated value ∆^^^ of ∆^ is thus: where ^ ^^^ the element of the Discrete Fourier Transform (DFT) which is in the continuous frequency domain of the angular frequency ^ ^ corresponds and accordingly the index ^ ^^^ the angular frequency 2^ ^ Ideally, ∆^ ^ ^ a purely real value, since it is the estimate of a real-valued physical quantity (voltage). Due to the above-mentioned additive noise ^[^] in ^^^^^, parasitic imaginary components also arise when determining ∆^^^ via DFT. A mapping of the generally complex estimate ∆^ ^^ to a real value while simultaneously reducing the noise components can be achieved by calculating the real part of ∆^ ^ ^ is formed, ie FH250205PCT-2025037744.DOCX 2023P66416 21 As defined in Eq. (4), the desired value ^^^^ can be estimated with the estimated value ∆^^ starting from a known operating point ^^^ = ^^^^^, ie ^^^^ ≈ ^^^^^ + ∆^^ .Determination of ^ ^ As can be seen from Equation (15), the estimation of ∆^^ by magnitude and sign requires knowledge of the phase angle ^^ = ^^ + ∆^, see Equation (6). This describes the effective phase shift of the sinusoidal pilot voltage ^^^^^^(^) relative to time ^ = 0. This temporal zero point can, in principle, be chosen arbitrarily, but then applies jointly to the pilot signal ^ ^^^^^ and for the estimated output power ^ ^ ^^^both in their continuous-time representation ^^^^^(^) and in the discrete-time representation as a sequence^ ^ ^^^[^]. This is illustrated in Fig. 8 above as an example for a sinusoidal pilot signal with ^^ = 0, which was chosen to be identical to zero for ^ < 0 for better illustration and without loss of generality. The time ^ = 0 was chosen to refer to the start of the pilot signal. In detail, Fig. 8 shows a diagram showing a temporal relationship between the pilot signal Vpilot and the estimated output power ^^^^^. The ordinates describe the amplitudes of the pilot signal Vpilot and the estimated output power ^^^^^, while the abscissas describe the respective times t and t'. In real causal systems, an unavoidable time delay ∆^ > 0 of ^^^^^(^) relative to ^^^^^^(^) always results, among other things, due to signal propagation times, e.g., in measuring devices, amplifiers, etc., see Fig. 8 below. The output signal assigned to the beginning of the pilot signal at time ^ = 0 therefore becomes effective in ^^^^^ at time ^ = ∆^ or^^ = 0.Since the estimate of ∆^^ is based on the evaluation of ^^^^^, its time axis is decisive. Based on the time axis ^. ^ from ^ ^ ^^^ the pilot signal^^^^^^(^) experiences a phase shift of ∆^ = ∆^ ∙ ^^ due to the (propagation time) delay ∆^. The effective phase ^^ with respect to the time axis of ^^^^^ is thus calculated as ^^ = ^^ + ∆^ ∙ ^^ , (17)FH250205PCT-2025037744.DOCX 2023P66416 22 where ^ ^can be freely selected. Compensation of the propagation delay ∆τ. Alternatively to the explicit determination of ^^, the propagation delay ∆^ can be taken into account by shifting the signal ^^^^^(^) or, after sampling, its discrete-time sequence ^^^^^[^] by ∆^ in the negative time direction (to the "left"). This shift by −∆^ causes ^^^^^ and ^^^^^^(^) to be temporally aligned with each other, resulting in ∆^ = 0. The signals after temporal shift of ^^^^^ by −∆^ are shown in Fig. 9. In detail, Fig. 9 shows a diagram showing a temporal relationship between the pilot signal Vpilot and the estimated output power ^^^^^ after compensation for the propagation delay. The ordinates describe the amplitudes of the pilot signal Vpilot and the estimated output power ^ ^ ^^^ , while the abscissas describe the time t. Thus, the running time ∆^ is compensated and ^^ = ^^ applies. If ^^ = 0 is chosen, then ^^ = 0 results. For this case, equation (11) simplifies to The spectral component at ^ = ^^ is thus purely imaginary, the component at ^ = 2^^purely real. Taking this fact into account, ∆^ is thus, after transformations, It turns out that in this way noise interference in the signal ^ ^ ^^^ effectively reduced, since these usually have both a real and an imaginary spectral partFH250205PCT-2025037744.DOCX 2023P66416 23 In the numerator and denominator of Eq. (20), corresponding noise disturbances at ^^ and 2^^ are halved again by the separate real and imaginary part formation. Practical aspects In practically realizable systems, the sequence ^^^^^[^] is of limited length ^. In order to concentrate as much energy as possible in the elements belonging to the corresponding (positive and negative) frequency in the discrete Fourier transform of a purely sinusoidal signal, the sequence to be transformed (in this case ^ ^ ^^^[^]) contain an integer multiple of signal periods. In the case of the present invention, this means that the sequence ^^^^^[^] with ^ elements contains an integer multiple ^ of pilot signal periods of length ^^ = 1⁄ ^^, ie^ ∙ ^^ = ^ ∙ ^^ with ^, ^ ∈ ℕ, (21)where the sampling interval according to ^^ = 1⁄ ^^ results from the sampling frequency of ^^^^^(^). 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 as well as a suitable length of the sequence ^ ^ ^^^ [ ^ ] realize.5. Summary / further embodimentsIn embodiments, the estimated value ∆^^ of the differential voltage to be estimated ∆^ = ^^^^ − ^^^^^ is not obtained as previously from the pure amplitude ratio of the fundamental wave and the first harmonic of the estimated optical output power ^ ^ ^^^ , but also the effective phase angle ^ ^The dither pilot signal is included in the calculation in exemplary embodiments. By taking 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 ^^^^^(^) is very noisy. The estimation of ∆^^ by magnitude and sign requires knowledge of the phase angle^^ of the dither pilot signal. In exemplary embodiments, together with the knowledge of the unavoidable time delay ∆^ > 0, i.e., the signal propagation time difference of ^^^^^(^) relative to ^^^^^^(^), which is measured in advance in an initialization step, and the FH250205PCT-2025037744.DOCX 2023P66416 24 Considering integer multiples of pilot signal periods, the initial phase angle ^ ^control. For the special case where ^^ = 0 applies to the phase angle of the dither pilot signal and integer multiples of pilot signal periods are considered in the DFT window, the estimation of ∆^^ according to Eq. (20) can be implemented particularly easily in exemplary embodiments, since only two real-valued spectral components are required there. By omitting the other components, which only contain noise components, the noise interference at ^^ and 2^^ is halved again by the separate real and imaginary part formation. Although some aspects have been described in connection with a device, it is 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 connection with 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 particular implementation requirements, embodiments of the invention may be implemented in hardware or in software.The implementation can be carried out using a digital storage medium, for example 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 another magnetic or optical storage device, on which electronically readable control signals are stored that can interact or interact with a programmable computer system such that the respective method is carried out. Therefore, the digital storage medium can be computer-readable. Some embodiments according to the invention thus comprise a data carrier having electronically readable control signals that are capable of interacting with a programmable FH250205PCT-2025037744.DOCX. 2023P66416 25Computer system to cooperate in such a way that one of the methods described herein is carried out. In general, embodiments of the present invention can be implemented as a computer program product with a program code, wherein the program code is effective to carry out one of the methods when the computer program product runs on a computer. The program code can, for example, also be stored on a machine-readable medium. Other embodiments comprise the computer program for carrying out one of the methods described herein, wherein the computer program is stored on a machine-readable medium. In other words, one 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 carrying out one of the methods described herein is recorded. The data carrier, the digital storage medium or the computer-readable medium are typically tangible and / or non-perishable or non-transitory. A further embodiment of the method according to the invention is thus a data stream or a sequence of signals which represents or represent the computer program for carrying out one of the methods described herein. The data stream or the sequence of signals can, for example, be configured to be transferred via a data communication connection, for example via the Internet.A further embodiment includes a processing device, such as a computer or a programmable logic device, configured or adapted to perform one of the methods described herein. FH250205PCT-2025037744.DOCX. 2023P66416 26A further embodiment comprises a computer on which the computer program for carrying out one of the methods described herein is installed. A further embodiment according to the invention comprises a device or a system designed to transmit a computer program for carrying out at least one of the methods described herein to a recipient. The transmission can be electronic or optical, for example. The recipient can be a computer, a mobile device, a storage device, or a similar device, for example. The device or system can 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) can be used to perform some or all of the functionalities of the methods described herein.In some embodiments, a field-programmable gate array may cooperate 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 device, or using a computer, or using a combination of a hardware device 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 (a 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. FH250205PCT-2025037744.DOCX. 2023P66416 27The 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. FH250205PCT-2025037744.DOCX 2023P66416 28 List of Abbreviations: EOM electro-optical modulator; DFT discrete Fourier transform; MZM Mach-Zehnder modulator; SNR signal-to-noise ratio; FH250205PCT-2025037744.DOCX 2023P66416 29 Designations =^^^^ − ^^^^^: differential voltage to be estimated propagation delay^^: frequency of a sinusoidal pilot or dither signal^ ^^: Imbalance factor for modeling a non-ideal MZM^{∗}, ^^^{∗}: Fourier-transformed, discrete Fourier transform^∗,^^ , ^∗,^^ : real-valued scaling factors of the various spectral components with associated phases^ = ^^^^ / ^^^: Transfer function from control voltage to optical output power (normalized to optical input power)^^ = 1⁄ ^^ : Duration of a pilot signal period^ ^ ^ ^^: Inherent phase difference between the two modulator branches, which, without the application of an electric field, characterizes the position of the first maximum of the transfer function.^^: 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).^^^^^^: Drift shift caused by refractive index changes.^^^^^^^^, ^^^^^^: Pilot tone signal.^^ = ^^^ + ^^^: Electrical control voltage applied to the electrodes.FH250205PCT-2025037744.DOCX 2023P66416 30^^^: DC bias applied to bias electrode^^^: high-frequency modulation signal applied to RF electrode^^^^ = ^^ − ^^: bias voltage leading to the minimum achievable optical output power ^ ^^^,^^^ leads^^^^^^, ∆^^, ∆^^^ , ^^^^^: measured values ​​to be estimated^^: control voltage of the pilot or dither signal^^ = 2^^^ : angular frequency of the pilot or dither signal^^ = ^^ + ∆^: phase angle^^: initial phase∆^: phase difference due to propagation time FH250205PCT-2025037744.DOCX 2023P66416 31Literaturverzeichnis[1] S. Sun, M. He, M. Xu, S. Gao, Z. Chen, X. Zhang, Z. Ruan, X. Wu, L. Zhou, L. Liu, C.Lu, C. Guo, L. Liu, S. Yu, and X. Cai, “Bias-drift-free Mach-Zehnder modulators basedon a heterogeneous silicon and lithium niobate platform”, Photonics Research, Vol.8, No.12, Dez.2020.[2] H. Yu, D. Tu, X. Huang, Y. Yin, Z. Yu, H. Guan, L. Jiang, and Z. Li, “A Novel SiliconForward-Biased PIN Mach-Zehnder Modulator with Two Operating States”,Micromachines 2023, 14, 1608[3] K. Sekine, C. Hasegawa, N. Kikuchi and S. Sasaki, “A novel bias control technique forMZ modulator with monitoring power of backward light for advanced modulationformats”, Proc. OSA / OFC / NFOEC, März 2007, S.1-3, Papier OTuH5.[4] H. G. Choi, Y. Takushima, H. Y. Choi, J. H. Chang and Y. C. Chung, “Modulation-format-free bias control technique for MZ modulator based on differential phasemonitor”, Proc. OSA / OFC / NFOEC, 2011, S.1-3, Papier JWA33.[5] US 5,003,624[6] US 5,400,417[7] US 5,812,297[8] US 6,046,838[9] US 6,317,247

[0010] US 7,369,290

[0011] US 7,729,621FH250205PCT-2025037744.DOCX. 2023P66416 32

[0012] C. H. Cox, III, E. I. Ackerman, G. E. Betts and J. L. Prince, “Limits on the performanceof RF-over-fiber links and their impact on device design” IEEE Trans. Microw. Theory Tech., Vol. 54, No. 2, S.906–920, Feb.2006.

[0013] A. Karim and J. Devenport, “High dynamic range microwave photonic links for RF signaltransport and RF-IF conversion,” J. Lightw. Technol., Vol.26, No.15, S.2718–2724, Aug.2008

[0014] US 2010 / 0119239 A1

[0015] S. Shi, J. Yuan, Q. Huang, C. Shi, X. Luo, S. Lu, P. Yuan, H. Yu and Q. Yue, “Biascontroller of Mach-Zehnder modulator for electro-optic analog-to-digital converter”, MDPI micromachines, Nov.2019.

[0016] Y. Li, Y. Zhang and Y. Huang, “Any bias point control technique for Mach-Zehndermodulator”, IEEE Photonics Technology Letters, Vol.25, No.24, Dez.2013.

[0017] L. L. Wang and T. Kowalcyzk, “A versatile bias control technique for any-point lockingin lithium niobate Mach-Zehnder modulators,” J. Lightwave Technol., Vol.28, No.11,S.1703–1706, Jun.2010.

[0018] US 2009 / 0003840 A1FH250205PCT-2025037744.DOCX.

Claims

2023P66416 33Patent claims1. Device for generating a bias voltage VDC for an electro-optical modulator, wherein the device is configured to generate an output power signal dependent on an optical output power of the electro-optical modulator ^ ^ ^^^ to obtain, wherein the device is designed to determine the bias voltage VDC as a function of an amplitude ratio between the fundamental wave and the first harmonic of the output power signal ^ ^ ^^^and depending on a phase angle of a pilot signal applied to the electro-optical modulator.

2. Device according to the preceding claim, wherein the device is configured to generate the pilot signal Vpilot for the electro-optical modulator.

3. Device according to one of the preceding claims, wherein the device is configured to apply the bias voltage VDC and / or the pilot signal Vpilot to at least one control input of the electro-optical modulator.

4. Device according to one of the preceding claims, wherein the device is configured to superimpose or combine the bias voltage VDC with the pilot signal Vpilot to obtain a control voltage VC for the electro-optical modulator.

5. Device according to claim 4, wherein the device is configured to control voltage V C to a control input of the electro-optical modulator. FH250205PCT-2025037744.DOCX 2023P66416 346. Device according to one of claims 1 to 3, wherein the device is configured to bias the voltage V DC to a first control input of the electro-optical modulator and the pilot signal V pilot to a second control input of the electro-optical modulator.

7. Device according to one of the preceding claims, wherein the output power signal ^ ^ ^^^describes an estimated optical output power of the electro-optical modulator.

8. Device according to one of the preceding claims, wherein the pilot signal Vpilot is a time-limited, sinusoidal signal.

9. Device according to one of the preceding claims, wherein the electro-optical modulator is a Mach-Zehnder modulator.

10. 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 Pout of the electro-optical modulator in order to obtain the output power signal.

11. Device according to one of the preceding claims, wherein the device is configured to, depending on the amplitude ratio between the fundamental wave and the first harmonic of the output power signal ^ ^ ^^^and to estimate a differential voltage ∆V depending on the phase angle of the pilot signal applied to the electro-optical modulator, wherein the differential voltage ∆V describes a difference between the current bias voltage =^^^^^ and a target bias voltage Vmin at which the optical output power P out of the electro-optical modulator has a minimum value. FH250205PCT-2025037744.DOCX 2023P66416 35 12. The device of claim 11, wherein the device is configured to bias the voltage V DC depending on the estimated differential voltage ∆V towards the target bias voltage V min 13. The device according to claim 11 or 12, wherein the device is configured to determine the differential voltage ∆V based on the following equation: where ^ ^ ^^^^ (^^) is a Fourier transform of the fundamental wave of the output power signal, where ^ ^^^^^ (2^^) is a Fourier transform of the first harmonic of the output power signal, where Vd is an amplitude of the pilot signal, and where ^ ^ the phase angle of the pilot signal.

14. Apparatus according to claim 11 or 12, wherein the output power signal ^ ^ ^^^ discretely sampled, the device being configured to estimate the differential voltage ∆V based on the following equation: where ∆^ ^ ^ is an estimate of the differential voltage ∆V, where ^^^^^^^^^^^^^^^^ is a discrete Fourier transform of the fundamental of the discretely sampled output power signal, where ^^^^^^^^^^^^^^^^ is a discrete Fourier transform of the first harmonic of the discretely sampled output power signal, where Vd is an amplitude of the pilot signal, and FH250205PCT-2025037744.DOCX 2023P66416 36 where ^ ^the phase angle of the pilot signal.

15. Device according to one of the preceding claims, wherein the device is configured to estimate the phase angle as a function of a signal propagation time difference between the output power signal and the pilot signal.

16. Device according to one of the preceding claims, wherein the pilot signal Vpilot is sinusoidal, wherein the output power signal is a sampled output power signal ^ ^ ^^^ [^] with K samples, where K is an integer multiple N of periods of the pilot signal V pilot 17. Device according to one of the preceding claims, wherein the device is configured to shift the output power signal or a sampled version of the output power signal by a propagation delay Δ^ of the pilot signal in order to obtain a propagation-related phase difference Δφ of the phase angle ^ ^to compensate.

18. Device according to the preceding claim, wherein the device is configured to determine the differential voltage ∆V based on the following equation: where ∆^^ is an estimate of the differential voltage ∆V, where ^^^^^^^^^^^^^^^^^^^^ is an imaginary part of a discrete Fourier transform of the fundamental wave ^ ^^^ of the discretely sampled output power signal ^^ ^^^ is, where ^^^^^^^^^^^^^^^^^^^^ is a real part of a discrete Fourier transform of the first harmonic ^ ^^^ of the discretely sampled output power signal ^^ ^^^ is, FH250205PCT-2025037744.DOCX 2023P66416 37 where V d is an amplitude of the pilot signal.

19. Optical arrangement, comprising: a light source, and a device according to one of the preceding claims. FH250205PCT-2025037744.DOCX

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