Method for monitoring optical signals extracted from an optical resonator

Abstract

1. A method for monitoring an optical signal (15) extracted from an optical resonator (42), such as a laser resonator (22), comprising the steps of: receiving at least a portion of the optical signal (15) as an incident signal (1); propagating the incident signal (1) to a detector (2); and converting the incident signal (1) into an error signal (9) by application of a conversion arrangement comprising: deriving an error signal (9) based on a geometric beam shape, in particular a beam shape ellipticity, of the propagated incident signal (1) to the detector (2), e.g. proportional to the beam shape ellipticity, the error signal (9) having a minimum, in particular a minimum value (13), and a maximum, in particular a maximum value (14), the minimum and the maximum delimiting an interval (59) of values ​​of the error signal (9), the conversion arrangement configuring the interval (59) such that a zero crossing (58) of the error signal (9) occurs when the optical resonator (42) is in a target state.

Description

[Technical field]

[0001] Optical resonators are one of the main components in modern science and technology, enabling a huge number of applications. Optical resonators are used, for example, in lasers and in a wide range of measurement instruments, such as microscopes and spectroscopes. Applications range from, for example, the detection of gravitational waves, to laser welding and facial recognition technology. To ensure the desired behavior of optical resonators, e.g., lasers, it is necessary to monitor their properties. Also, to control lasers, it is necessary to measure and quantify the laser's power output and obtain an estimate of the deviation of the laser's actual state from a nominal or reference state. [Background technology]

[0002] For example, laser frequency stabilization, which can be achieved by stabilizing and / or controlling the optical cavity of a laser, is essential in atomic timekeeping, gravitational wave detection, relativity experiments, atomic interferometers, and quantum control of a variety of systems such as atoms, nanoparticles, mechanical vibrators, etc. As an example, modern atomic clocks require millihertz linewidth lasers to detect long-lived optical atomic transitions, and the precision level of laser frequency stabilization to a reference optical cavity plays a crucial role in reaching state-of-the-art performance levels.

[0003] Another aspect crucial to quantifying the deviation of a target state of an optical resonator from a reference state is locking a laser to the optical resonator, where the deviation from the reference state may be relative to the state of the laser to be locked to the optical resonator.

[0004] Towards the challenge of finding a suitable measure of the state of the optical cavity and a suitable estimator for quantifying the deviation from the reference state, numerous methods have been developed over the past decades. For example, estimators can suggest the detuning of a laser with respect to the optical cavity. These are exploited, for example, by side-of-fringe intensity methods, polarization-based methods, frequency modulation techniques similar to Pound-Drever-Hall (PDH), and finally spatial mode interferometry methods such as tilt locking.

[0005] "Laser-cavity locking at the 10^(-7) instability scale utilizing beam ellipticity" by Diorico et al. -7 In the paper "Archival preprint arXiv:2203.04550 (2022)" a method is presented to lock a laser to an optical cavity. A laser beam is directed into the optical cavity. The laser frequency is in resonance (or nearly in resonance) with the cavity. The reflected beam is focused onto a quadrant photodiode. The method relies on the interference between the fundamental TEM00 mode of the optical cavity and two spatiotemporal modes, resulting in an elliptical beam on the detector. Based on the ellipticity an error signal is derived, which is used as feedback to achieve laser-stable laser cavity locking. This method is called "squash locking". The paper states that the general-purpose squash locking method can be applied to stabilizing laser injection locking. However, the paper does not give any hints on how to apply the disclosed method in different contexts.

[0006] Ottaway, DJ et al., "Stabilization of injection-locked lasers using spatial mode interference," IEEE journal of quantum electronics 37.5 (2001) : 653-657, discloses a method to stabilize injection-locked lasers by using tilt locking to generate an error signal. The technique relies on the interference between the TEM00 mode and the first spatial mode of the laser output, which is measured with a quadrant photodiode. The quadrant photodiode is the sum of the top two quadrants and the bottom two quadrants. Thus, the quadrant photodiode effectively has two active areas, each consisting of the summed sensor area. At zero frequency difference (resonance case), the two outputs of the photodiode are equal in amplitude but opposite in phase. These outputs are subtracted, giving a zero output as a result. If the frequency difference is non-zero, the TEM 00 The field is TEM 01 It interferes more constructively with one of the beams and more destructively with the other. Thus, the signal depends on the shift or intensity of the beam on the detector as well as the alignment of the detector with respect to the beam. However, the error signal does not contain any information of the beam shape. For example, if the beam shape changes from circular to elliptical, the two outputs remain equal regardless of the rotation of the ellipse as long as the center of the ellipse is on the detector. Furthermore, the incident laser beam must be misaligned with respect to the optical axis of the resonator of the slave laser. In summary, the error signal is sensitive to misalignment of the optics, especially the detector. This signal is fed back to the piezoelectric transducer to control the incidence of the master signal to stabilize the injection lock.

[0007] Further details of the tilt locking technique are disclosed in "Frequency locking a laser to an optical cavity by use of spatial mode interference" by Shaddock, DA, MB Gray, and DE McClelland, Optics letters 24.21 (1999):1499-1501.

[0008] Known methods for monitoring and quantifying the state of optical resonators rely on complex mechanisms and are sensitive to drift, especially alignment drift. Summary of the Invention [Problem to be solved by the invention]

[0009] It is therefore an object of the present invention to provide a method for measuring and quantifying the optical signal extracted from an optical resonator to derive an estimator of the deviation of the actual state of the optical resonator relative to a nominal or reference state, which has reduced complexity, is purely passive, is particularly insensitive to alignment drifts, and / or has a more robust implementation. [Means for solving the problem]

[0010] This is accomplished by a method for monitoring an optical signal extracted from an optical resonator, such as a laser resonator, which comprises: receiving at least a portion of an optical signal as an incident signal; - converting the incident signal into an error signal by application of a conversion arrangement comprising propagating the incident signal to a detector and deriving an error signal based on the geometric beam shape, in particular the beam shape ellipticity, of the propagated incident signal to the detector, for example proportional to the beam shape ellipticity, where the error signal has a minimum, in particular a minimum, and a maximum, in particular a maximum, which minimum and maximum delimit an interval of the error signal, the conversion arrangement constituting an interval having a zero crossing of the error signal when the optical resonator is in the target state.

[0011] The optical resonator may be, for example, a linear resonator or a ring resonator. The optical resonator may be, for example, a two-mirror resonator or a three-mirror resonator. However, any number of mirrors and any shape of resonator may be used. The optical resonator may be a free-space resonator or a non-free-space resonator, such as a microchip-based optical resonator. For example, the optical resonator may be a whispering gallery resonator, where light circulates around (inside the material) a sphere or toroid. The optical resonator may be a fiber ring, into which light can be evanescently coupled.

[0012] The optical signal extracted from the optical resonator may be, for example, light leaking from the optical resonator. For example, the optical resonator may be a cavity of a laser, i.e., an optical resonator. In this case, the optical signal may be an output signal of the laser. The optical signal extracted from the optical resonator may be an optical signal reflected from the optical resonator. For example, when a laser is locked to an optical resonator, the laser may emit light onto or into the optical resonator. Part of the laser light may be reflected from the resonator. Optionally, the optical signal may include both a signal leaking from the optical resonator and a signal reflected from the optical resonator.

[0013] The output signal can be split, for example by a beam splitter, to receive as the incident signal a portion of the optical signal extracted from the optical resonator, or optionally the entire output signal can be used as the incident signal.

[0014] As a result, a transformation arrangement is applied to the incident signal. The transformation arrangement includes propagating the incident signal to a detector and deriving an error signal based on the geometric beam shape of the propagated incident signal. The transformation arrangement thus includes the manipulation of the incident beam, e.g., by optical components, the detection and quantification of the incident signal, and the derivation of an error signal. The error signal is an estimator of the deviation of the actual state of an optical resonator, e.g., an optical resonator of a laser, from a reference state. The transformation arrangement uses the incident signal as an input and outputs an error signal based on the incident signal.

[0015] To understand how the error signal is generated, one should recall that optical resonators, such as laser resonators, support Hermite-Gaussian (HG) spatial modes. These are called transverse electromagnetic modes (TEM) mn ), which a resonator, e.g., a laser resonator, can support. The subscripts m and n are positive integers representing orthogonal axes in the transverse direction of beam propagation. The sum m+n is called the mode order and can be odd or even. Starting with a perfectly aligned optical mode entering the optical resonator, odd modes are induced by moving the input beam out of alignment, either by tilting or shifting the input beam. On the other hand, even modes are induced by focusing / mode mismatch and are independent of the tilt or shift of the input beam.

[0016] Early spatial interference implementations such as tilt locking were used to tilt / misalign the input beam entering the optical cavity to achieve first-order (TEM) 01 and TEM 10 ) mode. We are interested in the fundamental HG mode, denoted as "00". 00 and the specific second-order HG mode, denoted by "+", the second-order TEM 11The two modes are similar modes, both of which are even modes. The even modes are induced by mode misalignment, making the presented method insensitive to misalignment drifts that generate first-order odd modes. For example, a slightly elliptical beam with horizontal / vertical orientation can be mathematically resolved into a major "00" component and a small "+" component. In such a (laser) beam, the phase difference between these two modes encodes information about the ellipticity. Near the "00" resonance, the two modes experience a differential phase shift because the modes of different orders are generally non-degenerate, so that only one of them resonates. This mechanism allows the output signal of the optical resonator to acquire different geometric beam shapes, in particular inverse ellipticities, on opposite sides of the optical resonator resonance. To exploit this effect, the geometric beam shapes, in particular the beam shape ellipticity or beam mode shape ellipticity, of the output signal of the optical resonator, for example a laser optical resonator, can be measured. In general, it is possible to monitor the optical signal extracted from the optical resonator using a sensor that is sensitive to the second spatial mode of the output signal of the optical resonator. In principle this works for any beam deflection and regardless of the actual beam deflection.

[0017] These modes generally have different resonant frequencies. As for the cavity modes, the spatial resolution of the diagonally oriented slightly elliptical beam is mostly “TEM” 00 " mode, "TEM 11 "The contribution of the second mode is small. In such a beam, the phase difference between these two modes encodes information about the departure from circularity. As a result, the shape of the output signal changes. By this mechanism, the output signal is made to acquire inverse ellipticity on either side of the resonance, so that a determinable error signal can be generated. To exploit this property, an error signal proportional to the beam shape ellipticity of the output signal or the ellipticity of the beam mode shape can be used.

[0018] In general, the derivation of the error signal based on the geometric beam shape of the incident beam means that the error signal is a function, at least one parameter of which is a parameter characterizing the geometric beam shape of the incident beam. In particular, the error signal can be directly proportional to the geometric beam ellipticity. The beam ellipticity can be defined according to ISO11145:2018. Alternatively, the eccentricity can be used.

[0019] Before the method is used, a start-up or initial calibration of the method is performed. A set point is set by starting with the optical resonator in a target state. The terms "set point", "reference point" and "target state" are used interchangeably for the purposes of this disclosure. For this set point, an error signal is derived that indicates deviation from the set point. As explained above, the relative change in beam shape relative to the beam shape at the set point can be used to quantify deviation from the set point.

[0020] The error signal has a minimum, in particular a minimum, and a maximum, in particular a maximum, which delimit an interval of the error signal value. The conversion configuration defines an interval with a zero crossing of the error signal when the optical resonator is in a target state. The error signal is zero at a set value when the optical resonator is in the target state. A zero error signal indicates that the actual state of the optical resonator does not deviate from the target state. The error signal has a sign and a magnitude, the sign optionally indicating the direction of deviation of the actual state from the target state, i.e. whether an operating parameter of the optical resonator, in particular a laser resonator, should be increased or decreased in order to reach the target state. The magnitude of the error signal can indicate the magnitude of the required change of at least one operating parameter in order to reach the set value.

[0021] For example, at least one operating parameter of the optical resonator is one or more of the following: the temperature of the optical resonator; the path length of the optical resonator. In general, the path length of the optical resonator is crucial for the mode distribution of the optical resonator. Temperature affects the path length, since the path length can increase or decrease depending on the temperature change due to thermal expansion. The temperature can be optionally changed by a Peltier element or a separate heating element, e.g., an electric heater element, and a separate cooling element, e.g., a fan.

[0022] The zero crossing of the error signal is within the interval between the minimum and maximum values ​​of the error signal. The zero crossing refers to the error signal having a value of zero. A non-zero error signal may indicate that the optical resonator is detuned from the target state. For example, the error signal may have a positive value when the optical path length in the optical resonator is greater than the optical path length in the reference state. Conversely, the error signal may have a negative value when the optical path length in the optical resonator is less than the optical path length in the reference state.

[0023] Optionally, the minimum is a minimum value. Optionally, the maximum is a maximum value. Optionally, there are no further maximum values ​​between the minimum and maximum values. The error signal includes an interval going from the minimum to the maximum value. The minimum and maximum values ​​thus delimit an interval of the error signal value.

[0024] The detector may be a photodiode, for example an avalanche photodiode. The detector may be or may consist of a phase plate or a mode converter. The detector may convert the incident signal into an electrical signal. However, the detector is not limited to electrical signals. For example, the detector may be configured to generate an optical signal based on the incident signal. In general, the invention is not limited to a particular form of physical signal that the detector generates based on the incident signal.

[0025] For example, the detector may be a photodiode array of at least three photodiodes, and deriving the error signal includes determining the geometric beam shape of the incident signal. Using the photodiode array, the geometric beam shape of the incident signal can be estimated. To estimate the geometric beam shape, the use of at least three photodiodes is necessary for differentiation of the first and second (or higher) order spatial modes. In principle, the more photodiodes used, the better the signal shape can be approximated. In general, determining the geometric beam shape of the incident signal means determining a parameter value of at least one parameter that characterizes the geometric beam shape. Preferably, the photodiodes are arranged in a rectangular structured grid. Minimizing the distance between each photodiode is beneficial to make the measurement most sensitive towards changes in the beam shape.

[0026] Optionally, the conversion configuration configures an interval that includes an error signal value that corresponds to a substantially circular image of the propagating incident signal to the detector. Optionally, the error signal is substantially zero for a circular incident signal and / or a circular image of the incident signal on the detector. A substantially circular incident signal may thus correspond to the optical resonator being in a reference state. An incident signal that obtains an inverse ellipticity for an opposite deviation from the reference state may be utilized to determine the error signal. The determined error signal may be considered as an interference between a resonating "00" mode and a second order "+" mode, both of which may escape from the optical resonator. Optionally, the zero crossing of the error signal may be an inverse symmetric point of the error signal. Since the sign and magnitude of the error signal may provide an indication of the magnitude and direction of the detuning of the optical resonator relative to the set point, a substantially zero error signal value for a substantially circular incident signal may result in an error signal that is easy to interpret. For example, the temperature of the optical resonator and / or the path length of the optical resonator may be detuned, resulting in a non-zero error signal. The sign and magnitude of the error signal can provide an indication of in which direction and by how much an operating parameter, such as the temperature of the optical cavity, should be adjusted to reach a set point.

[0027] Optionally, the detector may be a quadrant photodiode, and deriving the error signal includes determining a diagonal signal of the quadrant photodiode. This diagonal signal is composed of the difference of the sum of the signals from the diagonal sensor areas of the quadrant photodiode. In this way, the beam shape of the incident signal, and in particular the beam ellipticity, can be particularly easily evaluated. In particular, a quadrant photodiode is a photodiode with four divided sensor areas. The gap between the sensor areas can be optimized to be as small as possible. In particular, a quadrant photodiode (QPD) generates a current proportional to the input light from each of the four sensor areas. The signals from each sensor area are labeled A, B, C, D, and on the quadrant photodiode the sensor areas are arranged in the order A, D, B, C (clockwise). This allows the following calculations: SUM=A+B+C+D, LR=(A+C)-(B+D), UD=(A+D)-(C+B), DIAG=(A+B)-(C+D). The beam ellipticity can be measured using the DIAG calculation. A zero DIAG signal indicates no beam ellipticity and implies a circular beam shape at the incident signal. A non-zero DIAG signal implies an elliptical beam on the quadrant photodiode. An opposite sign of the DIAG signal indicates an opposite ellipticity. The quadrant photodiode may be an InGaAs-based or silicon-based detector. The signal can be converted to a voltage (e.g., using standard analog electronics with an op amp).

[0028] Propagation of the incident signal may include, for example, adjusting the angle of incidence of the incident signal on the detector. The angle of incidence affects the beam shape of the incident signal on the detector. For example, an incident signal with a circular cross section perpendicular to the axis of propagation of the incident signal will appear to have an elliptical beam shape on the detector when the angle of incidence is different from 90°. An incident signal with an elliptical cross section can be made to have a circular beam shape on the detector with the appropriate angle of incidence. If the detector is a QPD, the angle setting can make the DIAG signal zero. The angle of incidence setting is done once during calibration by adjusting the tilt angle of the detector with respect to the optical axis of the incident signal and is kept constant during operation.

[0029] Optionally, the propagation of the incident signal includes shaping the incident signal, in particular by means of a shaping lens, preferably by means of at least one lens, in particular by means of at least one cylindrical lens. Shaping the incident signal allows for calibrating an initial error signal at the target state of the optical resonator. The error signal can be initially set to zero. If the detector is a QPD, shaping the incident signal allows for zeroing the DIAG signal. This can be achieved, for example, if the detector is a QPD and the incident signal is shaped such that the image of the incident signal on the detector is circular. This results in a zero DIAG signal. The DIAG signal can optionally be used as the error signal without further manipulation. The position and alignment of the beam shaping optics for shaping the incident beam, in particular the at least one lens, preferably the at least one shaping lens, in particular the at least one cylindrical lens, is set once during calibration and kept constant during operation. Shaping of the incident signal can also be performed by a prism or any other optical beam shaping device.

[0030] Optionally, the propagation of the incident signal includes defining an axial distance between the detection lens and the detector such that a transmission peak of the incident signal coincides with a zero crossing of the error signal. The transmission peak refers to a resonance between the external optical signal and the optical resonator, e.g., an external optical seed signal and an injection-locked laser. Upon resonance, the reflected signal at the optical resonator is minimal. The axial distance between the detection lens and the detector represents their distance along the path of the incident signal. For the method, the positions of the detector and the detection lens are fixed during calibration and are kept constant during operation. Optionally, the axial distance may be adjusted in an adjustment section to determine the curve of the error signal and set the axial distance. The relative change in the geometric beam shape of the incident signal is most significant when the optical resonator is in resonance with the external signal, e.g., the optical seed signal. Thus, the method is most sensitive at the transmission peak of the incident signal, which is also the transmission peak of the external optical signal. Optionally, the axial distance may be adjusted in an adjustment section to determine the curve of the error signal and set the axial distance.

[0031] In another optional embodiment, propagating the incident signal includes determining an axial distance between the optical cavity and the detector such that the transmission peak of the incident signal coincides with a zero crossing of the error signal. The transmission peak corresponds to a reflection minimum of the reflected signal at the optical cavity, i.e., the SUM channel of the quadrant photodiode shows a dip (since the SUM channel of the quadrant photodiode specifically measures the reflection curve). The positions of the detector and the optical cavity are fixed during calibration and kept constant during operation. The distance between the laser and the detector is adjusted to the TEM. 00 This affects the Guoy phase between the mode and the second spatial mode. The image obtained on the detector can therefore be adjusted by adjusting the position of the detector with respect to the optical cavity. There is an axial distance where the error signal is purely dispersive and another axial distance where the error signal is purely absorptive or any signal shape between absorptive and dispersive. Preferably the error signal is dispersive.

[0032] Optionally, deriving the error signal includes providing an offset to the error signal to set the error signal to zero when the optical resonator is in the target state. The error signal is derived as follows: the incident signal on the detector generates a signal for each photodiode of the detector. These signals are used in response to an estimator of the shape of the incident signal. For example, if the detector is a QPD, a DIAG operation may be used to estimate the ellipticity of the signal. The error signal may be initialized to zero during calibration. For example, the DIAG operation applies an additional offset to initialize the error signal at the set point to zero, since the DIAG operation may result in a residual signal even if the incident signal is substantially circular.

[0033] In another embodiment, an imbalance detection operation can be used. For example, instead of the DIAG operation, the imbalance operation IDIAG can be used. IDIAG=ε 1 (A+B)-ε 2 (C+D), where ε 1 / ε 2 ≠ 1 and ε 1 and ε 2can be initially set to zero the error signal, allowing the background offset of the diagonal channels to be adjusted to provide a zero crossing in the error signal at a desired condition (e.g., at a set point).

[0034] In another embodiment, the propagation of the incident signal includes focusing the incident signal on the detector using a detection lens, preferably a convex spherical lens. Optionally, the incident signal is focused and / or diverged in at least a first plane and preferably in a second plane (non-parallel to the first plane), particularly at a point along the path of the incident signal different from the focus / divergence in the first plane. Focusing the incident signal refers to focusing the incident signal in at least one plane. The position and alignment of the detection lens, preferably a convex spherical lens, for focusing the incident signal on the detector is set once during calibration and kept constant during operation.

[0035] Optionally, adjusting the axial distance of the detection lens and the detector optionally includes adjusting the axial distance along the optical axis of the incident signal of the detection lens and the detector so that the in-plane and out-of-plane incident signal waists of the incident signal coincide with the beam waists of the resonant cavity modes of the optical resonator (in particular the average beam waist of the cavity modes, which is the average in-plane and out-of-plane beam waists) before adjusting the axial distance along the optical axis of the incident signal so that the transmission peaks of the optical signal reflected from the optical resonator coincide with the zero crossings of the error signal. Optionally, in addition or alternatively, the axial distance is adjusted so that the in-plane and out-of-plane beam waists of the incident signal reflected from the resonator (with each other) coincide. The baseline offset of the error signal is due to the axial displacement of the detector position from the position where the beam has a circular profile (ignoring point aberrations). The axial distance can be selected to eliminate this baseline offset. This beam waist coincidence can be used as a first approximation before the coincidence of the transmission peak and zero crossing, or it can be used by itself. If there is no astigmatism, these two will be the same result. In this case, zeroing the baseline will automatically bring you to the transmission plateau since there is no residual astigmatism in the mode. In the present method and apparatus for frequency locking a laser, it is preferred that the axial distance between the detector and the detection lens is such that the in-plane and out-of-plane beam waists of the incident signal reflected from the resonator coincide with the beam waist of the resonant cavity mode of the optical resonator, or are within ±5%, or ±2%, or ±1% of the beam waist of the resonant cavity mode of the optical resonator.

[0036] Optionally, the method further comprises centering the incident signal on the quadrant photodiode by translating the detection lens and / or the quadrant photodiode in a direction perpendicular to the optical axis of the incident signal such that left / right and / or top / bottom signals of the quadrant photodiode are minimized, where the left / right and top / bottom signals are each composed of the difference of sums of adjacent sensor areas of the quadrant photodiode, and the left / right and top / bottom signals use the sums of different halves of adjacent sensor areas. Optionally, this is performed iteratively. In this way, the incident signal can be perfectly aligned with the quadrant photodiode, and an optimal beam fit is achieved on the active area of ​​the quadrant photodiode to generate the most accurate error signal. Optionally, the translation is performed in two directions, which are perpendicular to each other and perpendicular to the optical axis of the incident signal.

[0037] Optionally, the steps of adjusting the axial distance between the detection lens and the quadrant photodiode along the optical axis of the incident signal and centering the incident signal on the quadrant photodiode are performed iteratively so that the transmission peak of the incident signal coincides with the zero crossing of the error signal. Every axial distance adjustment can minimize the left-right and top-bottom signals. By repeating this process, the zero crossing can be optimized to exactly coincide with the transmission plateau.

[0038] The present invention further relates to a method for locking a laser to an optical cavity with the method of the present invention, which comprises: - providing a laser beam from a laser; - shaping the laser beam so that it acquires ellipticity, - directing a laser beam into an optical cavity, the laser beam reflected from the optical cavity being an incident signal that is propagated to a detector; and - locking the laser to the optical cavity based on the error signal; Further included are:

[0039] The method can be used with a wide range of laser beam wavelengths. For example, the wavelengths may be in the ultraviolet range between 190 nm and 400 nm, in the visible range between 400 nm and 800 nm, in the near infrared range between 800 nm and 1800 nm, and / or in the infrared range between 1800 and at least 3500 nm. Optionally, the error signal is directly proportional to the beam ellipticity of the focused incident signal, in this case the reflected laser beam. Optionally, the error signal is substantially zero for a circular laser beam. Optionally, the laser is locked to the optical cavity based on the error signal by feeding the error signal back to the laser current. Locking is based on monitoring the change in ellipticity of the laser beam reflected from the optical cavity. Locking can be achieved by closed loop control.

[0040] Under-shaping the laser beam so that it acquires ellipticity is understood to mean subsequently manipulating the laser beam such that its cross-section differs from a circular cross-section at at least one point, in particular such that at least one point the laser beam has different beam waists in two lateral directions. Optionally, the laser beam focuses and / or diverges at least in a first plane and preferably in a second plane (non-parallel to the first plane), in particular at a point along the path of the laser beam that differs from the focus / divergence in the first plane.

[0041] Shaping the laser beam so that it acquires ellipticity can optionally be performed by a pair of cylindrical lenses with focusing axes oriented non-parallel, in particular perpendicular to each other, and / or by a combination of lenses, wedge prisms, and / or anamorphic prism pairs, whereby an elliptical or astigmatic beam is generated.

[0042] Optionally, the two cylindrical lenses have the same focal length, and further optionally, one is oriented horizontally and the other vertically. Optionally, the focal length is such that when the two lenses are arranged coaxially (as if they formed a single spherical lens), the laser beam is focused on the TEM 00 It is chosen to be perfectly mode-matched to the cavity mode. The distance between the pair of cylindrical lenses determines the amount of light in the "+" mode, which can amount to, for example, 10% of the total power (for example, 400 μW). One cylindrical lens can be oriented with its axis in-plane and the other out-of-plane. By adjusting the position of the cylindrical lenses, the input mode can be manipulated to induce up to second order modes. TEM 00 , TEM 02 , TEM 20 It is possible to align the latter two modes so that their transmission levels are equal, suppressing all other modes except for TEM. 02 and TEM 20 To equalize the balance of the TEM, the position of the cylindrical lens can be set / adjusted so that the in-plane and out-of-plane beam waists converge and diverge. The average resonator waist is placed exactly where the in-plane and out-of-plane beam waists coincide. When the beam diverges outside the cavity, its average waist can also coincide with the cavity mode average waist. 00 TEM for 02 and TEM 20 The balance of the amount of can be adjusted by adjusting the distance between the two cylindrical lenses while maintaining the center of gravity. Each member of the pair is translated symmetrically in opposite directions from this reference configuration, so that the TEM 02 Mode and TEM 20 The modes are of equal amplitude as observed in the cavity transmission spectrum. Physically, this action moves the beam waist in the horizontal and vertical planes around the cavity waist location. Alternatively, shaping the laser beam so that it acquires ellipticity can be achieved by a combination of lenses, wedge prisms, and / or anamorphic prism pairs, optionally resulting in elliptical or astigmatic beams.

[0043] Optionally, the laser beam is polarisation filtered and the incident signal is polarisation filtered prior to directing the laser beam into the optical resonator. Optionally, the optical resonator is birefringent.

[0044] A limitation of the method in terms of stability may still exist due to residual variations in the laser beam shape, for example due to changes in the second order mode components affecting the phase reference. To help alleviate this remaining limitation, a polarization pre-selection and post-selection procedure is available, which is explained using the weak value concept. For this purpose, optionally, the laser beam is polarization filtered before being directed to the optical resonator, and the laser beam reflected from the optical resonator is polarization filtered. Optionally, the optical resonator is birefringent. In this way, technical noise that limits the system performance can be coherently suppressed. The device preferably comprises a first polarizing beam splitter (or a first polarizer) for polarization filtering the laser beam before directing it to the beam shaper, and a second polarizing beam splitter (or a second polarizer) for polarization filtering after the laser beam emission from the optical resonator, in particular between the detection lens and the detector. Polarization filtering can be used as a method for frequency locking a laser and / or as a method for tuning the device for frequency locking a laser.

[0045] When the polarization degree of freedom is included, the cavity reflection coefficient is given by 1 The reflection operator that acts on > Replaced by JPEG2025502754000002.jpg76. State |ψ 2 When a post-selection is made to w is given by the weak value of the reflection operator. JPEG2025502754000003.jpg2277((Formula 1) where δ is the frequency detuning of the laser cavity, κ is the total cavity linewidth, and γ' = Aγ, where γ is a dimensionless parameter characterizing the round-trip losses in the cavity. The "amplification" parameter A depends on the particular a posteriori selection of the linear polarization state |ψ 1 > and |ψ 2 > is a real value. It is a function of the polarizer angle, but operationally it is the residual power ratio after post-selection when the light is off-resonance |<ψ 2 |ψ 1 >| 2 ≒1 / (2A+1) 2 can be related to the reflection from a conventional high-finesse resonator. The right-hand side of Eq. 1 has the same form (for a fixed value of κ) as the reflection from a conventional high-finesse resonator, except for the variable loss parameter γ'. Note that γ<1, γ=1, γ>1, and γ>2 are identified as the under-coupled, impedance-matched, over-coupled, and cavity-gain regions, respectively. The variability of γ' can be experimentally demonstrated by observing the post-selection dependence of the reflected light from the resonator. The error signal can be given by the imaginary part of the reflection coefficient in Eq. 1. Adjusting the post-selection to an impedance-matched configuration reduces noise due to frequency stabilization. In this configuration, the signal-producing portion is less attenuated, while the non-signal-producing beam components incident on the detector are strongly attenuated, effectively increasing the slope of the error signal and reducing its sensitivity to beam shape and intensity fluctuations.

[0046] Optionally, the method further comprises adjusting the optical resonator to reach the target state by minimizing the error signal.

[0047] For example, tuning of the optical cavity can be done by - adjusting the temperature of the optical resonator; - adjusting the optical path length of the optical resonator; Contains one or more of the following.

[0048] Tuning the optical resonator, for example by adjusting the temperature and / or the optical path length, can affect the optical properties of the optical resonator and, consequently, the geometric beam shape of the incident signal. The error signal is based on the geometric beam shape of the incident beam.

[0049] In an optional embodiment, the optical cavity is the optical cavity of a laser, in particular the optical cavity of a semiconductor laser. The laser is, for example, a semiconductor laser, and the pumping of the laser is performed by an electric current passing through the laser cavity. The semiconductor laser is, for example, a FP (Fabry-Perot) laser, a DFB (Distributed Feedback) laser, a DBR (Distributed Bragg Reflector) laser, or a VCSEL (Vertical Cavity Surface Emitting) laser. Other classes of lasers compatible with the method include ECDL (External Cavity Diode Laser) or ECL (External Cavity Laser). The pumping of the semiconductor laser can be performed by an electric current passing through an active material in the laser cavity.

[0050] In general, the path length of the laser cavity, i.e. the optical cavity of the laser, is crucial for the mode distribution of the laser. Optional pumping of the optical cavity is an operating parameter of the optical cavity. Pumping of the laser depends on the laser technology and can be done, for example, by optical pumping via an additional light source, e.g. an additional laser. Adjusting the pumping can include, for example, adjusting the current, changing the intensity and / or frequency of the optical pump light.

[0051] Optionally, tuning the optical cavity includes tuning a current through a semiconductor laser. Pumping of the semiconductor laser can be achieved by a current through the laser cavity. Tuning of the pumping can be achieved by tuning the current.

[0052] Particular embodiments of the present invention relate to a closed-loop control method for controlling an injection-locked laser and a closed-loop control system for controlling an injection-locked laser, which are outlined below.

[0053] The linewidth of a laser is mainly determined by unwanted effects that do not contribute to the stimulated emission of the laser. A laser can be made to copy the phase correlation of another injected "seed" laser beam, which has a much narrower linewidth. This ideally results in a "slave" laser with the same linewidth as the seed laser. Thus, injection-locked lasers are an important part of modern technology, since injection locking allows for narrow linewidth laser output. Furthermore, the injection seed frequency can be changed. The "slave" laser follows the frequency of the injected seed. Thus, a shift in the output frequency of the laser is possible.

[0054] To obtain stable injection locking and a stable laser output, it is beneficial to control the injection-locked laser. To control the laser, it is necessary to measure and quantify the laser output and derive an estimator of the deviation of the laser's real state relative to a nominal or reference state. To address the challenge of finding a suitable measurement and estimator, many methods have been developed over the decades. These include, for example, side-of-fringe intensity methods, polarization-based methods, frequency modulation techniques similar to Pound-Drever-Hall (PDH), and finally spatial mode interferometry methods such as tilt locking.

[0055] "Laser-cavity locking at the 10^(-7) instability scale utilizing beam elipticity" by Diorico, F. et al. -7 In the paper "Laser Cavity Locking on the Instability Scale of 100 kV" (archived preprint arXiv:2203.04550 (2022)), a method is presented for locking a laser to an optical cavity. A laser beam is directed into the optical cavity. The laser frequency is in resonance (or nearly in resonance) with the cavity. The reflected beam is focused by a lens onto a quadrant photodiode. This method is the fundamental TEM technique for optical cavities. 00It relies on the interference between the ellipticity and two spatiotemporal modes, resulting in an elliptical beam on the detector. An error signal is derived based on the ellipticity, which is used as feedback to achieve laser-stable laser cavity locking. This method is called "squash locking". The paper states that the general-purpose squash locking method is applicable to stabilizing laser injection locking. However, the paper does not provide any hints to generalize the disclosed method, nor does it show how to transfer the disclosed method to laser injection locking.

[0056] In an embodiment of the present invention (see embodiment 1), a method and apparatus for closed-loop control of an injection-locked laser is provided, which is robust to alignment drift and oscillations.

[0057] This is accomplished by a closed loop control for controlling the injection locked laser. - injecting a laser with an optical seed signal, - propagating at least a portion of the optical output signal from the laser as an incident signal onto a detector; - deriving an error signal based on a geometric beam shape of the incident signal on the detector; - controlling at least one operating parameter of the laser sensitive to the error signal; Includes.

[0058] Furthermore, this is achieved by a closed-loop control system for controlling the injection-locked laser (see embodiment 13). a laser that is injected with an optical seed signal during operation; - detector, an optical component for propagating at least a portion of the optical output signal from the laser as an incident signal to a detector; a controller for deriving an error signal based on a geometric beam shape of a signal incident on the detector; wherein the controller is configured to control at least one operating parameter sensitive to the error signal.

[0059] A laser can use its own emission to enhance itself. This method is commonly called self-injection or self-feedback locking. By extracting a small portion of the laser light and frequency filtering it before feeding it back into itself, the laser emission is further enhanced into a narrower frequency band. This results in a narrower laser linewidth. Closed-loop control methods and systems for controlling injection-locked lasers can be used for both injection-locked lasers using an external source seed signal and self-injection-locked lasers.

[0060] The method and system are started up or initially calibrated prior to using the method. The set points or reference points of the control loops (including, for example, PID controllers) are set by starting with a locked laser and optical seed signal. The term "locked" should be understood as meaning that the coupling of the optical seed signal to the laser is what is desired. For example, if the optical seed signal is coupled to a TEM 00 The laser mode may be perfectly mode-matched and / or the center frequency of the optical seed signal is resonant with the laser. In general, a set point defines the relationship of the optical seed signal to the laser. The laser should be controlled such that the system is stabilized at the set point and the relationship between the laser and the optical seed signal is as constant as possible. This is achieved by deriving an error signal indicative of deviation from the set point. As explained above, the relative change in beam shape to the beam shape at the set point can be used to quantify the deviation from the set point. In general, the present closed loop control method is not limited to any particular type of control transfer function. Thus, it is not necessary to define a particular set point for controlling at least one operating parameter of the laser. Instead, it may be sufficient to drive the laser back within a tolerance range of possible operating conditions when the error signal indicates that such range remains and in which direction.

[0061] In general, deriving an error signal based on the geometric beam shape of the incident beam means that the error signal is a function, at least one parameter of which is a parameter characterizing the geometric beam shape of the incident beam. In particular, the error signal can be directly proportional to the geometric beam ellipticity. The beam ellipticity can be defined according to ISO11145:2018. Alternatively, the eccentricity can be used. The error signal can optionally be zero at a set value. The error signal has a sign and a magnitude, the sign optionally indicating the direction of control, i.e. whether the operating parameter of the laser should be increased or decreased. The magnitude of the error signal can indicate the magnitude of the required change of the at least one operating parameter to reach the set value.

[0062] Optionally, the zero crossing of the error signal is in an interval between the minimum and maximum values ​​of the error signal, which includes error signal values ​​corresponding to the determined beam ellipticity of a substantially circular beam, and an incident signal that obtains the inverse ellipticity on the other side of the resonance is available for determining the error signal, which can be used to lock and stabilize the laser beam. The determined error signal can be thought of as the interference between the resonant "00" mode that escapes from the laser and a second order "+" mode that reflects directly from the laser and forms the phase reference. This detection scheme is insensitive to drifts and variations in alignment, since small misalignments will simply generate the first order mode in the modal decomposition. The zero crossing refers to the error signal having a value of zero.

[0063] Optionally, the error signal includes a minimum, optionally a minimum. Optionally, the error signal includes a maximum, optionally a maximum. Optionally, there are no further extrema between the minimum and maximum. Optionally, the error signal includes an interval going from the minimum to the maximum, with a value in the interval between the minimum and maximum corresponding to the laser being locked to the seed signal. The slope of the error signal causes the error signal to be used for laser control to achieve locking of the laser to the optical seed signal.

[0064] The detector may be a photodiode, for example an avalanche photodiode. The detector may be or may consist of a phase plate or a mode converter. The detector may convert the incident signal into an electrical signal. However, the detector is not limited to electrical signals. For example, the detector may be configured to generate an optical signal based on the incident signal, and the controller may derive an error signal based on the optical signal from the detector. In general, the invention is not limited to a particular form of a physical signal that the detector generates based on the incident signal.

[0065] For example, at least one operating parameter of the laser cavity may be - the temperature of the laser cavity, - the path length of the laser cavity, -Pumping of the laser cavity, the phase of the optical seed signal, . In general, the path length of the laser resonator is crucial for the mode distribution of the laser. The temperature influences the path length, especially in the case of semiconductor lasers, since due to thermal expansion the path length can increase or decrease depending on the temperature change. The pumping influences the temperature. Hence also in this case the path length can be influenced by pumping. The temperature can be optionally changed by a Peltier element or a separate heating element, e.g. an electric heater element, and a separate cooling element, e.g. a fan. The pumping of the laser depends on the laser technology and can be done, e.g., by optical pumping via a further light source, e.g. a further laser. Adjusting the pumping includes, e.g., adjusting the current, changing the intensity and / or the frequency of the optical pump light. The phase of the optical seed signal can be adjusted, e.g., by adjusting the light source of the optical seed signal or by adjusting the optical path length of the optical seed signal.

[0066] In any embodiment, the laser is a semiconductor laser, and the pumping of the laser is performed by current flow through the laser cavity. Semiconductor lasers include, for example, FP (Fabry-Perot) lasers, DFB (Distributed Feedback) lasers, DBR (Distributed Bragg Reflector) lasers, or VCSEL (Vertical Cavity Surface Emitting) lasers. Other classes of lasers compatible with the method include ECDL (External Cavity Diode Lasers) or ECL (External Cavity Lasers). Semiconductor lasers can be pumped by current flow through an active material in the laser cavity. The pumping can be adjusted by adjusting the current.

[0067] For example, the detector may be a photodiode array of at least three photodiodes, and deriving the error signal includes determining the geometric beam shape of the incident signal. Using the photodiode array, the geometric beam shape of the incident signal can be estimated. To estimate the geometric beam shape, the use of at least three photodiodes is necessary for differentiation of the first and second (or higher) order spatial modes. In principle, the more photodiodes used, the better the signal shape can be approximated. In general, determining the geometric beam shape of the incident signal means determining a parameter value of at least one parameter that characterizes the geometric beam shape. Preferably, the photodiodes are arranged in a rectangular structured grid. Minimizing the distance between each photodiode is beneficial to make the measurement most sensitive towards changes in the beam shape.

[0068] The detector may be a quadrant photodiode, and deriving the error signal may include determining a diagonal signal of the quadrant photodiode. This diagonal signal is composed of the difference of the sum of the signals from the diagonal sensor areas of the quadrant photodiode. In this way, the beam shape of the incident signal, in particular the beam ellipticity, can be particularly easily evaluated. In particular, a quadrant photodiode is a photodiode with four divided sensor areas. The gap between the sensor areas can be optimized to be as small as possible. In particular, a quadrant photodiode generates a current proportional to the input light from each of the four sensor areas. The signals from each sensor area are denoted A, B, C, D, and on the quadrant photodiode the sensor areas are arranged (clockwise) as A, D, B, C. This allows the following calculations: SUM=A+B+C+D, LR=(A+C)-(B+D), UD=(A+D)-(C+B), DIAG=(A+B)-(C+D). The beam ellipticity can be measured using the DIAG calculation. A zero DIAG signal indicates no beam ellipticity, which means the laser beam is circular. A non-zero DIAG signal means an elliptical beam on the quadrant photodiode. An opposite sign of the DIAG signal indicates opposite ellipticity. The quadrant photodiode may be an InGaAs-based or silicon-based detector. The signal can be converted to a voltage (e.g., using standard analog electronics with an operational amplifier).

[0069] As mentioned above, before the method is used, a start-up or initial calibration of the method and system is performed. During control of the injection-locked laser, the initialized or calibrated characteristics are constant. The error signal can be initialized by the following parameters:

[0070] For example, propagation of the incident signal involves applying an angle of incidence of the incident signal to the detector. The angle of incidence affects the beam shape of the incident beam on the detector. For example, a circular beam will appear to have an elliptical beam shape on the detector if the angle of incidence is different from 90°. An incident beam with an elliptical cross section can be made to have a circular beam shape on the detector with the appropriate angle of incidence. If the detector is a QPD, the applied angle can set the DIAG signal to zero. The applied angle is set once during calibration and is kept constant during operation.

[0071] In an optional embodiment, the propagation of the incident signal includes shaping the incident signal, in particular shaping the incident signal with at least a lens, preferably at least a shaping lens, in particular at least a cylindrical lens. Shaping the incident signal allows to calibrate an initial error signal when the laser is locked to the optical seed signal. The error signal can be initially set to zero. Shaping the incident signal allows to zero the DIAG signal if the detector is a QPD. This can be achieved, for example, if the detector is a QPD and the incident signal is shaped such that the image of the incident signal on the detector is circular. In this way, the DIAG signal is zero. The DIAG signal can optionally be used as the error signal without further manipulation. The position and alignment of the beam shaping optics for shaping the incident beam, in particular the at least one lens, preferably the at least one shaping lens, in particular the at least one cylindrical lens, are set once during calibration and kept constant during operation.

[0072] In another embodiment, the propagation of the incident signal includes focusing the incident signal on the detector using a detection lens, preferably a convex spherical lens. Optionally, the laser beam is focused and / or diverged in at least a first plane and preferably in a second plane (non-parallel to the first plane), in particular at a point along the path of the laser beam different from the focus / divergence in the first plane. Focusing the laser beam refers to focusing the incident signal in at least one plane. The position and alignment of the detection lens, preferably a convex spherical lens, for focusing the incident signal on the detector is set once during calibration and kept constant during operation.

[0073] Optionally, propagating the incident signal includes defining an axial distance between the detection lens and the detector such that a transmission peak of the incident signal coincides with a zero crossing of the error signal. The transmission peak represents a resonance of the optical seed signal with the laser. The axial distance between the detection lens and the detector represents their distance along the path of the incident signal. For the present method and apparatus, the positions of the detector and the detection lens are fixed during calibration and are kept constant during operation. The relative change in the geometric beam shape of the incident signal is most noticeable when the laser resonates with the optical seed signal. Thus, the present method is most sensitive at the transmission peaks of the incident signal, which are also the transmission peaks of the optical seed signal. Optionally, the axial distance may be adjusted in an adjustment section to determine the curve of the error signal and set the axial distance.

[0074] In another optional embodiment, propagating the incident signal includes determining an axial distance between the laser and the detector such that a transmission peak of the incident signal coincides with a zero crossing of the error signal. The positions of the detector and the laser are fixed during calibration and are kept constant during operation. The relative change in the geometric beam shape of the incident signal is most noticeable when the laser is resonant with the optical seed signal. Thus, the method is most sensitive at the transmission peaks of the incident signal, which are also the transmission peaks of the optical seed signal. The distance between the laser and the detector is determined by the TEM 00This affects the Guoy phase between the laser and the second spatial mode. The image obtained on the detector can therefore be adjusted by adjusting the position of the detector relative to the laser. There is an axial distance where the error signal is purely dispersive and another axial distance where the error signal is purely absorptive or any signal shape between absorptive and dispersive. Preferably the error signal is dispersive.

[0075] Optionally, deriving the error signal includes applying an offset to the error signal. The error signal is derived as follows: the incident signal on the detector generates a signal for each photodiode of the detector. These signals are used in response to an estimator of the incident signal shape. For example, if the detector is a QPD, a DIAG calculation can be used to estimate the ellipticity of the signal. The error signal can be initialized to zero during calibration. Thus, the sign of the error signal indicates the direction of control, e.g., whether an operating parameter of the laser should be increased or decreased. For example, the DIAG calculation applies an additional offset to initialize the error signal at the set point to zero, since this can result in a residual signal even if the incident signal is substantially circular.

[0076] In another embodiment, an imbalance detection operation can be used. For example, instead of the DIAG operation, the imbalance operation IDIAG can be used. IDIAG=ε 1 (A+B)-ε 2 (C+D), where ε 1 / ε 2 ≠ 1 and ε 1 and ε 2 can be initially set to zero the error signal, allowing adjustment of the background offset of the diagonal channels to provide a zero crossing in the error signal at a desired condition (e.g., at a set point).

[0077] The closed loop control method of an injection-locked laser and the closed loop control system for controlling the injection-locked laser can be used to continuously shift the wavelength of the output signal of the injection-locked laser by the further step of shifting the wavelength of the optical seed signal, where adjusting the wavelength of the optical seed signal results in a change in the error signal. The shift in the wavelength of the optical signal disturbs the locking of the optical seed signal with the laser, which changes the error signal. The change in the error signal provides a feedback to the laser in response in the control step of the method, whereby the relationship between the laser and the optical seed signal is restored at the shifted wavelength of the optical seed signal. The laser output then tracks the wavelength of the optical seed signal. The error signal indicates whether the laser is operating at its optimum operating parameters for the injected laser frequency. With this information, it is possible to add an additional layer of feedback to dynamically adjust the operating parameters of the injection laser, whereby the laser can track the injection frequency throughout its entire operating range. This optimum range allows for continuous tuning of the laser frequency without interruption. This uninterrupted frequency tuning range is often referred to as the "mode hop free" region. This is a unique property of semiconductor lasers, which can range from a few GHz to tens of GHz. This optimum locking range is usually determined by the set operating current and temperature of the unpackaged semiconductor laser. The initial operating parameters are usually set before or during system initialization or calibration, so the range is already predetermined at the center frequency. This center frequency can be dynamically tuned beyond the mode-hop free region if the optimum locking range information of the laser is available. Since the natural output mode of a semiconductor laser is highly elliptical and the injected mode is generally circular (or different levels of ellipse), there exists an error signal that continuously monitors whether the laser parameters are still in the optimum injection range or at the optimum set value. With this, the current or temperature or both can be dynamically adjusted to track the injection signal frequency. This allows the generation of a widely tunable laser frequency over its entire operating range.

[0078] Optionally, at least the laser, optics, and preferably the detector are integrated on a micro-optics or silicon photonics platform, preferably a silicon chip. This can be achieved, for example, by heterogeneous integration of individual miniaturized elements on a single system-on-chip (SoC). For example, the system can be constructed using state-of-the-art optical packaging techniques and precision assembly.

[0079] In any embodiment, the system is integrated into a TO-Can package or butterfly mount (BTF) for easy compatibility with common technologies.

[0080] Optionally, the method further comprises centering the output signal on the quadrant photodiode by translating the detection lens and / or the quadrant photodiode in a direction perpendicular to the optical axis of the incident signal such that left / right and / or top / bottom signals of the quadrant photodiode are minimized, where the left / right and top / bottom signals are composed of the difference of the sums of adjacent sensor areas of the quadrant photodiode, respectively, and the left / right and top / bottom signals use the sums of different halves of adjacent sensor areas. Optionally, this is performed iteratively. In this way, the incident signal can be perfectly aligned with the quadrant photodiode, and an optimal beam fit is achieved on the active area of ​​the quadrant photodiode to generate the most accurate error signal. Optionally, the translation is performed in two directions, which are perpendicular to each other and perpendicular to the optical axis of the laser beam.

[0081] Constraints on stability still exist due to residual fluctuations in the input signal that may be due, for example, to changes in the second-order modal components affecting the phase reference. To help alleviate this remaining constraint, polarization pre- and post-selection procedures are available, which are explained using the weak value concept. For this purpose, optionally, the optical seed signal is polarization filtered before it is directed to the laser, and the optical output signal is polarization filtered. Optionally, the laser comprises a birefringent resonator. In this way, technical noise that limits the system performance can be coherently suppressed. The device preferably comprises a first polarizing beam splitter (or a first polarizer) for polarization filtering the optical seed signal before it is directed to the laser. A second polarizing beam splitter (or a second polarizer) is used for polarization filtering the input signal, in particular before the detector. Polarization filtering can be used in closed-loop control methods for controlling injection-locked lasers and in closed-loop control systems for controlling injection-locked lasers.

[0082] When the polarization degree of freedom is included, the cavity reflection coefficient is given by 1 The reflection operator that acts on > Replaced by JPEG2025502754000004.jpg86. State |ψ 2 When a post-selection is made to w is given by the weak value of the reflection operator. JPEG2025502754000005.jpg1863((Formula 1) where δ is the frequency detuning of the optical seed signal to the laser, κ is the total linewidth of the laser, and γ' = Aγ. γ is a dimensionless parameter that characterizes the round-trip losses in the laser. The "amplification" parameter A depends on the particular a posteriori selection of the linear polarization state |ψ 1 > and |ψ 2 > is a real value. It is a function of the polarizer angle, but operationally it is the residual power ratio after post-selection when the optical seed signal is off-resonance |<ψ 2 |ψ 1 >|2 ≒1 / (2A+1) 2 can be related to the reflection from a conventional high-finesse cavity. The right-hand side of Eq. 1 has the same form (for a fixed value of κ) as the reflection from a normal high-finesse cavity, except for the variable loss parameter γ'. Note that γ<1, γ=1, γ>1, and γ>2 are identified as the under-coupled, impedance-matched, over-coupled, and cavity-gain regions of the laser, respectively. The variability of γ' can be experimentally demonstrated by observing the post-selection dependence of the reflected power from the laser. The error signal can be given by the imaginary part of the reflection coefficient in Eq. 1. Adjusting the post-selection to an impedance-matched configuration reduces noise due to frequency stabilization. In this configuration, the non-signal-producing beam components incident on the detector are strongly attenuated while the signal-producing parts are only weakly attenuated, effectively increasing the slope of the error signal and reducing its sensitivity to beam shape and intensity fluctuations.

[0083] The optional detection lens may be a convex spherical lens.

[0084] The optical seed signal may be shaped before being injected into the laser, for example so that the optical seed signal has an elliptical geometric beam shape or a circular beam shape.

[0085] The methods and devices can be used with a wide range of laser wavelengths, for example, the wavelengths may be in the ultraviolet range between 190 nm and 400 nm, the visible range between 400 nm and 800 nm, the near infrared range between 800 nm and 1800 nm, and / or the infrared range between 1800 and at least 3500 nm.

[0086] Optionally, the error signal is directly proportional to the geometric beam ellipticity of the incident signal.

[0087] Optionally, the error signal is substantially zero for a circular incident signal and / or a circular image of the incident signal on the detector.

[0088] Specific embodiments of the present invention relate to a method for frequency locking a laser to an optical resonator, a method for adjusting an apparatus for frequency locking a laser to an optical resonator, and an apparatus for frequency locking a laser to an optical resonator, which are outlined below.

[0089] A number of methods have been developed over the decades to address the challenge of locking a laser into a cavity. These include side-of-fringe intensity methods, polarization-based methods, frequency modulation techniques including transmission modulation and Pound-Drever-Hall (PDH) reflection methods, and finally, spatial mode interferometry.

[0090] While certain techniques may be preferred depending on the application, the PDH technique - which uses high frequency modulation / demodulation of an optical carrier - has become the prevailing standard due to its stability and versatility. 5 -10 6 To achieve the most demanding lock stability of 1 / 1000, additional layers of active feedback mechanisms were further required to reduce the residual amplitude (RAM) modulation that typically limits the lock point stability of PDH systems. A purely passive, low-complexity method that can compete with these highly engineered configurations would be beneficial for all applications, including space-based ones, where electronic and acousto-optical devices used for modulation are undesirable due to power consumption and risk of failure.

[0091] In one embodiment of the present invention, a method and device for frequency locking a laser to an optical cavity and a method for tuning such a device are provided, which has reduced complexity, is purely passive and / or achieves good laser stability, is especially less susceptible to alignment drifts and / or has a more robust implementation. In particular, the arrangement of the detection lens and detector in such a device and a method for achieving better stabilization are disclosed (see embodiment 15).

[0092] This is achieved by a method of frequency locking a laser to an optical cavity (see embodiment 15) in the following steps. - providing a laser beam from a laser; - shaping the laser beam so that it acquires ellipticity; - directing a laser beam into an optical resonator; - focusing the laser beam reflected from the optical resonator using a detection lens; - determining, by means of a detector, an error signal proportional to the beam ellipticity of the focused laser beam; - locking the laser to the optical cavity based on the error signal. Here, the axial distance between the detection lens and the detector is such that when the laser provides a laser beam at the resonant frequency of the optical resonator, the zero crossing of the error signal may be in an interval between a minimum and a maximum value of the error signal, the interval including an error signal value corresponding to the determined beam ellipticity of the substantially circular laser beam.

[0093] This is further achieved by a method for tuning an apparatus for frequency locking a laser to an optical cavity (see embodiment 17), where the apparatus comprises: a beam shaper for shaping the laser beam so that it acquires ellipticity; an optical resonator for receiving the laser beam after beam shaping; a detection lens for focusing the laser beam reflected from the optical resonator; a detector for determining an error signal proportional to the beam ellipticity of the laser beam focused on a detection lens; The method includes: - providing a laser beam at a resonant frequency of an optical resonator; - shaping the laser beam with a beam shaper so that the laser beam acquires ellipticity; - directing a laser beam into an optical resonator; - focusing the laser beam reflected from the optical resonator using a detection lens; - determining an error signal proportional to the beam ellipticity of the focused laser beam using a detector; - adjusting an axial distance along the optical axis of the laser beam between the detection lens and the detector such that a zero crossing of the error signal is in an interval between a minimum and a maximum value of the error signal, this interval including an error signal value corresponding to the determined beam ellipticity of the substantially circular laser beam; Includes.

[0094] Furthermore, this can be achieved by an apparatus for frequency locking a laser to an optical cavity (see embodiment 27), which comprises: a beam shaper for shaping the laser beam so that it acquires ellipticity; an optical resonator for receiving the laser beam after beam shaping; a detection lens for focusing the laser beam reflected from the optical resonator; a detector for determining an error signal proportional to the beam ellipticity of the laser beam focused by the detection lens; wherein the axial distance between the detection lens and the detector is such that, when providing the device, in particular the beam shaper, with a laser beam at the resonant frequency of the optical resonator, a zero crossing of the error signal is in an interval between a minimum and a maximum value of the error signal, said interval including an error signal value corresponding to the determined beam ellipticity of the substantially circular laser beam.

[0095] The present disclosure further relates to a method of manufacturing an apparatus for frequency locking a laser to an optical cavity, the method comprising the steps of: - providing a beam shaper for shaping the laser beam so that it acquires ellipticity, - providing an optical resonator for receiving the laser beam after beam shaping, - providing a detection lens for focusing the laser beam reflected from the optical resonator; - providing a detector for determining an error signal proportional to the beam ellipticity of the laser beam focused by the detection lens; - providing a laser beam at a resonant frequency of an optical resonator; - shaping the laser beam with a beam shaper so that the laser beam acquires ellipticity; - directing a laser beam into an optical resonator; - focusing the laser beam reflected from the optical resonator using a detection lens; - determining an error signal proportional to the beam ellipticity of the focused laser beam using a detector; - adjusting an axial distance along the optical axis of the laser beam between the detection lens and the detector such that a zero crossing of the error signal is in an interval between a minimum and a maximum value of the error signal, this interval including an error signal value corresponding to the determined beam ellipticity of the substantially circular laser beam.

[0096] To understand the primary stabilization method, one should recall that optical resonators support Hermite-Gaussian (HG) spatial modes. These cavities can support a set of transverse electromagnetic modes (TEM) mn ), where the subscripts m and n are positive integers representing each orthogonal axis in the transverse direction of beam propagation. The sum m+n is called the mode order and can be even or odd. Starting with a perfectly aligned optical mode entering the optical cavity, odd modes are induced by moving the input beam out of alignment, either by tilting or shifting the input beam. On the other hand, even modes are induced by focusing / mode mismatch and are independent of the beam tilt or shift.

[0097] Previous implementations of spatial mode interference have focused on the first order (TEM) by tilting / shifting the input beam. 01 and TEM 10 ) mode. We are interested in the fundamental HG mode, denoted as "00", in the TEM 00and a specific second-order HG mode, denoted as "+", and a second-order TEM11-like mode, all of which are even modes. The even modes are induced by mode misalignment, making the method insensitive to misalignment drifts that generate the first-order odd modes. A slightly elliptical beam with horizontal / vertical orientation can be mathematically resolved into a major "00" component and a small "+" component. In such a laser beam, the phase difference between these two modes encodes information about the ellipticity. When incident on an optical resonator near the "00" resonance, the two modes experience a differential phase shift upon reflection, since modes of different orders are generally non-degenerate, so only one of them will resonate. By this mechanism, the reflected signal is made to acquire the opposite ellipticity on the other side of the optical resonator resonance. To exploit this effect, the ellipticity of a laser beam reflecting from an optical resonator is measured after focusing.

[0098] These modes generally have different resonant frequencies. As for the cavity modes, the spatial resolution of the diagonally oriented slightly elliptical beam is mostly “TEM” 00 " mode, "TEM 11"The contribution of the second mode is small. For such a beam, the phase difference between these two modes encodes information about the deviation from circularity. When the resonator is incident at a frequency close to one of its fundamental frequencies, the two modes naturally undergo a differential phase shift, since only one of them is resonating. This results in a change in the shape of the reflected beam. By this mechanism, the reflected signal is made to acquire an inverse ellipticity on the other side of the resonance, which can generate a determinable error signal. To utilize this property, an error signal proportional to the beam ellipticity of the reflected laser beam must be used and the axial distance between the detector and the detection lens must be appropriately adjusted. By setting the axial distance so that the beam shaper is presented with a laser beam at the resonant frequency of the optical resonator, the zero crossing of the error signal is in an interval between the minimum and maximum values ​​of the error signal, which interval contains the error signal value corresponding to the determined beam ellipticity of a substantially circular beam, the reflected signal that acquires an inverse ellipticity on the other side of the resonance can be used to determine the deviation and / or drift of the laser beam, which can be used to lock and thereby stabilize the laser beam. The determined error signal can be considered as the interference between the resonant "00" mode leaking out of the resonator and the second-order "+" mode reflecting directly from the optical resonator and forming the phase reference. This detection scheme is insensitive to drifts and variations in alignment, since small misalignments will simply generate the first-order mode in the modal decomposition.

[0099] The optical resonator may be a linear resonator or a ring resonator. For example, a ring resonator can be used where the incident and reflected beams follow separate paths. The optical resonator may be, for example, a two-mirror resonator or a three-mirror resonator. However, any number of mirrors and any shape of resonator can be used. When a two-mirror (standing wave) resonator is used as the optical resonator, the reflected beam can be separated from the incident path, for example, by using a Faraday circulator or a quarter wave plate and a polarizing beam splitter. The optical resonator may be a free-space resonator or a non-free-space resonator, such as a microchip-based optical resonator. For example, the optical resonator may be a whispering gallery mode resonator, where the light circulates around (inside the material) a sphere or toroid. The optical resonator may be a fiber ring into which light can be evanescently coupled. This can be done, for example, by using a prism, where the evanescent tail during total internal reflection from the prism couples the light in and out of the optical resonator. Thus, the prism effectively interfaces a waveguide-like structure into the free-space optical system. Although the prism may look like the input mirror of a free-space resonator, it has a highly elliptical mode structure. In this case, input mode shaping is required to prepare a more squashed beam profile for the eigenmodes of the microresonator-prism assembly.

[0100] The method and apparatus can be used with a wide range of laser beam wavelengths. For example, the wavelengths may be in the ultraviolet range between 190 nm and 400 nm, the visible range between 400 nm and 800 nm, the near infrared range between 800 nm and 1800 nm, and / or the infrared range between 1800 and at least 3500 nm. Optionally, the error signal is directly proportional to the beam ellipticity of the focused laser beam. Optionally, the error signal is substantially zero for a circular laser beam. Optionally, the laser is locked to the optical cavity based on the error signal by feeding back the error signal to a laser current. Optionally, the apparatus includes a laser. Locking is based on monitoring changes in the ellipticity of the laser beam reflecting from the optical cavity.

[0101] Focusing a laser beam refers to focusing the laser beam in at least one plane. Providing a laser beam at the resonant frequency of the optical resonator preferably includes a step of tuning the laser (in particular finding the resonant frequency). Optionally, the method of tuning the device (without beam shaping, i.e. before inserting the beam shaper into the device) includes: - (completely) coupling a laser beam into the resonator and measuring the characteristics of the reflected beam, in particular with a CCD camera, - inserting a beam shaper into the device, Optionally, the method of adjusting the apparatus includes measuring an in-plane beam waist and an out-of-plane beam waist of the laser beam reflecting from the optical cavity at two or more distances.

[0102] The axial distance between the detection lens and the detector refers to their distance along the path of the laser beam. With respect to the method and apparatus for frequency locking a laser to an optical resonator, the positions of the detector and the detection lens can be fixed. With respect to the method for tuning and manufacturing the apparatus, the detector and / or the detection lens can be movable. The zero crossing refers to the error signal having a value of zero. Optionally, the minimum is a global minimum and / or the maximum is a global maximum. Optionally, there are no further extrema between the minimum and maximum. Optionally, the interval spans from the minimum to the maximum. Between the interval that includes the error signal corresponding to a circular laser beam (providing the laser beam with a frequency at the resonant frequency) and the minimum and maximum due to the slope of the error signal, the error signal can be used to lock the laser. Optionally, the resonant frequency of the optical resonator is determined by a TEM 00 The frequency of resonance, or TEM 02 or TEM 20 or TEM 11 is the resonant frequency of the resonance. Optionally, the axial distance may be adjusted in an adjustment interval to determine the curve of the error signal and set the axial distance.

[0103] Preferably, the detector is a quadrant photodiode and the error signal is determined from the diagonal signals of the quadrant photodiode. This diagonal signal is composed of the difference of the sums of the signals from the diagonal sensor areas of the quadrant photodiode. In this way, an error signal corresponding to the beam ellipticity is particularly easily achievable. In particular, a quadrant photodiode is a photodiode with four divided sensor areas. The gap between the sensor areas can be optimized to be as small as possible. In particular, a quadrant photodiode generates a current proportional to the input light from each of the four sensor areas. The signals from each sensor area are denoted A, B, C, D, and on the quadrant photodiode the sensor areas are arranged in the order A, D, B, C (clockwise). This allows the following calculations: SUM=A+B+C+D, LR=(A+C)-(B+D), UD=(A+D)-(C+B), DIAG=(A+B)-(C+D). The beam ellipticity can be measured using the DIAG calculation. A zero DIAG signal indicates no beam ellipticity, which means the laser beam is circular. A non-zero DIAG signal means there is an elliptical beam on the quadrant photodiode. The quadrant photodiode can be an InGaAs-based or silicon-based detector. The signal can be converted to a voltage (e.g., using standard analog electronics with an op amp).

[0104] Constraints on stability still exist due to residual variations in the incoming beam shape, for example due to changes in the second order mode components affecting the phase reference. To help alleviate this remaining constraint, polarization pre- and post-selection procedures are available, which are explained using the weak value concept. For this purpose, optionally, the laser beam is polarization filtered before being directed to the optical resonator, and the laser beam reflected from the optical resonator is polarization filtered. Optionally, the optical resonator is birefringent. In this way, technical noise that limits the system performance can be coherently suppressed. The device preferably comprises a first polarizing beam splitter (or a first polarizer) for polarization filtering the laser beam before directing it to the beam shaper, and a second polarizing beam splitter (or a second polarizer) for polarization filtering after the laser beam emission from the optical resonator, in particular between the detection lens and the detector. Polarization filtering can be used as a method for frequency locking a laser and / or as a method for tuning the device for frequency locking a laser.

[0105] When the polarization degree of freedom is included, the cavity reflection coefficient is given by 1 The reflection operator that acts on > When post-selecting to the state, the effective reflection coefficient r w is given by the weak value of the reflection operator. JPEG2025502754000007.jpg2069((Formula 1) where δ is the frequency detuning of the laser cavity, κ is the total cavity linewidth, and γ' = Aγ, where γ is a dimensionless parameter characterizing the round-trip losses in the cavity. The "amplification" parameter A depends on the particular a posteriori selection of the linear polarization state |ψ 1 > and |ψ 2 > is a real value. It is a function of the polarizer angle, but operationally it is the residual power ratio after post-selection when the light is off-resonance |<ψ 2 |ψ 1 >|2 ≒1 / (2A+1) 2 can be related to the reflection from a conventional high-finesse resonator. The right-hand side of Eq. 1 has the same form (for a fixed value of κ) as the reflection from a conventional high-finesse resonator, except for the variable loss parameter γ'. Note that γ<1, γ=1, γ>1, and γ>2 are identified as the under-coupled, impedance-matched, over-coupled, and cavity-gain regions, respectively. The variability of γ' can be experimentally demonstrated by observing the post-selection dependence of the reflected light from the resonator. The error signal can be given by the imaginary part of the reflection coefficient in Eq. 1. Adjusting the post-selection to an impedance-matched configuration reduces noise due to frequency stabilization. In this configuration, the signal-producing portion is less attenuated, while the non-signal-producing beam components incident on the detector are strongly attenuated, effectively increasing the slope of the error signal and reducing its sensitivity to beam shape and intensity fluctuations.

[0106] Turning now to the method of adjusting the device, it is preferred that adjusting the axial distance of the detection lens and the detector along the optical axis of the laser beam such that the zero crossing of the error signal is located in the interval between the minimum and maximum values ​​of the error signal (wherein this interval includes the error signal value corresponding to the determined beam ellipticity of the substantially circular laser beam) comprises adjusting the axial distance of the detection lens and the detector along the optical axis of the laser beam such that the transmission peak of the laser beam reflected from the resonator coincides with the zero crossing of the error signal. The transmission peak corresponds to the reflection minimum, i.e. the SUM channel of the quadrant photodiode shows a dip (the SUM channel of the quadrant photodiode is specifically for measuring the reflection curve). In this way, the baseline offset of the error signal due to the natural astigmatism of the optical resonator can be corrected. The baseline offset also causes a shift of the transmission peak from the zero crossing of the error signal. It is therefore possible to adjust the axial position of the QPD until the transmission peak and the error signal zero crossing coincide. If this coincidence does not occur, cross-coupling of the intensity noise into the frequency noise occurs. The zero crossing or the error signal can be made to coincide with the transmission plateau by moving the detector closer to or further away from the detection lens and vice versa. This position can be determined experimentally, for example iteratively, i.e. by fine-tuning the position of the detector and / or the detection lens. It is also preferred in the context of the method and device for frequency locking a laser if the axial distance between the detection lens and the detector along the optical axis of the laser beam, when the laser provides a laser beam at the resonant frequency of the optical resonator, is such that the transmission peak of the laser beam reflected from the resonator substantially coincides with the zero crossing of the error signal, or within ±5%, or within ±2%, or within ±1%. This can also be substituted for the feature: "The axial distance between the detection lens and the detector is such that, when the laser provides a laser beam at the resonant frequency of the optical resonator, the zero crossing of the error signal is in the interval between the minimum and maximum values ​​of the error signal, the interval including the error signal value corresponding to the determined beam ellipticity of the substantially circular laser beam". There can also be a baseline offset of the error signal.Without experience and knowledge of astigmatism, this may not seem to give the best results, however, this criterion helps to produce the best possible results, surpassing the Pound-Drever-Hall method.

[0107] Optionally, adjusting the axial distance between the detection lens and the detector along the optical axis of the laser beam such that the zero crossing of the error signal is in the interval between the minimum and maximum values ​​of the error signal (this interval includes the error signal value corresponding to the determined beam ellipticity of the substantially circular laser beam) comprises adjusting the axial distance such that the in-plane and out-of-plane signal waists of the laser beam reflected from the resonator match the beam waists of the resonant cavity modes of the optical resonator (in particular the average beam waist of the cavity modes, which is the average beam waist of the in-plane and out-of-plane beam waists). Optionally, this is done before adjusting the axial distance between the detection lens and the detector along the optical axis of the laser beam such that the transmission peaks of the laser beam reflected from the resonator match the zero crossings of the error signal. Optionally additionally or alternatively, the axial distance is adjusted such that the in-plane and out-of-plane beam waists of the laser beam reflected from the resonator match (to each other). The baseline offset of the error signal may be due to a displacement of the detector position from the position where the beam has a circular profile (ignoring astigmatism). The axial distance can be selected to eliminate this baseline offset. This beam waist coincidence can be used as a first approximation before the coincidence of the transmission peak and zero crossing, or it can be used by itself. If there is no astigmatism, these two will be the same result. In this case, zeroing the baseline will automatically enter the transmission plateau since there is no residual astigmatism in the mode. In the present method and apparatus for frequency locking a laser, it is also preferred that the axial distance of the detector and detection lens is such that the in-plane and out-of-plane beam waists of the laser beam reflected from the resonator coincide with the beam waist of the resonant cavity mode of the optical resonator, or are within ±5%, or ±2%, or ±1% of the beam waist of the resonant cavity mode of the optical resonator.This may replace the feature "the axial distance between the detection lens and the detector is such that when the laser provides a laser beam at the resonant frequency of the optical resonator, the zero crossing of the error signal is in an interval between a minimum and a maximum value of the error signal, the interval including an error signal value corresponding to the determined beam ellipticity of the substantially circular laser beam."

[0108] Optionally, the detector is a quadrant photodiode and the error signal is determined from a diagonal signal of the quadrant photodiode, the diagonal signal being comprised of a difference in sums of signals from diagonal sensor areas of the quadrant photodiode.

[0109] Optionally, the method further comprises centering the laser beam on the quadrant photodiode by translating the detection lens and / or the quadrant photodiode in a direction perpendicular to the optical axis of the laser beam such that the left / right and / or up / down signals of the quadrant photodiode are minimized, where the left / right (LR) and up / down (UD) signals are each composed of the difference between the sums of adjacent sensor areas of the quadrant photodiode, and the left / right and up / down signals use the sums of different halves of adjacent sensor areas. Optionally, this is performed iteratively. In this way, the laser beam can be perfectly aligned with the quadrant photodiode, and an optimal beam fit is achieved on the active area of ​​the quadrant photodiode to generate the most accurate error signal. Optionally, the translation is performed in two directions, which are perpendicular to each other and to the optical axis of the laser beam.

[0110] Optionally, the steps of adjusting the axial distance between the detection lens and the quadrant photodiode along the optical axis of the laser beam and centering the laser beam on the quadrant photodiode are performed iteratively (i.e., alternately or sequentially) so that the transmission peak of the laser beam reflected from the resonator coincides with the zero crossing of the error signal. Every axial distance adjustment can minimize the left-right and up-down signals. By repeating this process, the zero crossing can be optimized to exactly coincide with the transmission plateau.

[0111] For all methods and apparatus disclosed herein, it is preferred if the detection lens is a convex spherical lens.

[0112] Furthermore, before shaping the laser beam so that it acquires ellipticity, the laser beam is preferably propagated in a polarizing optical fiber and optionally exits a fiber collimator, which may have an adjustable focus.

[0113] Shaping the laser beam so that it acquires ellipticity is preferably performed by at least one pair of cylindrical lenses, which optionally have mutually non-parallel, in particular vertically oriented, focus axes. Optionally, the two cylindrical lenses have the same focal length, and further optionally, one is oriented horizontally and the other vertically. The focal length is optionally such that the two lenses are arranged coaxially (as if they formed a spherical lens) such that the incident beam is focused on the TEM. 00 It is chosen to be perfectly mode-matched to the cavity mode. The distance between the pair of cylindrical lenses determines the amount of light in the "+" mode and can range, for example, to 10% of the total power (e.g., 400 μW). One cylindrical lens can be oriented with its axis in-plane and the other out-of-plane. By adjusting the position of the cylindrical lens and, optionally, the fiber collimator, the input mode can be manipulated to induce up to second order modes. TEM 00 , TEM 02 , TEM 20It is possible to align the latter two modes so that all other modes are suppressed and the transmission levels of the latter two are equal. 02 and TEM 20 To equalize the balance of the TEM, the position of the cylindrical lens can be set / adjusted so that the in-plane and out-of-plane beam waists converge and diverge. The average resonator waist is placed exactly where the in-plane and out-of-plane beam waists coincide. When the beam diverges outside the cavity, its average waist can also coincide with the cavity mode average waist. 00 TEM for 02 and TEM 20 The balance of the amount of can be adjusted by adjusting the distance between the two cylindrical lenses while maintaining the center of gravity. Each member of the pair is translated symmetrically in opposite directions from this reference configuration, so that the TEM 02 Mode and TEM 20 The modes are of equal amplitude as observed in the cavity transmission spectrum. Physically, this action moves the beam waist in the horizontal and vertical planes before and after the cavity waist position. Alternatively, shaping the laser beam so that it acquires ellipticity is performed by a combination of lenses, wedge prisms, and / or anamorphic prism pairs, optionally producing an elliptical or astigmatic beam. Next, with regard to the device for frequency locking the laser to the optical cavity, it is preferred that the detector is a quadrant photodiode. Optionally, the detector is configured to determine an error signal by a diagonal signal of the quadrant photodiode, the diagonal signal being composed of the difference of the sum of the signals from the diagonal sensor areas of the quadrant photodiode.

[0114] Optionally, the beam shaper comprises a pair of cylindrical lenses, which have focusing axes oriented in non-parallel directions, in particular perpendicular to each other.

[0115] By way of example, the present disclosure is further described below with respect to selected embodiments 1 to 29 and illustrated in the drawings, which should not be construed as limiting the present disclosure.

[0116] 1. A closed-loop control method for controlling an injection-locked laser, comprising: - injecting a laser with an optical seed signal, - propagating at least a portion of the optical output signal from the laser as an incident signal onto a detector; - deriving an error signal based on a geometric beam shape of the signal incident on the detector; - controlling at least one operating parameter of the laser sensitive to the error signal; The method includes:

[0117] 2. The method according to embodiment 1, wherein at least one operating parameter of the laser is - the temperature of the laser cavity, - the path length of the laser cavity, -Pumping of the laser cavity, the phase of the optical seed signal, A method that is one or more of the following:

[0118] 3. A method according to embodiment 1 or embodiment 2, wherein the laser is a semiconductor laser and pumping of the laser is performed by an electric current through the laser resonator.

[0119] 4. A method according to any one of embodiments 1 to 3, wherein the detector is a photodiode array consisting of at least three photodiodes, and deriving the error signal includes determining a geometric beam shape of the incident signal.

[0120] 5. A method according to any one of embodiments 1 to 3, wherein the detector is a quadrant photodiode and deriving the error signal includes determining a diagonal signal of the quadrant photodiode, the diagonal signal being comprised of a difference between sums of signals from diagonal sensor areas of the quadrant photodiode.

[0121] 6. The method according to any one of embodiments 1 to 5, wherein propagating the incident signal includes imparting an angle of incidence of the incident signal to the detector.

[0122] 7. A method according to any one of embodiments 1 to 6, wherein the propagation of the incident signal includes shaping the incident signal, in particular using at least one lens, preferably at least one shaping lens, in particular at least one cylindrical lens.

[0123] 8. A method according to any one of embodiments 1 to 7, wherein propagating the incident signal includes focusing the incident signal onto the detector using a detection lens (33), preferably a convex spherical lens.

[0124] 9. A method according to embodiment 8, wherein propagating the incident signal includes determining an axial distance between the detection lens and the detector such that a transmission peak of the incident signal coincides with a zero crossing of the error signal.

[0125] 10. A method according to any one of embodiments 1 to 9, wherein propagating the incident signal includes determining an axial distance between the laser and the detector such that a transmission peak of the incident signal coincides with a zero crossing of the error signal.

[0126] 11. A method according to any one of embodiments 1 to 10, wherein deriving the error signal includes applying an offset to the error signal.

[0127] 12. A method for continuously shifting the wavelength of an output signal of an injection-locked laser using a method according to any one of the first to twelfth embodiments, comprising: - shifting the wavelength of the optical seed signal wherein adjusting the wavelength of the optical seed signal (10) varies the error signal.

[0128] 13. A closed loop control system for controlling an injection-locked laser, comprising: a laser that is injected with an optical seed signal during operation; - detector, an optical component for propagating at least a portion of the optical output signal from the laser as an incident signal to a detector; a controller for deriving an error signal based on a geometric beam shape of a signal incident on the detector; wherein the controller is configured to control at least one operating parameter sensitive to the error signal.

[0129] 14. A control system according to embodiment 13, wherein at least the laser, the optical components, and preferably the detector are integrated on a micro-optics or silicon photonics platform, preferably on a silicon chip.

[0130] 15. A method for frequency locking a laser to an optical cavity, comprising the steps of: - providing a laser beam from a laser; - shaping the laser beam so that it acquires ellipticity, - directing a laser beam into an optical resonator; - focusing the laser beam reflected from the optical resonator using a detection lens; - determining an error signal proportional to the beam ellipticity of the focused laser beam using a detector; - locking the laser to the optical cavity based on the error signal; wherein an axial distance between the detection lens and the detector is such that, when the laser provides a laser beam at a resonant frequency of the optical resonator, a zero crossing of the error signal is in an interval between a minimum and a maximum value of the error signal, the interval including an error signal value corresponding to a determined beam ellipticity of the substantially circular laser beam.

[0131] 16. A method according to embodiment 15, wherein the detector is a quadrant photodiode and the error signal is determined from a diagonal signal of the quadrant photodiode, the diagonal signal being composed of the difference between the sums of signals from diagonal sensor areas of the quadrant photodiode.

[0132] 17. A method of adjusting an apparatus for frequency locking a laser to an optical cavity, the apparatus comprising: a beam shaper for shaping the laser beam so that it acquires ellipticity; an optical resonator for receiving the laser beam after beam shaping; a detection lens for focusing the laser beam reflected from the optical resonator; a detector for determining an error signal proportional to the beam ellipticity of the laser beam focused by the detection lens; The method includes: - providing a laser beam at a resonant frequency of an optical resonator; - shaping the laser beam with a beam shaper so that the laser beam acquires ellipticity; - directing a laser beam into an optical resonator; - focusing the laser beam reflected from the optical resonator using a detection lens; - determining an error signal proportional to the beam ellipticity of the focused laser beam using a detector; - adjusting the axial distance along the optical axis of the laser beam between the detection lens and the detector such that the zero crossing of the error signal is in the interval between the minimum and maximum values ​​of the error signal; wherein the interval includes error signal values ​​corresponding to a determined beam ellipticity of a substantially circular laser beam.

[0133] 18. A method according to embodiment 17, wherein adjusting the axial distance between the detection lens and the detector includes adjusting the axial distance between the detection lens and the detector along the optical axis of the laser beam so that a transmission peak of the laser beam reflected from the optical resonator coincides with a zero crossing of the error signal.

[0134] 19. A method according to embodiment 17 or embodiment 18, wherein adjusting the axial distance between the detection lens and the detector optionally includes adjusting the axial distance so that the in-plane beam waist and the out-of-plane beam waist of the laser beam reflected from the optical resonator match the beam waist of a resonant cavity mode of the optical resonator before adjusting the axial distance along the optical axis of the laser beam between the detection lens and the detector so that the transmission peak of the laser beam reflected from the optical resonator matches a zero crossing of the error signal.

[0135] 20. A method according to any one of embodiments 17 to 19, wherein the detector is a four-part photodiode and the error signal is determined from a diagonal signal of the four-part photodiode, the diagonal signal being composed of the difference between the sums of signals from diagonal sensor areas of the four-part photodiode.

[0136] 21. The method according to embodiment 20, - centering the laser beam on the quadrant photodiode by translating the detection lens and / or the quadrant photodiode in a direction perpendicular to the optical axis of the laser beam such that signals to the left and right and / or above and below the quadrant photodiode are minimized. wherein the left / right signal and the top / bottom signal each comprise a difference between sums of adjacent sensor areas of a quadrant photodiode, and the left / right signal and the top / bottom signal use sums of different halves of adjacent sensor areas.

[0137] 22. A method according to embodiment 7, in which the steps of adjusting the axial distance along the optical axis of the laser beam between the detection lens and the quadrant photodiode and centering the laser beam on the quadrant photodiode are performed iteratively so that the transmission peak of the laser beam reflected from the optical resonator coincides with the zero crossing of the error signal.

[0138] 23. The method according to any one of embodiments 15 to 22, wherein the detection lens is a convex spherical lens.

[0139] 24. A method according to any one of embodiments 15 to 23, wherein the laser beam is propagated in a polarizing optical fiber and optionally emitted from a fiber collimator before shaping the laser beam so that it acquires ellipticity.

[0140] 25. A method according to any one of embodiments 15 to 24, wherein shaping the laser beam so that it acquires ellipticity is performed by a pair of cylindrical lenses whose focusing axes are oriented non-parallel, in particular perpendicular to each other, and / or shaping the laser beam so that it acquires ellipticity is performed by a combination of lenses, wedge prisms, and / or anamorphic prism pairs, in which an elliptical or astigmatic beam is generated.

[0141] 26. A method according to any one of embodiments 15 to 25, wherein the laser beam is polarized filtered before directing the laser beam towards the optical resonator, and the laser beam reflected from the optical resonator is polarized filtered, and optionally the optical resonator is birefringent.

[0142] 27. An apparatus for frequency locking a laser to an optical resonator comprising: a beam shaper for shaping the laser beam so that it acquires ellipticity; an optical resonator for receiving the laser beam after beam shaping; a detection lens for focusing the laser beam reflected from the optical resonator; a detector for determining an error signal proportional to the beam ellipticity of the laser beam focused by the detection lens; wherein the axial distance between the detection lens and the detector is such that, when providing a laser beam to the beam shaper at a resonant frequency of the optical resonator, a zero crossing of the error signal is in an interval between a minimum and a maximum value of the error signal, the interval including an error signal value corresponding to a determined beam ellipticity of the substantially circular laser beam.

[0143] 28. An apparatus according to embodiment 27, wherein the detector is a quadrant photodiode, and optionally the detector is configured to determine an error signal according to a diagonal signal of the quadrant photodiode, the diagonal signal being composed of the difference between the sums of signals from diagonal sensor areas of the quadrant photodiode.

[0144] 29. An apparatus according to any one of embodiments 27 to 28, wherein the beam shaper comprises a pair of cylindrical lenses having focusing axes oriented non-parallel to each other, in particular perpendicular to each other.

[0145] By way of example, the present disclosure will be further described with respect to some selected embodiments shown in the drawings for illustrative purposes, but which should not be construed as limiting the present disclosure. Figures 1 to 6 relate to embodiments 1 to 14, and Figures 7 to 15c relate to embodiments 15 to 29. [Brief description of the drawings]

[0146] [Figure 1] FIG. 1 is a schematic diagram of beam ellipticity detection using a quadrant photodiode. [Diagram 2] 1 shows the error signal as a function of the detuning of the laser relative to the optical seed signal. [Diagram 3] FIG. 1 is a schematic diagram of an example of a closed-loop control system for controlling an injection-locked laser with an external optical seed signal. [Figure 4] 1 is a schematic diagram of an example of a closed-loop control system for controlling a self-injection-locked laser that uses a portion of the laser's output signal as an optical seed signal. [Diagram 5] FIG. 1 illustrates an example of a closed loop control system for controlling a self-injection locked laser with an external cavity providing a reflected signal for self-injection. [Figure 6] FIG. 13 illustrates an example of a closed loop control system for controlling self-injection locking with an external cavity providing a reflected signal for self-injection, with a second detector between the laser and the external cavity. [Figure 7] FIG. 1 is a schematic diagram of one embodiment of an apparatus for frequency locking a laser to an optical cavity. [Figure 8a] Schematic of a triangular ring resonator geometry and its associated Hermite-Gaussian mode resonances. [Figure 8b] FIG. 1 illustrates beam ellipticity detection using a quadrant photodiode. [Figure 8c] FIG. 4 is a diagram showing the frequency dependence of an error signal. [Figure 9] FIG. 8 illustrates the positions of relevant elements in the beam path and the resulting beam characteristics in the embodiment of FIG. [Figure 10] FIG. 8 illustrates the error signal and resonator transmission in the embodiment of FIG. [Figure 11] FIG. 1 shows an error signal and transmission. [Figure 12] FIG. 2 is a schematic diagram of the axial distance between the detection lens and the detector. [Figure 13a] FIG. 15 illustrates the use of a first polarizer and a second polarizer in the embodiments of FIGS. 7 and 14. [Figure 13b] FIG. 13 shows the normalized reflected power measured for different post-selection configurations. [Figure 13c] FIG. 13 shows the real and imaginary parts of the obtained weak value. [Figure 14] FIG. 1 shows two copies of the lock-up configuration used to demonstrate the performance of the stabilization scheme. [Figure 15a] FIG. 15 illustrates the lock instability over the averaging time of the experimental setup shown in FIG. 14 in comparison with other stabilization schemes. [Figure 15b] FIG. 13 illustrates the achieved instabilities with and without post-selection, and the corresponding beat note time traces at the bottom. [Figure 15c] FIG. 1 shows estimated limits for stability. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0147] FIG. 1 shows the beam ellipticity detection of an incident signal 1 using a detector 2 and the mode decomposition of a slightly elliptical beam. The detector 2 is a quadrant photodiode (QPD) 3 with four sensor areas. The quadrant photodiode 3 generates a current proportional to the incident light from each of the four sensor areas 4. The derivation of the error signal 9 (not shown in FIG. 1, see FIG. 2) involves the determination of the diagonal signal of the quadrant photodiode 3. This diagonal signal is composed of the difference between the sums of the signals from the diagonal sensor areas 4 of the quadrant photodiode 3. The signals from each sensor area 4 are labeled A, B, C, D, and on the quadrant photodiode 3, the sensor areas 4 are arranged in the following order (clockwise): A, D, B, C. This allows the following calculations: SUM=A+B+C+D, LR=(A+C)-(B+D), UD=(A+D)-(C+B), and DIAG=(A+B)-(C+D). The beam ellipticity (or a parameter directly proportional to it; it depends on the rotational alignment with the detector and the resonator that determines the relevant mode) can be measured using the DIAG calculation as can be seen in FIG. 2. A zero DIAG signal indicates no beam ellipticity, which means that the geometric beam shape of the incident signal 1 is circular. A non-zero DIAG signal indicates the presence of an elliptical incident signal 1 on the quadrant photodiode 3. An opposite sign of the DIAG signal indicates an opposite ellipticity. The quadrant photodiode 3 can be an InGaAs-based or silicon-based detector. The signal is further converted to a voltage. Furthermore, an error signal 9 is generated using a transimpedance filter (AMP) 5, e.g. using standard analog electronics with an operational amplifier, and a controller 6 (not shown in FIG. 1). The angles of incidence, i.e. the angles of incidence P7 and Y8 of the incident optical signal 1 on the detector 2, can be changed by tilting the detector 2 with respect to the optical axis of the incident signal 1 or by tilting the optical axis of the incident signal 1 with respect to the main development plane of the detector 2 (e.g. by tilting the mirror upstream of the quadrant photodiode 3). The relative change in the geometric beam shape of the incident signal 1 is most accentuated when the laser light is resonant with the optical seed signal 10 (not shown in FIG. 1, see FIG. 3).Therefore, the method is most sensitive at the transmission peaks of the incident signal 1 which are also the transmission peaks of the optical seed signal 10. The distance between the laser 11 and the detector 2 is determined by the TEM. 00 This affects the Guoy phase between the axial mode and the second spatial mode. Therefore, the image obtained on detector 2 can be adjusted by adjusting the position of detector 2 relative to laser 11. The detector 2 and laser 11 positions are fixed during calibration and kept constant during operation. There are axial distances where the error signal 9 is purely dispersive, and other axial distances where the error signal 9 is purely absorptive or any signal shape between absorptive and divergent.

[0148] FIG. 2 shows the frequency dependence of the error signal 9 with respect to the optical seed signal 10 (not shown in FIG. 2) in the vicinity of the '00' mode resonance of the laser 11. Here, the DIAG signal is used as the error signal 9. This error signal 9 is divergent. Note that in this case the native output of the laser 11 has a circular geometric beam shape and the optical seed signal 10 also has a circular beam shape. The error signal 9 has a zero crossing at zero detuning of the laser 11 with respect to the optical seed signal 10. This also corresponds to a substantially circular incident signal 1 on the detector 2. Depending on the detuning, the geometric shape of the incident signal 1 changes from circular to elliptical resulting in a non-zero error signal 9. The sign of the error signal 9 indicates the direction of the elliptical incident signal 1. At zero detuning, the transmission 12 of the incident signal 1 through the resonator of the laser 11 has a peak. The zero crossing of the error signal 9 is in the interval between the minimum 13 and maximum 14 of the error signal 9. The interval has a value of the error signal 9 that corresponds to the determined ellipticity of the incident signal 1. The incident signal 1 has an inverse ellipticity on the opposite side of the peak in transmission 12. This is used to determine the error signal 9, which is then used to control the laser 11 to stabilize the laser output signal 15. The incident signal 1 can be thought of as the interference between a resonant "00" mode leaking out of the laser 11 and a second order "+" mode reflecting directly off the laser 11 to form a phase reference. With this detection modality, the error signal 9 is insensitive to drifts and variations in alignment, since small misalignments will generate (essentially only) a first order mode in the modal decomposition of the incident signal 1. The slope of the error signal 9 allows it to be used to control the laser 11 to achieve locking of the laser to the optical seed signal 10.

[0149] FIG. 3 shows a schematic example of a closed loop control system 16 for controlling an injection-locked laser with an external optical seed signal 10. The laser 11 is a semiconductor laser 17 and generates an optical output signal 15. An optical isolator 18 having a seed injection port 19 is used through which the external optical seed signal 10 is injected into the laser 11. The frequency of the injected optical seed signal 10 is at or near the resonance of the laser 11. The optical isolator 18 prevents the optical output signal 15 from the laser 11 from re-injecting into the laser 11. The system includes an optical component 20 for propagating at least a portion of the output signal 15 from the laser 11 as an incident signal 1 to the detector 2. The optical component 20 is a beam splitter 21, which is used to split the output signal 15. The portion reflected by the beam splitter 21 is further directed to the detector 2 and is used as the incident signal 1 propagated onto the detector 2. The axial distance between the laser 11 and the detector 2 is such that the transmission peak 12 of the incident signal 1 coincides with a zero crossing of the error signal 9.

[0150] The beam splitter 21 reflects, for example, less than 1% of the incident intensity. The detector 2 is tilted with respect to the optical axis of the incident signal 1. The detector 2 is a QPD 3 as described in relation to FIG. 1. The controller 6 is configured to receive the signal from the sensor area 4 of the QPD 3 and derive an error signal 9 based on the geometric beam shape, in this case the ellipticity. The derivation of the error signal 9 includes applying an offset to the error signal 9 such that at a set point the error signal 9 is zero. The set point is determined by a target state of the laser cavity 22 of the laser 11. For example, the target state may be such that the optical seed signal 10 is at a TEM 00 This is achieved when the laser mode is perfectly mode-matched and / or the center frequency of the optical seed signal 10 is on resonance with the laser. The controller 6 is configured to control one or more operating parameters of the laser 11 based on the error signal 9 to stabilize the laser 11 at a set value. The operating parameters of the laser 11 with the laser cavity 22 include: the temperature of the laser resonator 22, the path length of the laser resonator 22, pumping of the laser cavity 22, The phase of the optical seed signal 10.

[0151] The system 16 can also be used to continuously shift the wavelength of the output signal 15 of the injection-locked laser 11 by shifting the wavelength of the optical seed signal 10. The wavelength of the optical seed signal 10 is adjusted to change the error signal 9. The change in the error signal 9 causes the controller 6 to control at least one operating parameter of the laser 11 to restore the relationship between the laser 11 and the optical seed signal 10, in other words to reach a set value. In this way, by shifting the wavelength of the optical seed signal 10, the wavelength of the optical output of the laser 11 can be shifted by tracing it with a closed control loop. The laser 11, the beam splitter 21 and the detector 2 can be integrated in a common micro-optics or silicon photonics platform (not shown).

[0152] The closed loop control system 16 implements a method for monitoring an optical signal 15 extracted from an optical cavity 42 (see, e.g., FIG. 7), which in this embodiment is a laser cavity 22. The method comprises: - receiving at least a portion of an optical signal (15) as an incident signal 1; - converting the incident signal 1 into an error signal 9 by application of a conversion arrangement, which comprises propagating the incident signal 1 to a detector (2) and deriving an error signal 9 based on the geometric beam shape of the propagated incident signal 1 to the detector 2 (see Figs. 1 and 2), in particular the beam shape ellipticity, for example proportional to the beam shape ellipticity (see Fig. 2), Here, the error signal 9 has a minimum value, in particular a minimum value 13 (see FIG. 2) and a maximum value, in particular a maximum value 14 (see FIG. 2), which minimum and maximum values ​​delimit an interval 59 of the error signal value (see, for example, FIG. 2 or FIG. 10), and the conversion arrangement configures the interval 59 such that the error signal 9 includes a zero crossing 58 (see FIG. 10) when the optical resonator 42 is in the target state.

[0153] FIG. 4 shows another closed loop control system 16 for controlling the self-injection-locked laser 11, in which a portion of the output signal 15 of the laser 11 is used to inject back into the laser 11. In comparison with the configuration of FIG. 3, a second beam splitter 23 is used to propagate a portion of the output signal 15 of the laser 11 to a frequency-selective back reflector 24. The optical isolator 18 is omitted. The second beam splitter 23 can have any splitting ratio, with a 50 / 50 splitting ratio being particularly useful in the illustrated configuration. The frequency-selective back reflector 24 can be any optical element that reflects at least a portion of the laser output back into the laser 11 for self-injection. It can be, for example, a mirror (which may not be frequency-selective in this case), an optical grating, a resonant optical cavity, or any optical element that reflects back at least a portion of the input optical power. The back reflector 24 preferably reflects back light of a specific wavelength in order to achieve a narrow linewidth of the optical seed signal 10 and thus a narrow linewidth of the output signal 15 of the laser 11. In this embodiment, it is beneficial that the optical axis of the incident signal 1 on the detector 2 has a detector impingement angle 7, 8 different from 0° with respect to the normal vector of the main extension of the active area of ​​the detector 2. This minimizes detector reflections that wrap around back into the laser 11. In addition, the impingement angles 7, 8 can be used to adjust the error signal 9 when calibrating the system 16. Alternatively, it is also possible to use the detector surface as a back reflector 24 for self-injection and use parameters other than the impingement angle to calibrate the error signal 9.

[0154] 5 shows another embodiment of a closed-loop control system 16 for controlling a self-injection-locked laser 11. The optical seed signal 10 is generated by propagating a portion of the output signal 15 of the laser 11 through an optical resonator 25 acting as a back reflector 24. The spectral transmittance of the optical resonator 25 depends on the optical path length of the optical resonator 25. Thus, by controlling the optical path length of the optical resonator 25, the optical seed signal 10 can be controlled, and thereby the output signal 15 of the laser 11 can be shifted.

[0155] The laser 11 has an output signal 15, which is split by a beam splitter 21. In this embodiment, the transmitted portion is further shaped by a lens 39 and propagates through an optical isolator 26. The beam splitter 21 reflects a portion of the output signal 15. A mirror 27 and a piezo mounted mirror 28 propagate a portion of the output signal 15 to an optical resonator 25. Prior to the optical resonator 25, the optical output signal 15 is mode matched to the optical resonator 25 using a lens 29. The optical resonator 25 is composed of a second beam splitter 30, a second mirror 31 and a second piezo mounted mirror 32. The second piezo mounted mirror 32 is used to determine the spectral transmission through the optical resonator 25. The optical resonator 25 is configured such that the reflected intensity at the second beam splitter 30 is propagated onto a detection lens 33 and a detector 2, which is a QPD 3. The detection lens 33 is a convex spherical lens 34. The transmitted intensity through the second beam splitter 30 and the optical cavity 25 is reflected back towards the laser 11, thereby providing the optical seed signal 10 for self-injecting the laser 11.

[0156] A controller 6 (not shown in FIG. 5, see FIG. 3) derives an error signal 9 and controls one or more operating parameters of the laser 11 that are sensitive to the error signal 9. These parameters include: the temperature of the laser resonator 22, the path length of the laser resonator 22, pumping of the laser cavity 22, The phase of the optical seed signal 10. The phase and injection of the optical seed signal 10 can be manipulated by a piezo-actuated mirror 28 .

[0157] In this example, the laser 11 is a standard Eagleyard DFB 780 nm 100 mW laser. The external optical cavity 25 assembly is formed of three mirrors: convex [r=10 cm], flat, and concave. The cavity has a linewidth of 10 MHz. The L-shaped geometry of the optical cavity 25 allows for easy mode matching and also allows the reflected beam to be accessed as the input signal 1 to the detector 2. However, many different cavity geometries can be used, including a confocal two-mirror cavity, and only light circulating within the cavity can be self-injected into the laser unless the reflected beam is directly reflected back into the laser.

[0158] FIG. 6 shows another embodiment of a closed loop control system 16 for controlling the self-injection locked laser 11. Compared to the embodiment of FIG. 5, a third beam splitter 35 is used and a second detector 36 is arranged to receive a second injection signal 37 transmitted through the third beam splitter 35. The second detector 36 is a QPD. The embodiment of FIG. 6 is a combination of the embodiments of FIG. 4 and FIG. 5. The addition of the second detector 36 leads to a second error signal 38 (not shown, but similar to the error signal 9, see FIG. 1 and FIG. 2 for details) that is used to monitor the injection condition of the laser 11 (as in the embodiment of FIG. 4). The resulting error signal 38 is used in particular by the controller 6 (not shown in FIG. 6, see FIG. 3) to control one or more of the following operating parameters of the laser 11: the temperature of the laser resonator 22, the path length of the laser resonator 22, - Pumping of the laser cavity 22.

[0159] In this embodiment, the error signal 9 of detector 2 is used by controller 6 to stabilize the phase of the optical seed signal 10. The signals incident on detector 2 and detector 36 are coupled to a resonant TEM of an optical cavity, such as a laser cavity. 00 In the case of detector 2, the signal can be understood as the interference of the TEM of the optical cavity 25 with a second order "+" mode which reflects directly from the optical cavity to generate a phase reference. 00The signal at the second detector 36 is based on the interference of the mode with the output signal 15 of the laser 11. However, the signal at the second detector 36 is based on the interference of the TEM of the laser cavity 22. 00 The phase reference is based on the interference of the mode with the transmitted signal through the optical cavity 25. This is reflected back to the laser 11 to act as the optical seed signal 10, and is also reflected from the laser cavity 22 to form a phase reference. It is therefore beneficial to use an error signal 9 based on the signal at the (first) detector 2 to adjust the phase between the laser cavity 22 and the optical cavity 25 (which is also the phase of the optical seed signal) by means of a piezo-mounted mirror 28, and a second error signal 38 based on the signal at the second detector 36 to control one or more remaining operating parameters of the laser 11. The use of two detectors 2 and 36 in this embodiment results in a more stable configuration, since more information on the injection locking is obtained and used to control the operating parameters.

[0160] FIG. 7 shows a schematic representation of an embodiment of an apparatus 40 for frequency locking a laser 41 to an optical cavity 42. In this embodiment, the apparatus 40 includes a laser 41, and the optical cavity 42 is formed as a three-mirror optical cavity. The laser beam from the laser 42 is coupled into a polarizing optical fiber 43 at an input collimator 44. The optical fiber 43 is configured with a 50:50 polarization-maintaining fiber splitter 45, and one of the outputs of the fiber splitter 45 is used as the "useful" light. Furthermore, a polarizing fiber (PZF) 46 for emitting the laser beam from the polarizing fiber 43 and a fiber collimator 47 are provided. The emitted laser beam is directed to a beam shaper 48 for shaping the laser beam so that it obtains ellipticity. In this embodiment, the beam shaper 48 comprises a pair of cylindrical lenses (CLP) 49, the focusing axes of which are oriented perpendicular to each other. The laser beam is then irradiated onto a first polarizing beam splitter (PBS) 50, which is rotatable.

[0161] The laser beam is then directed to the resonator input mirror 51 of the optical resonator 42. The laser beam reflected from the optical resonator 42 is directed to a detection lens (DL) 52, which may be a convex spherical lens (e.g. 100 mm). The detection lens 52 serves to focus the laser beam reflected from the optical resonator 42. The laser beam then passes through a second polarizing beam splitter 53. The device 40 further comprises a detector 54 for determining an error signal proportional to the beam ellipticity of the laser beam focused by the detection lens 52. In this embodiment, the detector 54 is a quadrant photodiode 55, which is configured to determine the error signal based on the diagonal signal of the quadrant photodiode 55. The diagonal signal consists of the difference between the sums of the signals from the diagonal sensor areas of the quadrant photodiode 55.

[0162] The device 40 according to the embodiment shown in Fig. 7 performs a method for monitoring an optical signal 15 extracted from an optical resonator 42. In this case, the optical signal 15 is a laser beam reflected from the optical resonator 42. The method comprises the following steps: - receiving at least a portion of the optical signal 15 as incident signal 1 (in this case the signal incident on the quadrant photodiode 55); - converting the incident signal 1 into an error signal 9 (see Figure 10) by application of a conversion arrangement that includes propagating the incident signal 1 to a detector 2 (in this case a quadrant photodiode 55) and deriving an error signal 9 based on the geometric beam shape (see Figure 8C) of the propagated incident signal 1 to the detector 2, in particular the beam shape ellipticity, for example proportional to the beam shape ellipticity. Here, the error signal 9 has a minimum, in particular a minimum value 13 (see FIG. 2 or FIG. 10) and a maximum, in particular a maximum value 14 (see FIG. 2 or FIG. 10), which minimum and maximum values ​​delimit an interval 59 of the error signal value (see, for example, FIG. 10), and the conversion arrangement configures the interval 59 such that the error signal 9 has a zero crossing 58 (see FIG. 10) when the optical resonator 42 is in a target state, which in this case refers to an optimal frequency lock of the laser 41 to the optical resonator 42.

[0163] Figures 8a, 8b and 8c conceptually show the generation of the error signal on which the laser is locked (the dotted lines in Figure 7 show the use of the error signal as feedback to lock the laser, e.g. by adjusting the laser current).

[0164] Figure 8a shows a schematic of a triangular ring resonator geometry and its associated Hermite-Gaussian mode resonances. In this embodiment, the optical resonator 42 consists of two flat mirrors and one convex mirror (e.g., with a radius of curvature of 5 cm). Furthermore, the inset shows the spatial amplitude structures of the "00" and "+" modes. Among the intrinsic cavity modes, the "+" mode appears due to the superposition of the "02" and "20" modes. The alignment drift belongs to the "01" and "10" modes. Here, ν FSR represents the free spectral range, and n represents an integer. In this example, the resonator parameters are: round trip length 7.01 cm, v FSR =4.28GHz, '00' mode finesse 195.

[0165] Figure 8b shows the beam ellipticity detection using a quadrant photodiode and the mode decomposition of a slightly elliptical beam, where ε is a small complex number. As can be seen in Figure 8b, the DIAG=(A+B)-(C+D) function of the quadrant photodiode 55 gives the beam ellipticity. Furthermore, an error signal is generated by the transimpedance filter (AMP).

[0166] FIG. 8c shows the frequency dependence of the error signal for light reflected from the cavity near the "00" mode resonance, where κ is the cavity total linewidth.

[0167] The arrangement of the detection lens 52 and detector 54 will be described below in conjunction with FIGS.

[0168] Figure 9 shows the position of the relevant elements in the beam path and the resulting beam characteristics, in particular the in-plane and out-of-plane beam waists and the cavity mode beam waist, for the embodiment of Figure 7. 00TEM at about 5% of the peak power of the mode 02 and T.E.M. 20 The configuration of the optical elements for exciting the mode is shown. The in-plane and out-of-plane beam waists of the laser beam coincide before reaching the beam shaper 48 with a pair of cylindrical lenses 49. As can be seen, the beam shaper 48 gives the laser beam ellipticity. The laser beam is focused at the detector lens 52. The detector 54 is shown at a position where the in-plane and out-of-plane beam waists coincide with the beam waist of the resonant mode. The average cavity mode waist may be 74 μm, the in-plane beam waist 84 μm, and the out-of-plane beam waist 65 μm. Furthermore, an adjustment section 56 is shown that allows adjustment of the axial distance 57 of the detector lens 52 and the detector 54 in order to use the error signal for frequency locking of the laser. The dimensions in the figure (e.g. positions in cm in FIG. 9) are chosen for illustration purposes and may take different values.

[0169] Figure 10 shows the error signal and resonator transmission for the embodiment of Figure 7. The axial distance 57 is adjusted such that the zero crossing 58 of the error signal is within an interval 59 between a minimum 60 and a maximum 61 of the error signal, and such that the error signal value that corresponds to a predefined beam ellipticity of a substantially circular laser beam is within the interval 59 when the laser provides a laser beam at the resonant frequency of the resonator. The interval 59 in frequency is converted into an adjustment interval 56 of the axial distance 57. The error signal shown in Figure 10 can be used to lock the laser 41 to the optical resonator 42.

[0170] A second laser system can also be locked onto the opposite side of the optical cavity 42, mirroring the left side. In principle, up to three lasers can be locked simultaneously into this three mirror cavity. This can be scaled up by increasing the number of mirrors. Alternatively, if the laser wavelengths are sufficiently different, multiple lasers can be locked onto the same input mirror, with subsequent separation of each wavelength by filters.

[0171] FIG. 11 shows the error signal and the transmission. Even if the background offset is zero or minimized when the in-plane and out-of-plane beam waists are exactly aligned with the cavity mode beam waists, the zero crossing 58 of the error signal does not exactly correspond to the plateau of the transmission peak of the optical cavity 42, as can be seen in FIG. 11. Therefore, it is further desirable to adjust the axial distance 57 of the detection lens 52 and the detector 54 so that the transmission peak of the laser beam reflected from the optical cavity 42 is aligned with the zero crossing 58 of the error signal. FIG. 11 shows the error signal when the beam waists are aligned ("initial") and when the transmission peak and the zero crossing 58 are aligned ("optimized").

[0172] FIG. 12 shows a schematic of the axial distance 57 between the detection lens 52 and the detector 57. It is also possible to center the laser beam on the detector 57. This is achieved by translating the detection lens 52 (and / or the detector 54) in a direction 62 perpendicular to the optical axis 63 ("LR" or left-right direction and "UD" or up-down direction) such that the left-right and up-down signals of the quadrant photodiode 55 are minimized. The left-right signal LR=(A+C)-(B+D) and the up-down signal UD=(A+D)-(C+B) each consist of the difference of the sums of adjacent sensor areas of the quadrant photodiode 55, and the left-right and up-down signals use the sums of different halves of the adjacent sensor areas. When the axial distance 57 is adjusted, it is desirable to recenter the laser beam. It is therefore desirable that these steps are performed reproducibly and repeatedly. That is, after each adjustment of the axial distance 57, the LR and UD signals should be minimized again. By repeating this process, the zero crossing 58 is optimized to exactly coincide with the transmission plateau, as shown in the "optimized" error signal of FIG.

[0173] Figures 13a-c show the effective resonator under weak value emphasis, i.e. pre- and post-selection of polarization. In particular, Figure 13a shows the use of first and second polarizers in the embodiment of Figure 7 and Figure 14 described below. The input polarizer 11 and the output polarizer 53 are used to generate a reflection operator JPEG2025502754000008.jpg86 weak value r w 13b shows the normalized reflected power measured for the different post-selection configurations. The curves correspond to the off-resonance residual power fraction |<ψ 2 |ψ 1 >| 2 The 1 / 2 curve is identical to the original resonator reflection curve, and the 1 / 60 curve is the impedance-matched (γ'=1) configuration. The dashed-dotted lines represent the intensity contribution levels from non-"00" mode components. Figure 13c shows the (theoretical) r w 4 shows the real and imaginary parts of , and illustrates the advantage of post-selection: an effective increase in the slope of the error signal and a minimization of the optical power to the detector 54 near the lock point are achieved.

[0174] Figure 14 shows two copies of the lock-up configuration used to demonstrate the performance of the stabilization scheme. The two copies share the same optical resonator 42, but otherwise each (uniquely) comprises the same elements as the device 40 described in the context of Figure 7. Therefore, reference is made to the description of Figure 7, where identical elements are designated by the same reference numbers. Two separate lasers 41 were stabilized to the two degenerate counter-propagating "00" modes of a triangular ring resonator 42. The distance between a pair of cylindrical lenses 49 determines the amount of light in the "+" mode, which corresponds to about 10% of the total power used in the experiment (about 400 μW).

[0175] Comparison between different optical frequency stabilization systems can be achieved by evaluating their performance as fractional instability versus cavity linewidth. Typically, absolute instability scales linearly with cavity linewidth, since a typical source of problematic instability comes from variations in the error signal shape. In this experiment, a 22-MHz full-linewidth cavity and two 780 nm external cavity diode lasers with intrinsic linewidth of 500 kHz were used.

[0176] Figure 15a shows the locking instability over the averaging time of the experimental setup shown in Figure 14 compared to other stabilization schemes. The performance of the implemented scheme while locked to the "00" mode resonance is shown. The performance was 10 times higher for averaging times between 0.25 and 100 seconds. -7 The lock instability region of 5×10 is reached, surpassing the previous state of the art, with a 10 second averaging time of 5×10 -7 An instability of 10 ... -Salomon (Salomon, C., Hils, D. & Hall, JL Laser stabilization at the millihertz level. J. Opt. Soc. Am. B5, 1576 (1988).): Transmission modulation method; -Slagmolen (Slagmolen, BJJ, Shaddock, DA, Gray, MB & McClelland, DE Frequency stability of spatial mode interference (tilt) locking. IEEE J. Quantum Electron. 38, 1521-1528 (2002).): Double-pass "tilt locking" method; -Matei (Matei, DGet al. 1.5μm Lasers with Sub-10mHz Linewidth). Phys. Rev. Lett. 118, 263202 (2017).): PDH method; -Kessler (Kessler, T. et al. A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity. Nat. Photonics 6, 687-692 (2012).): Out-of-loop noise floor characteristics of the PDH method with and without RAM stabilization. (arXiv:1112.3854). Data were extracted and expressed relative to the cavity linewidth used in each study.

[0177] Figure 15b shows a comparison of the achieved instabilities with and without post-selection at the top and the corresponding beat note traces at the bottom.

[0178] Figure 15c shows the estimated limits on stability for operation with and without post-selection. The main contribution from beam shape noise is shown in solid lines, and the secondary contributions in dashed lines. The secondary contributions are broken down as follows: the measured in-loop error signal noise (X), the detector 54 dark noise (◇), and the laser 41 intensity drift noise (◯).

[0179] More details of experiments carried out with the configuration shown in Figure 14 are given below, however it should be noted that different parameters may be used and should not be considered as limiting the invention.

[0180] The optical cavity 42 consists of one highly reflective curved mirror (5 cm radius of curvature) and two low reflectivity (-98.93 for p-polarized light at 30° incidence) flat mirrors. The reflectivities are chosen for simplicity so that the linewidth of the cavity is larger than that of the laser 41 (500 kHz). The mirrors are epoxy mounted to an aluminum block with three holes that form an approximately equilateral triangle. A fiber collimator 47 and a pair of cylindrical lenses 49 forming the beam shaper 48 are mounted in a cage assembly that sits on a five-axis tip-tilt translation stage. This assembly, along with the axial translation capability of the fiber collimator lenses, facilitates mode matching into the optical cavity 42. The optical cavity 42, along with the other optics, are all mounted directly on an optical table and enclosed in a metal box for stability against temperature and air variations. The aluminum cavity spacer is stabilized to within 1 mK to control the cavity length.

[0181] To experimentally determine the cavity mode reference profile (see also Figure 9), light is back-coupled into the "00" mode from one of the mirrors and the mode profile emerging from the resonator 42 (along a path to a quadrant photodiode (QPD) 55) is analyzed with a CMOS camera. By adjusting the position of the optical elements, the incident beam profile is adjusted to match the reference mode upon reflection from the resonator. Once aligned, all modes except the "00" and "02"-"20" modes are suppressed, and the latter two have equal transmission levels.

[0182] Following alignment of the input light, the position of the QPD 55 is aligned. To center the beam on the QPD 55, the detector lens (DL) 52 is translated while monitoring the left-right and up-down signals from the QPD 55. When the beam is centered, a baseline offset on the error signal is due to the axial displacement of the position of the QPD 55 from the position where the beam has a circular profile. The position of the QPD 55 can in principle be adjusted to eliminate this offset. However, the natural astigmatism of the cavity 42 also contributes to the baseline offset. This also results in a shift of the transmission peak from the error signal zero crossing 58. It is therefore possible to adjust the axial position of the QPD 55 until the transmission peak and the error signal zero crossing 58 are coincident. If this coincidence is not present, then mutual coupling of intensity noise into frequency noise will appear. Electronic backend:

[0183] The QPD55 and feedback circuit are homemade. Every quadrant of the QPD55 is amplified independently with a transimpedance gain of 380kΩ. Then, the operational amplifier circuit performs operations to simultaneously access the diagonal, left / right, top / bottom and sum signals. These respectively provide the cavity lock error signal, the beam centering signal and the total light intensity reflected from the cavity.

[0184] The signals from the diagonal channels go to an analog feedback circuit and are fed into the laser current to maintain resonance with the resonator of the laser 41 when locked. The feedback loop contains two integrators (1 / f2 response) from DC to 20 kHz and one integrator (1 / f response) from 20 kHz up to a locking bandwidth of 100 kHz, providing lock tightness without significant limitations on stability (see crosses in Figure 15c). All input and output signals to the feedback circuit are differential for ground loop elimination, which is critical for performance.

[0185] The QPD55 and implemented amplifier circuitry are shot-noise limited at frequencies above 300Hz. Below this frequency, the amplifier noise dominates. However, the lock stability observed on time scales greater than 10ms is not related to either shot noise or amplifier noise. This is typical for other experiments operating on such time scales. System performance, comparison, and noise characterization:

[0186] Assuming that the two copies of the laser cavity locking configuration have the same performance, the actual instability for one copy is given by 1 / √2 of the measured laser beat note instability, since the independent noises add in quadrature. This correction has already been incorporated in the description of Fig. 15 and is a standard procedure. This generally underestimates the instability that can be achieved, since if one of the two configurations had a higher performance, it would require dividing by a larger number to extract the performance of that configuration. In the present case, the analyzed beam shape noise, which limits the performance, was found to be equal for the two copies, so the performance of both copies is approximately equal.

[0187] For comparison with previous state-of-the-art, instability data was extracted from the studies cited in Figure 15 (see below) using a webplot digitizer, with the data expressed as a percentage of the cavity linewidth utilized in each study. In selecting specific publications for comparison, special consideration was given to studies that included two or more similar configurations in order to obtain actual measured stability via laser beat notes. Note that it is possible to infer lock stability from a single copy system by recording the in-loop error signal and utilizing an appropriate conversion factor. This method is not necessarily reliable since the feedback loop will suppress the error signal regardless of its physical origin. For example, the in-loop error signal in this study was continuously at 10 -8 The instability scale is averaged to , but the actual stability limit is imposed by the deviation of the error signal from the actual frequency value due to beam shape noise. Among the comparative data, the "Slagmolen" and "Salomon" curves, together with the results presented in this work, were obtained by locking two lasers to the same cavity, while the "Matei" curve was obtained by comparing two lasers locked to separate cavities and is limited by the thermal noise of the utilized cavity.

[0188] The estimated limits to stability shown in Figure 15c (see bottom panel) were performed ex situ. Beam shape noise was measured with the cavity modes physically blocked; therefore, there is no light accumulation in the resonator. In this configuration, the input mirror directs the incident light directly to the QPD, allowing the identification of beam ellipticity variations in the incident beam. No further modeling was performed to account for the shape noise filtering effects that the resonator would perform while the mode is unblocked. QPD dark noise was measured by blocking the incident light and monitoring the error signal. Intensity drift noise was measured by simultaneously monitoring the beat note frequency of the laser and the sum channel of the QPD 55 while the laser 41 was locked to the resonator 42. Except for the intensity noise, the slope of the error signal on resonance was used to convert the measurements to frequency. For the intensity drift noise, the measured laser intensity dependence of the beat note during the locking operation was used. Note that no active intensity stabilization was performed in this study.

[0189] It was verified that the stability limitations actually come from beam ellipticity variations and not indirectly from beam pointing variations combined with non-idealities of the QPD detection system. By translating the QPD position perpendicular to the beam axis, a crosstalk separation between beam position and beam ellipticity of more than 200 times was observed. On the other hand, the voltage noise in the beam position signal during operation was only one order of magnitude larger than the voltage noise in the ellipticity signal, ruling out beam pointing crosstalk as the dominant noise source. A narrower laser linewidth would in principle improve stability on short time scales, assuming there are no other dominant defect sources in the system. In the present case, artificially widening the laser linewidth from 500 kHz to 1 MHz (by injecting current noise) did not degrade performance, suggesting that the linewidth is not a limiting factor. We also note that in this parameter region, no reduction in the laser linewidth occurs, since the locking bandwidth (100 kHz) is smaller than the initial laser linewidth. The advantage of the locking method considered in this work compared to the suboptimal "transmission modulation" method (Fig. 15a) is the use of a reflected beam, a property also shared by the PDH method. In this mode of operation, the locking bandwidth is not directly limited by the cavity linewidth. This is important in the context of narrow-linewidth, high-finesse resonators when trying to narrow the laser linewidth in addition to stabilizing the center frequency.

[0190] As a further supplement, it has been shown that the fundamental limits of many frequency locking techniques using optical shot noise are the same within a numerical range of one order of magnitude, and the present method is no exception.

[0191] Advantages of post-selection: In addition to the suppression of sensitivity to intensity noise and incident beam shape variations already discussed, post-selection reduces the sensitivity to beam pointing variations. These variations, which only affect the signals of the diagonal QPD channels quadratically and do not appear to cause limitations in this work, are nevertheless further reduced. Another non-ideality suppressed by post-selection is the adverse effect of astigmatism of the triangular cavity modes. Astigmatism causes the zero crossing of the error signal to shift away from the maximum of the transmitted intensity. This results in reduced performance through intensity-to-frequency noise conversion. As discussed in the alignment procedure, fine-tuning of the alignment can reduce this effect, but post-selection nevertheless significantly suppresses these issues, making alignment easier.

[0192] Qualitatively, the improvement provided by post-selection can be thought of as coming from a differential measurement made on two orthogonally polarized beams, one entering the resonator and one reflected directly from the resonator. Through post-selection, this differential measurement appears as a coherent process that improves the slope of the error signal for a given amount of light incident on the QPD (making it more sensitive to smaller frequency shifts). The post-selection process improves performance as long as the resulting intensity on the QPD does not fall below the equivalent electronic noise floor.

[0193] Along with all the improvements, we also observe that post-selection introduces a small sensitivity of the frequency locking point to changes in cavity length. Preliminary characterization indicates that this increased sensitivity can be attributed to the change in polarization of the light reflected from the cavity with changes in optical frequency, effectively changing the post-selection angle. Although the cavity length is necessarily a well-controlled parameter in frequency stabilization applications, it is not the dominant limitation in our implementation. Adaptation to different cavity shapes:

[0194] In Figures 7 and 14, a ring resonator is utilized where the incident and reflected beams follow separate paths. Different applications may require different resonator designs. For example, very low vibration sensitivity generally requires the use of a two mirror standing wave resonator. In this configuration, nothing changes in terms of the main method. The reflected beam follows the same path as the incident beam, but can be easily separated, for example, by using a quarter wave plate together with a polarizing beam splitter (as is usually done in the PDH method). Also, note that the particular mode spectrum of the resonator is irrelevant for the implementation. The only requirement for proper operation is that the TEM 00 The advantage of this approach is that the modes are not spectrally degenerate with the second-order modes, and therefore the second-order modes can have any common resonant frequency. This primary method can also be used for waveguide-like resonators that require evanescent coupling, e.g., dielectric whispering gallery mode resonators or fiber rings. One well-established coupling strategy is to use a prism, where the evanescent tail from the prism during total internal reflection couples light into and out of the resonator. In this way, the prism effectively connects the waveguide-based structure to free-space optics and looks like the input mirror of a free-space resonator for the purposes of applying the method developed in this work. Postselected beam expansion and weak values ​​of reflection coefficients

[0195] We now outline the derivation of the effective reflection coefficients and associated expressions used above. First, we model the cavity reflection without the polarization degree of freedom in the quantum mechanical language used for weak value descriptions, and then we take polarization into account.

[0196] In the first case, the problem has two degrees of freedom: the "mode shape" and the "channel". The states propagating in the first degree of freedom are the various TEM modes, and the states propagating in the second degree of freedom are the "incident" JPEG2025502754000009.jpg613, "Reflection" JPEG2025502754000010.jpg613, "Transparent" It will be displayed as JPEG2025502754000011.jpg613. The initial state is the direct product state of two degrees of freedom: Let JPEG2025502754000012.jpg738, where the state 'Φ' represents an arbitrary input mode shape. To phenomenologically describe the reflection from the cavity close to the '00' resonance, we use the unitary operator This defines JPEG2025502754000013.jpg797. If the "mode shape" is in state |0>, then the "channel" state is transformed into another unitary operator Expand with JPEG2025502754000014.jpg89, otherwise, in the incident state JPEG2025502754000015.jpg58 in reflection state Directly expand to JPEG2025502754000016.jpg69, where JPEG2025502754000017.jpg86 is the identity operator for the "mode shape" degrees of freedom, and |0><0| is the projection operator for the "00" mode. If we only consider the reflected part of the beam, we can express the effective expansion operator R^, which acts only on the input "mode shape" state |Φ>, as the state expansion: This procedure defines the effective expansion operator You get JPEG2025502754000019.jpg751, where JPEG2025502754000020.jpg839 is the amplitude reflectance coefficient, which is quantified experimentally.

[0197] Next, when the polarization degree of freedom is included, the initial state A more general unitary operator is introduced to expand JPEG2025502754000021.jpg752, where |ψ 1 > is the initial state of the polarization degrees of freedom. Using the above procedure, the initial "mode shape" and "polarization" state The effective expansion operator R^=R^ that only works on JPEG2025502754000022.jpg727 H +R^V is derived, JPEG2025502754000023.jpg897 and JPEG2025502754000024.jpg890 are obtained. In these expressions, r H and r V are the amplitude reflection coefficients with respect to the horizontal polarization state |H> and the vertical polarization state |V> with respect to the main axis of the cavity. Here, a potential relative birefringence phase shift Φ is explicitly included for the direct reflection from the tilted cavity input mirror. Defining R^ post |Φ>=<ψ‘ 2 |R^|Φ>|ψ 1 >, and further projecting onto the polarization state |ψ‘ 2 >, the final post-selection evolution operator R^ post =<ψ 2 |ψ 1 [[ID=23>>[r w |0><0|+(I - |0><0|)] is obtained. This acts purely on the input mode state |Φ>. Here, r w =<ψ 2 |r^|ψ 1 > / <ψ 2 |ψ 1 > is the weak value of the reflection operator r^ = r H |H><H|+r V |V><V|, and the overlap coefficient <ψ post of R^ 2 |ψ 1 > further represents the lossiness of the post-selection operation. For clarity, note that the effect of the birefringence phase shift is included in the definition of the post-selection state: |ψ 2 >=(|H><H|+|V><V|e -iΦ )|ψ‘ 2 >. Above, a small birefringence phase was ignored, but the asymmetry remaining in the curve of Fig. 15b may be due to this.

[0198] Next, r w regarding this experimental configuration is evaluated. By using a linear polarizer, the pre-selection state and the post-selection state are |ψ 1 >=cosθ 1 |H>+sinθ 1 |V> and |ψ2 >=cosθ 2 |H>+e -iΦ sinθ 2 |V>. This means that the weak value is r w =r H ((1 / 2)+B)+r V ((1 / 2)-B), and the overlap factor is <ψ 2 |ψ 1 >=cos(θ 2 +θ 1 ) / (B+(1 / 2)+e -iΦ (B-(1 / 2))), where the complex number B=(1+e -iΦ tan θ 2 tan θ 1 ) -1 In this study, the cavity is polarization-resolved, and the vertically polarized component is considered to be a far-off resonance, r H =1-γ / (1-i δ / (κ / 2) ), and r v = 1, where γ, δ, and κ are as defined in the text. This allows the weak value to be r w =1-γ(B+(1 / 2)) / (1-i δ / (κ / 2) ) Finally, by applying the small birefringence phase approximation Φ<<1 and expanding Φ into a power series, we arrive at the formula used in the text. As a result, JPEG2025502754000025.jpg1446(2) JPEG2025502754000026.jpg14118(3) Here, γ'=Aγe -iΦ(A-1) And A=(1 / 2)+B| Φ=0 The successful posterior selection probability can be expressed as follows: JPEG2025502754000027.jpg970 Here, the relevant post-selection angle is cos(θ 2 +θ 1 ) 2 We have assumed that Φ = 0. The notation used above further assumes that Φ = 0.

Claims

1. A method for monitoring an optical signal (15) extracted from an optical resonator (42), such as a laser resonator (22), comprising the steps of: receiving at least a portion of the optical signal (15) as an incident signal (1); - converting said incident signal (1) into an error signal (9) by application of a conversion arrangement, comprising propagating said incident signal (1) to a detector (2) and deriving an error signal (9) based on a geometric beam shape of said propagated incident signal (1) on said detector (2), in particular the beam shape ellipticity, e.g. proportional to the beam shape ellipticity; Including, said error signal (9) having a minimum value, in particular a minimum value (13), and a maximum value, in particular a maximum value (14); The minimum value and the maximum value delimit a range (59) of values ​​of the error signal (9), The transformation arrangement configures the interval (59) such that the error signal (9) crosses zero (58) when the optical resonator (42) is in a target state.

2. 2. The method of claim 1, wherein the conversion arrangement is configured such that the interval (59) includes values ​​of the error signal (9) that correspond to a substantially circular image of the propagating incident signal (1) on the detector (2).

3. 3. The method according to claim 1 or 2, wherein the detector (2) is a quadrant photodiode (3) and the deriving of the error signal (9) comprises determining a diagonal signal of the quadrant photodiode (3), the diagonal signal being composed of the difference of sums of signals from diagonal sensor areas of the quadrant photodiode (3).

4. The method according to any one of claims 1 to 3, wherein the propagation of the incident signal (1) comprises adjusting an angle of incidence (7, 8) of the incident signal (1) onto the detector (2).

5. The method according to any one of claims 1 to 4, wherein the propagation of the incident signal comprises shaping the incident signal (1), in particular by means of a shaping lens, preferably by means of at least one lens, in particular by means of at least a cylindrical lens.

6. The method according to any one of claims 1 to 5, wherein deriving the error signal (9) comprises offsetting the error signal (9) such that the error signal (9) is set to zero when the optical resonator (42) is in a target state.

7. The method according to any one of claims 1 to 6, wherein propagating the incident signal (1) comprises focusing the incident signal (1) onto the detector (2) using a detection lens (33), preferably using a convex spherical lens (34).

8. 8. The method of claim 7, wherein propagating the incident signal (1) comprises adjusting an axial distance (57) between the detection lens and the detector (2) such that a transmission peak of the incident signal (1) coincides with a zero crossing of the error signal (9).

9. 9. The method of claim 8, wherein adjusting the axial distance between the detection lens (33) and the detector (2) optionally comprises adjusting the axial distance (57) such that an in-plane incidence signal waist and an out-of-plane incidence signal waist of the incident signal (1) coincide with a beam waist of a resonant cavity mode of the optical resonator (42) before adjusting the axial distance (57) between the detection lens (33) and the detector (2) along the optical axis of the incident signal (1) such that the transmission peak of the incident signal (1) coincides with a zero crossing (58) of the error signal (9).

10. The method according to any one of claims 3 and 7 to 9, further comprising the step of centering the incident signal (1) on the quadrant photodiode (3) by translating the detection lens (33) and / or the quadrant photodiode (3) in a direction perpendicular to the optical axis of the incident signal (1) such that the left-right signal and / or the top-bottom signal are composed of the difference of sums of adjacent sensor areas (4) of the quadrant photodiode (3), the left-right signal and the top-bottom signal using the sums of different halves of adjacent sensor areas (4).

11. 11. The method of claim 10, wherein the steps of adjusting the axial distance (57) between the detection lens (33) and the quadrant photodiode (3) along the optical axis of the incident signal (1) and centering the incident signal (1) on the quadrant photodiode (3) are performed iteratively so that the transmission peaks of the incident signal (1) coincide with the zero crossings of the error signal (9).

12. A method for frequency locking a laser (41) to an optical resonator (42) by a method according to any one of claims 7 to 11, comprising the steps of: providing a laser beam from said laser (41); shaping the laser beam so that it acquires ellipticity; directing the laser beam into the optical resonator (42), the laser beam reflected from the optical resonator (42) being the incident signal (1) that is propagated to the detector (2); locking the laser (41) to the optical resonator (42) based on the error signal (9); The method further comprising:

13. Shaping the laser beam so that it acquires ellipticity is performed by a pair of cylindrical lenses (49) whose focusing axes are oriented non-parallel to each other, in particular perpendicularly, or Shaping the laser beam so that it acquires ellipticity is performed by a combination of lenses, wedge prisms, and / or anamorphic prism pairs, whereby an elliptical or astigmatic beam is generated; The method according to claim 12, wherein the method is at least one of the above.

14. 14. The method of claim 12 or 13, wherein the laser beam is polarized filtered before directing the laser beam to the optical resonator (42), and the incident signal (1) is polarized filtered, and optionally the optical resonator (42) is birefringent.

15. The method according to any one of claims 1 to 11, further comprising the step of adjusting the optical resonator (42) to reach the target state by minimizing the error signal (9).

16. The adjustment of the optical resonator (42) is adjusting the temperature of the optical resonator (42); Adjusting the optical path length of the optical resonator (42); The method of claim 15, comprising one or more of the following:

17. The method according to any one of the preceding claims, wherein the optical resonator (42) is the laser resonator (22) of a laser (11), in particular the optical resonator (42) of a semiconductor laser (17).

18. The method according to claims 16 and 17, wherein tuning the optical cavity (42) comprises the step of adjusting the current of the semiconductor laser (17).

Images