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 a key component in modern science and technology, enabling a vast number of applications. They are used in a wide range of measuring instruments, such as lasers, microscopes, and spectrometers. Their applications are diverse, ranging from gravitational wave detection to laser welding and facial recognition technology. To ensure the desired behavior of an optical resonator, such as a laser resonator, monitoring of its characteristics is necessary. Furthermore, to control a laser, it is necessary to measure and quantify the laser output and determine an estimate of the deviation of the laser's actual state from the nominal or reference state. [Background technology]

[0002] For example, stabilizing the laser frequency (which can be achieved by stabilizing and / or controlling the laser's optical resonator) is essential in atomic time preservation, gravitational wave detection, relativity experiments, atomic interferometers, and the quantum control of various systems such as atoms, nanoparticles, and mechanical vibrations. 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 relative to a reference optical resonator plays a crucial role in achieving state-of-the-art performance levels.

[0003] Another crucial aspect of quantifying the deviation of the target state of an optical resonator from a reference state is locking the laser into the optical resonator. In this case, the deviation from the reference state can be relative to the state of the laser that should be locked into the optical resonator.

[0004] Over the past several decades, numerous methods have been developed to address the challenge of finding appropriate measurement techniques for the state of optical resonators and suitable estimators for quantifying deviations from a reference state. For example, these estimators may suggest laser detuning related to the optical resonator. These methods are utilized, for instance, by side-of-fringe intensity methods, polarization-based methods, frequency modulation techniques similar to Pound-Drever-Hall (PDH), and finally, spatial mode interferometry such as tilt-rocking.

[0005] Diorico et al.'s "Laser-cavity locking at the 10^(-7) instability scale utilizing beam ellipticity" -7 The archive preprint arXiv:2203.04550(2022) “Laser Cavity Locking at an Instability Scale” presents a method for locking a laser to an optical cavity. A laser beam is directed to the optical cavity. The laser frequency resonates (or nearly resonates) with the cavity. The reflected beam is focused onto a four-segment photodiode. This method relies on the interference between the fundamental TEM00 mode and two spatiotemporal modes of the optical cavity, resulting in an elliptic beam on the detector. An error signal is derived based on the ellipticity, and this is used as feedback to achieve a stable laser cavity lock. This method is called “squash lock.” The paper states that the general squash lock method is applicable to stabilizing laser injection locks. However, the paper does not provide hints on how to apply the disclosed method to different contexts.

[0006] In "Stabilization of injection-locked lasers using spatial mode interference" by Ottaway, DJ et al., IEEE journal of quantum electronics 37.5(2001):653-657, a method for stabilizing the injection lock of a laser by generating an error signal using tilt locking is disclosed. This technique relies on the interference between the TEM00 mode and the first spatial mode of the laser output, which is measured by a quad-segment photodiode. The quad-segment photodiode is composed of the sum of two upper quarters and two lower quarters. Therefore, the quad-segment photodiode has essentially two active regions, each consisting of a combined sensor region. At zero frequency difference (resonance), the two outputs of the photodiode have equal amplitude but opposite phase. These outputs are subtracted, resulting in a zero output. When the frequency difference is not zero, TEM 00 Field TEM 01 It interferes more constructively with one and more destructively with the other. Therefore, the signal depends on the beam shift or intensity on the detector, as well as the detector's alignment with the beam. However, the error signal does not contain any information about 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 needs to be offset with respect to the optical axis of the slave laser's resonator. In summary, the error signal is sensitive to the misalignment of optical components, especially the detector. This signal is fed back to the piezoelectric transducer to control the incidence of the master signal and 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" (Optics letters 24.21(1999):1499-1501) by Shaddock, DA, MBGray, and DEMcClelland.

[0008] Known methods for monitoring and quantifying the state of optical resonators rely on complex mechanisms and are sensitive to drift, particularly alignment drift. [Overview of the project] [Problems that the invention aims to solve]

[0009] Therefore, an object of the present invention is 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 from a nominal or reference state. This method reduces complexity, is purely passive, is particularly insensitive to alignment drift, and / or has a more robust implementation. [Means for solving the problem]

[0010] This is achieved by monitoring the optical signal extracted from an optical resonator, such as a laser resonator, and it is, - The step of receiving at least a portion of the optical signal as an incident signal, - The step of converting the incident signal to an error signal by applying a conversion configuration which includes propagating the incident signal to a detector and deriving an error signal based on the geometric beam shape of the incident signal propagating to the detector, particularly the beam shape ellipticity, for example proportional to the beam shape ellipticity. Here, the error signal has a local minimum, particularly a minimum value, and a local maximum, particularly a maximum value, where the local minimum and maximum value demarcate intervals in the error signal, and the conversion configuration constitutes intervals in which the error signal has zero-crossings when the optical resonator is in a target state.

[0011] Optical resonators include, for example, linear resonators and ring resonators. Optical resonators also include, for example, two-mirror resonators and three-mirror resonators. However, any number of mirrors and resonators of any shape can also be used. Optical resonators may be free-space resonators or non-free-space resonators such as microchip-based optical resonators. For example, an optical resonator may be a whispering gallery resonator, where light circulates around a sphere or ring (inside the material). An optical resonator may also be a fiber ring, in 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 the cavity of a laser, i.e., an optical resonator. In this case, the optical signal may be the output signal of the laser. The optical signal extracted from the optical resonator may be the optical signal reflected by the optical resonator. For example, when a laser is locked into an optical resonator, the laser can emit light onto or into the optical resonator. Some of the laser light may be reflected by the resonator. By option, the optical signal may include both the signal leaking from the optical resonator and the signal reflected by the optical resonator.

[0013] The output signal can be split, for example, by a beam splitter, and a portion of the optical signal extracted from the optical resonator can be received as the incident signal. It is also possible to use the entire output signal as the incident signal by choice.

[0014] As a result, a conversion configuration is applied to the incident signal. The conversion configuration includes propagating the incident signal to a detector and deriving an error signal based on the geometric beam shape of the propagated incident signal. Therefore, the conversion configuration includes manipulating the incident beam, for example by optical components, detecting and quantifying the incident signal, and deriving an error signal. The error signal is an estimator of the deviation of the actual state from the reference state of an optical resonator, such as a laser optical resonator. The conversion configuration takes the incident signal as input and outputs an error signal based on that incident signal.

[0015] To understand how error signals are generated, we should recall the Hermitian Gaussian (HG) spatial modes supported by optical resonators such as laser resonators. These are transverse electromagnetic modes (TEM). mn ) and a resonator, such as a laser resonator, can support it. 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 odd or even. Starting with perfectly aligned optical modes entering the optical resonator, odd modes are induced by either tilting or shifting the input beam, which deviates the input beam from its aligned state. Even modes, on the other hand, are induced by focusing / mode mismatch and are independent of the tilting or shifting of the input beam.

[0016] Early spatial interference implementations, such as tilt-lock, involved tilting / misaligning the input beam incident on the optical resonator to achieve primary interference (TEM). 01 and TEM 10 It depends on inducing the mode. What we are interested in here is the basic HG mode TEM, which is displayed as "00". 00 And, certain secondary HG modes and secondary TEM modes that display "+" 11(such) modes, all of which are even modes. The even modes are induced by mode mismatch, making the method of the present disclosure insensitive to misalignment drifts that generate first-order odd modes. For example, a slightly elliptical beam with horizontal / vertical orientation can be mathematically decomposed into a dominant "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, modes of different orders are generally non-degenerate, so that only one of them resonates, resulting in a differential phase shift between the two modes. This mechanism allows the output signal of the optical resonator to acquire different geometric beam shapes, in particular opposite ellipticities, on opposite sides of the optical resonator resonance. To utilize this effect, the geometric beam shape, in particular the beam shape ellipticity or the beam mode shape ellipticity, of the output signal of an 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-order 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. With respect to cavity modes, the spatial decomposition of a slightly elliptical beam oriented diagonally is mostly "TEM 00 " mode, and the contribution of the "TEM 11 " second-order mode is small. In such a beam, the phase difference between these two modes encodes information about the deviation from circularity. As a result, the shape of the output signal changes. This mechanism causes the output signal to obtain opposite ellipticities on both sides of the resonance, whereby a determinable error signal can be generated. To utilize 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] Generally, the derivation of an error signal based on the geometric beam shape of an incident beam means that the error signal is a function in which at least one parameter of the function characterizes the geometric beam shape of the incident beam. Specifically, the error signal can be directly proportional to the geometric beam ellipticity. Beam ellipticity can be defined according to ISO 11145:2018. Alternatively, eccentricity may be used.

[0019] Before the method is used, the method is started or initially calibrated. A setpoint is set by starting with an optical resonator in the target state. The terms “setpoint,” “reference point,” and “target state” are used synonymously for the purposes of this disclosure. An error signal is derived from this setpoint, indicating the deviation from that setpoint. As described above, the deviation from the setpoint can be quantified using the relative change in beam shape to the beam shape at the setpoint.

[0020] The error signal has local minimums, particularly minimums, and local maximums, particularly maximums, which delineate intervals in the error signal value. The transformation configuration constitutes an interval in which the error signal crosses zero when the optical resonator is in the target state. The error signal is zero at the setpoint when the optical resonator is in the target state. A zero error signal indicates that the actual state of the optical resonator is not deviating from the target state. The error signal has a sign and a magnitude; the sign arbitrarily indicates the direction of the deviation of the actual state from the target state, that is, whether the operating parameters of the optical resonator, particularly the laser resonator, should be increased or decreased to reach the target state. The magnitude of the error signal can indicate the magnitude of the necessary change of at least one operating parameter to reach the setpoint.

[0021] For example, at least one operating parameter of an optical resonator is one or more of the temperature of the optical resonator and the path length of the optical resonator. Generally, the path length of an optical resonator is extremely important for the mode distribution of the optical resonator. Temperature affects the path length because it can increase or decrease depending on temperature changes due to thermal expansion. The temperature can be selectively changed by individual heating elements such as a Peltier element or an electric heater element, and individual cooling elements such as a fan.

[0022] Zero-crossing of an error signal occurs when the error signal lies within the interval between its minimum and maximum values. Zero-crossing means the error signal has 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 can 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 can 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] By choice, the local minimum is the minimum value. By choice, the local maximum is the maximum value. By choice, there are no further extreme values ​​between the local minimum and the local maximum. The error signal includes the interval from the local minimum to the local maximum. The local minimum and local maximum therefore demarcate the interval of the error signal value.

[0024] The detector may be a photodiode, such as an avalanche photodiode. The detector may also be a phase plate or a mode converter, or may be composed of a phase plate or a mode converter. The detector is capable of converting 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 present invention is not limited to any particular form of physical signal that the detector generates based on the incident signal.

[0025] For example, the detector may be a photodiode array consisting of at least three photodiodes, and the derivation of an error signal includes determining the geometric beam shape of the incident signal. By using a 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 required to differentiate between first-order and second-order (or higher-order) spatial modes. In principle, the more photodiodes used, the better the signal shape can be approximated. Generally, determining the geometric beam shape of the incident signal means determining the parameter value of at least one parameter that characterizes the geometric beam shape. Preferably, the photodiodes are arranged in a rectangular structural grid. Minimizing the distance between individual photodiodes is beneficial to make the measurement most sensitive to changes in beam shape.

[0026] Optionally, the transformation configuration constitutes an interval containing error signal values ​​corresponding to a substantially circular image of the incident signal propagating to the detector. Optionally, the error signal is substantially zero with respect to the circular image of the incident signal on the detector and / or the incident signal on the detector. A substantially circular incident signal can therefore correspond to the optical resonator being in a reference state. The error signal can be determined using an incident signal that acquires an inverse ellipticity for the opposite deviation from the reference state. The determined error signal can be considered as interference between the resonant "00" mode and the second-order "+" mode, both of which may leak from the optical resonator. Optionally, the zero-crossing of the error signal may be the point of inversion symmetry of the error signal. Since the sign and magnitude of the error signal can provide an index of the magnitude and direction of the optical resonator's detuning relative to a setpoint, a substantially zero error signal value for a substantially circular incident signal can result in an easily interpretable error signal. For example, the temperature of the optical resonator and / or the path length of the optical resonator can detune and result in a non-zero error signal. The sign and magnitude of the error signal can provide indication of which direction and by how much operating parameters, such as the temperature of the optical resonator, should be adjusted to reach the set value.

[0027] Optionally, the detector may be a quad-segment photodiode, and the derivation of the error signal involves determining the diagonal signals of the quad-segment photodiode. These diagonal signals are composed of the difference between the sums of the signals from the diagonal sensor regions of the quad-segment photodiode. Thus, the beam shape of the incident signal, particularly the beam ellipticity, can be evaluated quite easily. Specifically, a quad-segment photodiode is a photodiode having four divided sensor regions. The gaps between the sensor regions can be optimized to be as small as possible. Specifically, a quad-segment photodiode (QPD) generates a current proportional to the input light from each of the four sensor regions. The signals from each sensor region are denoted as A, B, C, and D, and on the quad-segment photodiode, the sensor regions 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, meaning the incident signal has a circular beam shape. A non-zero DIAG signal means an elliptic beam is present on the four-segment photodiode. A reversed sign in the DIAG signal indicates inverse ellipticity. The four-segment photodiode may be an InGaAs-based or silicon-based detector. The signal can be converted to a voltage (e.g., using standard analog electronic equipment with an operational amplifier).

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

[0029] Optionally, the propagation of the incident signal includes shaping the incident signal, particularly using shaping lenses, preferably at least one lens, especially at least a cylindrical lens. Shaping the incident signal allows calibration of the initial error signal in the target state of the optical resonator. The error signal can be set to zero initially. If the detector is a QPD, shaping the incident signal can make the DIAG signal zero. 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 case, the DIAG signal becomes zero. The DIAG signal can optionally be used as an error signal without further operation. The position and alignment of the beam shaping optical system for shaping the incident beam, particularly at least one lens, preferably at least one shaping lens, especially at least one cylindrical lens, are set once during calibration and are 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 determining the axial distance between the detection lens and the detector such that the transmission peak of the incident signal coincides with the zero-crossing of the error signal. The transmission peak refers to the resonance between the external optical signal and the optical resonator, for example, an external optical seed signal and an injection-locked laser. When resonance occurs, the reflected signal in the optical resonator is minimized. The axial distance between the detection lens and the detector represents their distance along the path of the incident signal. In this method, the positions of the detector and detection lens are fixed during calibration and maintained constant during operation. Optionally, the axial distance may be set by adjusting the error signal curve in an adjustment interval. The relative change in the geometric beam shape of the incident signal is most pronounced at the time of resonance between the optical resonator and the external signal, for example, the optical seed signal. Therefore, this 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 set by adjusting the error signal curve in an adjustment interval.

[0031] In another optional embodiment, the propagation of the incident signal involves determining the axial distance between the optical resonator and the detector such that the transmitted peak of the incident signal coincides with the zero-crossing of the error signal. The transmitted peak corresponds to the minimum reflection of the reflected signal in the optical resonator. That is, the SUM channel of the four-segment photodiode shows a dip (the SUM channel of the four-segment photodiode is specifically for measuring the reflection curve). The positions of the detector and the optical resonator are fixed during calibration and maintained constant during operation. The distance between the laser and the detector is TEM 00 This affects the Guoy phase between the mode and the second-order spatial mode. Therefore, the image obtained on the detector can be adjusted by adjusting the position of the detector relative to the optical resonator. There is an axial distance at which the error signal is purely dispersive, and another axial distance at which the error signal is purely absorptive or has any signal shape between absorptive and dispersive. Preferably, the error signal is dispersive.

[0032] Optionally, deriving the error signal involves offsetting the error signal to set it 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 in the detector. These signals are used depending on the estimator of the shape of the incident signal. For example, if the detector is a QPD, the ellipticity of the signal can be estimated using a DIAG calculation. The error signal can be initialized to zero during calibration. For example, since a DIAG calculation can produce a residual signal even if the incident signal is substantially circular, an additional offset is applied to initialize the error signal to zero at the set point.

[0033] In another embodiment, an imbalance detection operation is available. For example, instead of the DIAG operation, an IDIAG imbalance operation can be used. IDIAG = ε1(A+B) - ε2(C+D), where ε1 / ε2 ≠ 1, and ε1 and ε2 can be initialized to set the error signal to zero. This makes it possible to adjust the background offset of the diagonal channel to give the error signal a zero crossing at a desired state (e.g., at a set point).

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

[0035] Optionally, adjusting the axial distance between the detection lens and the detector optionally includes adjusting the axial distance along the optical axis of the incident signal between the detection lens and the detector so that the in-plane and out-of-plane incident signal waists of the incident signal match the beam waist of the resonant cavity mode of the optical resonator (in particular, the average beam waist of the cavity mode, which is the average beam waist of the in-plane and out-of-plane mode), before adjusting the axial distance along the optical axis of the incident signal so that the transmission peak of the incident signal matches the zero-crossing of the error signal. Optionally, this is done before adjusting the axial distance between the detection lens and the detector along the optical axis of the incident signal so that the transmission peak of the optical signal reflected from the optical resonator matches the zero-crossing of the error signal. Optionally, additionally 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 (mutually) match. The baseline offset of the error signal is due to the axial displacement of the detector position from the position where the beam takes a circular profile (ignoring point aberration). The axial distance can be selected so as 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 on its own. If there is no astigmatism, these two will yield the same result. In this case, since no astigmatism remains in the mode, setting the baseline to zero automatically leads to a transmission plateau. In this method and apparatus for laser frequency locking, it is preferable 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%, ±2%, or ±1% of the beam waist of the resonant cavity mode of the optical resonator.

[0036] Optionally, the method further includes the step of centering the incident signal on the quadrilateral photodiode by translating the detection lens and / or quadrilateral photodiode in a direction perpendicular to the optical axis of the incident signal, such that the left-right and / or up-down signals of the quadrilateral photodiode are minimized. Here, the left-right and up-down signals are each composed of the difference in the sum of adjacent sensor regions of the quadrilateral photodiode, and the left-right and up-down signals use the sum of different halves of adjacent sensor regions. Optionally, this is performed iteratively. In this way, the incident signal can be perfectly aligned with the quadrilateral photodiode, achieving optimal beam fit on the active region of the quadrilateral photodiode and generating the most accurate error signal. Optionally, 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 process involves iteratively adjusting the axial distance between the detection lens and the four-segment photodiode along the optical axis of the incident signal, and centering the incident signal on the four-segment photodiode, so that the transmission peak of the incident signal coincides with the zero-crossing of the error signal. All axial distance adjustments minimize the left-right and up-down signals. Repeating this process optimizes the zero-crossing so that it precisely coincides with the transmission plateau.

[0038] The present invention further relates to a method for locking a laser into an optical resonator using the method of the present invention. - Steps of supplying a laser beam from a laser, - A step of shaping the laser beam so that it acquires ellipticity, - A step of directing a laser beam into an optical resonator, wherein the laser beam reflected from the optical resonator is an incident signal propagated to a detector, and - A step of locking the laser into the optical resonator based on the error signal, It also includes [something else].

[0039] This method is applicable to a wide range of laser beam wavelengths. For example, the wavelengths may be in the ultraviolet region between 190 nm and 400 nm, the visible region between 400 nm and 800 nm, the near-infrared region between 800 nm and 1800 nm, and / or the infrared region between 1800 nm 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 an optical resonator based on the error signal by feeding the error signal back into the laser current. Locking is based on monitoring changes in the ellipticity of the laser beam reflected from the optical resonator. Locking can be performed by closed-loop control.

[0040] Undershaping a laser beam so that it acquires ellipticity is understood to mean manipulating the laser beam so that the cross-section of the laser beam is not circular at at least one point, and in particular so that the laser beam has two different beam waists in the transverse direction at at least one point. Optionally, the laser beam is focused and / or diverged to at least a first plane, preferably a second plane (non-parallel to the first plane), particularly along the path of the laser beam, at a point different from the focus / divergence in the first plane.

[0041] Shaping a laser beam to acquire ellipticity can be done, by choice, using a pair of cylindrical lenses whose focusing axes are non-parallel, particularly perpendicular to each other. And / or shaping a laser beam to acquire ellipticity can be done using a combination of lenses, wedge prisms, and / or pairs of anamorphic prisms, which produce an elliptical or astigmatic beam.

[0042] By choice, the two cylindrical lenses have the same focal length, and further optionally, one is oriented horizontally and the other vertically. By choice, the focal length is such that when the two lenses are arranged coaxially (as if forming a single spherical lens), the laser beam is TEM 00 The system is selected to perfectly match the cavity mode. The distance between the pair of cylindrical lenses determines the light intensity of the "+" mode, which accounts for, for example, 10% of the total power (e.g., 400 μW). One cylindrical lens can be oriented 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 secondary modes. TEM 00 TEM 02 TEM 20 All other modes are suppressed, and the latter two modes can be aligned so that their transparency levels are equal. TEM 02 and TEM 20 To balance these forces, the cylindrical lens position can be set / adjusted so that the in-plane and out-of-plane beam waists converge and diverge. The average resonator waist is precisely positioned where the in-plane and out-of-plane beam waists coincide. If the beam diverges out of the cavity, its average waist may also coincide with the cavity-mode average waist. TEM 00 TEM 02 and TEM 20 The balance of the quantities can be adjusted by the distance between the two cylindrical lenses while maintaining the center of gravity. Each of the paired components is translated symmetrically in the opposite direction from this reference configuration, so TEM 02 Mode and TEM 20 The modes become equiamplitude, as observed in the cavity transmission spectrum. Physically, this action causes the beam waist in the horizontal and vertical planes to move forward and backward from the cavity waist position. Alternatively, shaping the laser beam to acquire ellipticity is done by a combination of lenses, wedge prisms, and / or anamorphic prism pairs, which optionally produce an elliptical or astigmatic beam.

[0043] Optionally, the laser beam is polarized before being directed into the optical resonator, and the incident signal is also polarized. Optionally, the optical resonator is birefringent.

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

[0045] When polarization degrees of freedom are included, the cavity reflection coefficient is the reflection operator acting on the input polarization state |ψ1>. Replaced with JPEG0007933876000001.jpg76. When performing a post-selection to state |ψ2>, the effective reflection coefficient r obtained is w This is given by the weak value of the reflection operator. JPEG0007933876000002.jpg2277((Formula 1) Here, δ is the frequency detuning of the laser cavity, κ is the total linewidth of the cavity, and γ'=Aγ. γ is a dimensionless parameter characterizing the round-trip loss within the cavity. The "amplification" parameter A depends on a specific post-selection and is real for linearly polarized states |ψ1> and |ψ2>. It is a function of the polarizer angle, but operationally, it is the residual power ratio after post-selection when the light is out of resonance |<ψ2|ψ1>|. 2 ≈ 1 / (2A+1) 2 It is possible to relate this to the following. The right-hand side of Equation 1 has the same form as the reflection from a normal high-finesse resonator, except for the variable loss parameter meter γ' (at a fixed value κ). Note that γ<1, γ=1, γ>1, and γ>2 are identified as the under-coupling, impedance matching, over-coupling, 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 Equation 1. By adjusting the post-selection to an impedance matching configuration, noise is reduced for frequency stabilization. In this configuration, the attenuation of the signal-generating portion is small, while the non-signal-generating beam component incident on the detector is strongly attenuated, so the slope of the error signal is effectively increased and sensitivity to beam shape and intensity variations is reduced.

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

[0047] For example, tuning an optical resonator involves, - The step of adjusting the temperature of the optical resonator, - Steps to adjust the optical path length of the optical resonator, Includes 1 or more of the above.

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

[0049] In an optional embodiment, the optical resonator is a laser optical resonator, particularly a semiconductor laser optical resonator. The laser is, for example, a semiconductor laser, and the laser pumping is performed by a current passing through the laser resonator. Examples of semiconductor lasers include FP (Fabry-pero) lasers, DFB (distributed feedback) lasers, DBR (distributed Bragg reflection) lasers, or VCSEL (vertical surface emission) lasers. Other classes of lasers compatible with this method include ECDL (external resonator diode lasers) or ECL (external resonator lasers). Pumping of the semiconductor laser can be performed by a current passing through an active material in the laser resonator.

[0050] In general, the path length of a laser resonator, i.e., an optical resonator of a laser, is extremely important for the mode distribution of the laser. Optionally, the pumping of the optical resonator is an operating parameter of the optical resonator. Laser pumping depends on the laser technology and can be performed, for example, by optical pumping via an additional light source, such as an additional laser. Adjusting the pumping includes, for example, adjusting the current and changing the intensity and / or frequency of the optical pump light.

[0051] Optionally, the tuning of the optical resonator may include a step of adjusting the current of the semiconductor laser. Pumping of the semiconductor laser can be performed by the current passing through the laser resonator. Pumping can be adjusted by adjusting the current.

[0052] Specific embodiments of the present invention relate to a closed-loop control method and a closed-loop control system for controlling an injection-synchronous laser. These are outlined below.

[0053] The linewidth of a laser is primarily determined by unwanted effects that do not contribute to the laser's stimulated emission. A laser can be made to copy the phase correlation of another incident, much narrower-linewidth "seed" laser beam. Ideally, this results in a "slave" laser with the same linewidth as the seed laser. Therefore, injection-locked lasers play a crucial role in modern technology because injection locking enables narrow-linewidth laser outputs. Furthermore, the injection seed frequency is variable; the "slave" laser follows the frequency of the injected seed. Thus, a shift in the laser's output frequency is possible.

[0054] Controlling an injection-locked laser is beneficial to obtain stable injection locking and stable laser output. 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 actual state from the nominal or reference state. To address the challenge of finding appropriate measurements and estimators, many methods have been developed over several 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 such as tilt-rocking.

[0055] "Laser-cavity locking at the 10^(-7) instability scale utilizing beam elipticity" by Diorico, F. et al. -7 The archive preprint "Laser Cavity Locking at an Instability Scale" (arXiv:2203.04550(2022)) presents a method for locking a laser to an optical cavity. A laser beam is directed towards the optical cavity. The laser frequency resonates (or nearly resonates) with the cavity. The reflected beam is focused by a lens onto a four-part photodiode. This method is a fundamental TEM for optical cavities. 00The interference between the mode and the two-space-time modes results in an elliptic beam on the detector. An error signal is derived based on the ellipticity, and this is used as feedback to achieve a stable laser resonator lock. This method is called "squash lock." This paper states that the general squash lock method is applicable to stabilizing laser injection locks. However, this paper does not provide hints for generalizing the disclosed method, nor does it show how to adapt the disclosed method to laser injection locks.

[0056] Embodiments of the present invention (see Embodiment 1) provide a method and apparatus for controlling an injection-synchronous laser in a closed loop, which is robust against alignment drift and vibration.

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

[0058] Furthermore, this is achieved by a closed-loop control system for controlling the injection-synchronous laser (see Embodiment 13). -A laser injected along with the optical seed signal during operation, - detector, - Optical components for propagating at least a portion of the light output signal from a laser as an incident signal to a detector, - A controller for deriving error signals based on the geometric beam shape of the incident signal to the detector. This includes the following: The controller is configured to control at least one operating parameter that is sensitive to error signals.

[0059] It is possible to enhance the laser itself by utilizing its own light emission. This method is generally called self-injection or self-feedback locking. By extracting a small portion of the laser light, frequency filtering it, and feeding it back to the laser, the laser emission into a narrower frequency band is further enhanced. As a result, the laser linewidth is narrowed. Closed-loop control methods and closed-loop control systems for controlling injection-locked lasers can be used for both injection-locked and self-injection-locked lasers that use a seed signal from an external light source.

[0060] This method and system are started or initially calibrated before use. The setpoints or reference points of the control loop (including, for example, a PID controller) 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 desired. For example, if the optical seed signal is TEM 00 The laser mode can be perfectly mode-matched, and / or the center frequency of the optical seed signal resonates with the laser. Generally, at a setpoint, the relationship between the optical seed signal and the laser is defined. The laser should be controlled such that the system stabilizes at the setpoint and the relationship between the laser and the optical seed signal is as constant as possible. This is achieved by deriving an error signal that indicates the deviation from the setpoint. As described above, the deviation from the setpoint can be quantified using the relative change in beam shape relative to the beam shape at the setpoint. In general, this closed-loop control method is not limited to any particular type of control transfer function. Therefore, it is not necessary to define a specific setpoint to control at least one operating parameter of the laser. Instead, it may suffice to drive the laser back within the acceptable range of possible operating states when the error signal indicates that such a range remains and in which direction.

[0061] Generally, deriving an error signal based on the geometric beam shape of an incident beam means that the error signal is a function in which at least one parameter of the function characterizes the geometric beam shape of the incident beam. Specifically, the error signal can be directly proportional to the geometric beam ellipticity. Beam ellipticity can be defined according to ISO 11145:2018. Alternatively, eccentricity may be used. The error signal may be zero at the setpoint by choice. The error signal has a sign and a magnitude; the sign optionally indicates the direction of control, i.e., whether the laser operating parameter should be increased or decreased. The magnitude of the error signal can indicate the magnitude of the necessary change of at least one operating parameter to reach the setpoint.

[0062] By choice, the zero-crossing of the error signal lies in an interval between the minimum and maximum values ​​of the error signal, which contains error signal values ​​corresponding to the determined beam ellipticity of a substantially circular beam, and an incident signal that acquires the opposite ellipticity on the opposite side of the resonance can be used to determine the error signal, which can then 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 leaking from the laser and the second-order "+" mode that reflects directly from the laser and forms the phase reference. This detection scheme is insensitive to alignment drift and variation, because small misalignments simply generate first-order modes in the mode decomposition. A zero-crossing means that the error signal has a value of zero.

[0063] By choice, the error signal contains a local minimum, which is by choice the minimum value. By choice, the error signal contains a local maximum, which is by choice the maximum value. By choice, there are no further extreme values ​​between the minimum and maximum values. By choice, the error signal contains an interval from the minimum to the maximum value. Within that interval is a value corresponding to the laser being locked to the seed signal, which is between the minimum and maximum values. The slope of the error signal allows the error signal to be used for laser control, achieving locking of the laser to the optical seed signal.

[0064] The detector may be a photodiode, such as an avalanche photodiode. The detector may be a phase plate or a mode converter, or may consist of a phase plate or a mode converter. The detector can convert an 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 present invention is not limited to any particular form of physical signal that the detector generates based on the incident signal.

[0065] For example, at least one operating parameter of a laser resonator is: -Temperature of the laser resonator, - Laser resonator path length, - Pumping of laser resonators, - Phase of optical seed signal, It is 1 or greater. Generally, the path length of a laser resonator is extremely important for the mode distribution of the laser. Because the path length can increase or decrease depending on the temperature due to thermal expansion, temperature affects the path length, especially in the case of semiconductor lasers. Pumping affects temperature. Therefore, in this case as well, the path length can be affected by pumping. The temperature can be selectively changed by individual heating elements such as a Peltier element or an electric heater element, and individual cooling elements such as a fan. Laser pumping depends on the laser technology and can be done, for example, by optical pumping via an additional light source, for example, an additional laser. Pumping adjustments include, for example, adjusting the current, and changing the intensity and / or frequency of the optical pump light. The phase of the optical seed signal can be adjusted, for example, 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 laser pumping is performed by a current passing through a laser resonator. Examples of semiconductor lasers include FP (Fabry-pero) lasers, DFB (distributed feedback) lasers, DBR (distributed Bragg reflection) lasers, or VCSEL (vertical surface-emitting) lasers. Other classes of lasers compatible with this method include ECDL (external resonator diode lasers) or ECL (external resonator lasers). Pumping of the semiconductor laser can be performed by a current passing through an active material in the laser resonator. Pumping can be adjusted by adjusting the current.

[0067] For example, the detector may be a photodiode array consisting of at least three photodiodes, and the derivation of an error signal includes determining the geometric beam shape of the incident signal. By using a 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 required to differentiate between first-order and second-order (or higher-order) spatial modes. In principle, the more photodiodes used, the better the signal shape can be approximated. Generally, determining the geometric beam shape of the incident signal means determining the parameter value of at least one parameter that characterizes the geometric beam shape. Preferably, the photodiodes are arranged in a rectangular structural grid. Minimizing the distance between individual photodiodes is beneficial to make the measurement most sensitive to changes in beam shape.

[0068] The detector may be a quad-segment photodiode, and the derivation of the error signal may involve determining the diagonal signals of the quad-segment photodiode. These diagonal signals are composed of the difference between the sums of the signals from the diagonal sensor regions of the quad-segment photodiode. Thus, the beam shape of the incident signal, particularly the beam ellipticity, can be easily evaluated. Specifically, a quad-segment photodiode is a photodiode having four divided sensor regions. The gaps between the sensor regions can be optimized to be as small as possible. Specifically, a quad-segment photodiode generates a current proportional to the input light from each of the four sensor regions. The signals from each sensor region are denoted as A, B, C, and D, and on the quad-segment photodiode, the sensor regions 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, meaning the laser beam is circular. A non-zero DIAG signal indicates an elliptic beam on the four-segment photodiode. A reversed sign in the DIAG signal indicates inverse ellipticity. The four-segment photodiode may be an InGaAs-based or silicon-based detector. The signal can be converted to a voltage (e.g., using standard analog electronic equipment with an operational amplifier).

[0069] As previously described, the method and system are started or initially calibrated before the method is used. During control of the injection-locked laser, the initialized or calibrated characteristics remain constant. The error signal can be initialized by the following parameters.

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

[0071] In an optional embodiment, the propagation of the incident signal includes shaping the incident signal, particularly using at least a lens, preferably at least a shaping lens, and especially at least a cylindrical lens. Shaping the incident signal allows for calibration of the initial error signal when the laser is locked to the optical seed signal. The error signal can be initially set to zero. If the detector is a QPD, shaping the incident signal can make the DIAG signal zero. 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 an error signal without further operation. The position and alignment of the beam shaping optical system for incident beam shaping, particularly at least one lens, preferably at least one shaping lens, and especially at least one cylindrical lens, are set once during calibration and are kept constant during operation.

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

[0073] Optionally, the propagation of the incident signal includes determining the axial distance between the detection lens and the detector such that the transmission peak of the incident signal coincides with the zero-crossing of the error signal. The transmission peak represents the resonance between the optical seed signal and the laser. The axial distance between the detection lens and the detector represents their distance along the path of the incident signal. In this method and apparatus, the positions of the detector and detection lens are fixed during calibration and maintained constant during operation. The relative change in the geometric beam shape of the incident signal is most pronounced when the laser resonates with the optical seed signal. Therefore, this method is most sensitive at the transmission peak of the incident signal, which is also the transmission peak of the optical seed signal. Optionally, the axial distance may be set by adjusting the error signal curve in an adjustment interval.

[0074] In another optional embodiment, the propagation of the incident signal involves determining the axial distance between the laser and the detector such that the transmission peak of the incident signal coincides with the zero-crossing of the error signal. The positions of the detector and laser are fixed during calibration and maintained constant during operation. The relative change in the geometric beam shape of the incident signal is most pronounced when the laser resonates with the optical seed signal. Therefore, this method is most sensitive at the transmission peak of the incident signal, which is also the transmission peak of the optical seed signal. The distance between the laser and the detector is TEM 00This affects the Guoy phase between the mode and the secondary spatial mode. Therefore, the image obtained on the detector can be adjusted by adjusting the position of the detector relative to the laser. There is an axial distance at which the error signal is purely dispersive, and another axial distance at which the error signal is purely absorptive or has any signal shape between absorptive and dispersive. Preferably, the error signal is dispersive.

[0075] By choice, deriving the error signal involves 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 in the detector. These signals are used depending on the estimater of the shape of the incident signal. For example, if the detector is a QPD, the ellipticity of the signal can be estimated using DIAG calculations. The error signal can be initialized to zero during calibration. Thus, the sign of the error signal indicates the direction of control, for example, whether the operating parameter of the laser should be increased or decreased. For example, because DIAG calculations can produce residual signals even if the incident signal is substantially circular, an additional offset is applied to initialize the error signal to zero at the set point.

[0076] In another embodiment, an imbalance detection operation is available. For example, instead of the DIAG operation, an IDIAG imbalance operation can be used. IDIAG = ε1(A+B) - ε2(C+D), where ε1 / ε2 ≠ 1, and ε1 and ε2 can be initialized to set the error signal to zero. This allows adjustment of the background offset of the diagonal channel to give the error signal a zero crossing at a desired state (e.g., at a set point).

[0077] A closed-loop control method for injection-locked lasers, and a closed-loop control system for controlling injection-locked lasers, can be used to continuously shift the wavelength of the output signal of the injection-locked laser by a further step of shifting the wavelength of the optical seed signal. Here, 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 disrupts the lock between the optical seed signal and the laser, thereby changing the error signal. The change in the error signal results in feedback to the laser as a response in the control step of this method. This restores the relationship between the laser and the optical seed signal at the shifted wavelength of the optical seed signal. Thus, the laser output tracks the wavelength of the optical seed signal. The error signal indicates whether the laser is operating at its optimal operating parameters for the injected laser frequency. With this information, it is possible to add an additional layer of feedback for dynamically adjusting the operating parameters of the injection laser. This allows the laser to track the injection frequency throughout the entire operating range. This optimal range allows for continuous adjustment of the laser frequency without interruption. This uninterrupted frequency adjustment range is often called the "mode-hop free" region. This is a unique property of semiconductor lasers, which range from several GHz to tens of GHz. This optimal lock range is typically determined by the set operating current and temperature of an unpackaged semiconductor laser. Since the initial operating parameters are usually set before or during system initialization or calibration, the range is already predetermined by the center frequency. This center frequency can be dynamically adjusted beyond the mode-hop free region if optimal lock range information for the laser is available. Because the intrinsic output modes of a semiconductor laser are highly elliptic, and the injected modes are generally circular (or elliptic at different levels), there is an error signal that continuously monitors whether the laser parameters are still within the optimal injection range or at the optimal setpoint. With this, the current, temperature, or both can be dynamically adjusted to track the injection signal frequency. This makes it possible to generate a wide range of adjustable laser frequencies across its entire operating range.

[0078] Optionally, at least a laser, optical components, and preferably a detector are integrated on a micro-optical device or silicon photonics platform, preferably a silicon chip. This can be achieved, for example, by heterogeneously integrating individual miniaturized elements on a single system-on-a-chip (SoC). For example, the system can be constructed using state-of-the-art optical packaging technology and high-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 includes the step of centering the output signal on the quadrilateral photodiode by translating the detection lens and / or quadrilateral photodiode in a direction perpendicular to the optical axis of the incident signal, such that the left-right and / or up-down signals of the quadrilateral photodiode are minimized. Here, the left-right and up-down signals are each composed of the difference in the sum of adjacent sensor regions of the quadrilateral photodiode, and the left-right and up-down signals use the sum of different halves of adjacent sensor regions. Optionally, this is performed iteratively. In this way, the incident signal can be perfectly aligned with the quadrilateral photodiode, and the optimal beam fit is achieved on the active region of the quadrilateral photodiode, generating 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] Stability constraints still exist, for example, due to residual fluctuations in the incident signal that can result from changes in the second-mode component affecting the phase reference. To help mitigate these remaining constraints, pre-selection and post-selection procedures for polarization are available, which are described using the weak-value concept. For this purpose, optionally, the optical seed signal is polarized before being directed to the laser, and the optical output signal is polarization-filtered. Optionally, the laser is equipped with a birefringent resonator. In this way, technical noise that constrains system performance can be coherently suppressed. The apparatus preferably includes a first polarization beam splitter (or first polarizer) for polarization filtering the optical seed signal before it is directed to the laser. A second polarization beam splitter (or second polarizer) is used to polarization filter the incident signal, particularly before the detector. Polarization filtering can be used in closed-loop control methods and closed-loop control systems for controlling injection-synchronous lasers.

[0082] When polarization degrees of freedom are included, the cavity reflection coefficient is the reflection operator acting on the input polarization state |ψ1>. Replaced with JPEG0007933876000003.jpg86. When performing a post-selection to state |ψ2>, the effective reflection coefficient r obtained is w This is given by the weak value of the reflection operator. JPEG0007933876000004.jpg1863((Formula 1) Here, 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 characterizing the round-trip loss in the laser. The "amplification" parameter A depends on a specific post-selection and is real for linearly polarized states |ψ1> and |ψ2>. 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 out of resonance |<ψ2|ψ1>|. 2 ≈ 1 / (2A+1) 2This can be related to the following. The right-hand side of Equation 1 is the same form as the reflection from a normal high-finesse resonator, except for the variable loss parameter meter γ' (at a fixed value κ). Note that γ<1, γ=1, γ>1, and γ>2 are identified as the under-coupling, impedance matching, over-coupling, 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 Equation 1. By adjusting the post-selection to an impedance matching configuration, noise is reduced for frequency stabilization. In this configuration, the attenuation of the signal-generating portion is small, while the non-signal-generating beam component incident on the detector is strongly attenuated, so the slope of the error signal is effectively increased and sensitivity to beam shape and intensity variations is reduced.

[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] This method and apparatus can be used with a wide range of laser wavelengths. For example, the wavelengths may be in the ultraviolet region between 190 nm and 400 nm, the visible region between 400 nm and 800 nm, the near-infrared region between 800 nm and 1800 nm, and / or the infrared region between 1800 nm and at least 3500 nm.

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

[0087] By choice, the error signal is substantially zero with respect to the circular incident signal and / or the 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, the outlines of which are described below.

[0089] Over the decades, numerous methods have been developed to address the challenge of locking lasers into resonators. 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 specific methods may be preferable depending on the application, the PDH method—which utilizes high-frequency modulation / demodulation of an optical carrier—has become the general standard due to its stability and versatility. (Cavity linewidth 10) 5 -10 6 To achieve the most stringent lock stability of 1 / 1, additional multi-layer active feedback mechanisms were required to reduce residual amplitude (RAM) modulation, which generally limits the stability of the lock point in PDH schemes. A purely passive and less complex method that can rival the configurations of these advanced designs would be beneficial for all applications, especially space-based ones where the electronic and acousto-optical devices used for modulation are undesirable due to power consumption and failure risk.

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

[0092] This is achieved by frequency-locking the laser to an optical resonator in the following steps (see Embodiment 15). - Steps to provide a laser beam from a laser; - A step of shaping the laser beam so that it acquires ellipticity; - The step of directing the laser beam towards the optical resonator; - A step of focusing the laser beam reflected from the optical resonator using a detection lens; - A step of determining an error signal proportional to the beam ellipticity of the focused laser beam using a detector; - A step of locking the laser into an optical resonator 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 can occur in the 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 laser beam.

[0093] Furthermore, this is achieved by a method of adjusting the apparatus for frequency-locking the laser to an optical resonator (see Embodiment 17). Here, the apparatus is - A beam shaper for shaping a 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 a laser beam focused on a detection lens. This method includes, - Steps include providing a laser beam at the resonant frequency of an optical resonator. - A step of shaping the laser beam with a beam shaper so that the laser beam acquires ellipticity, - The step of directing the laser beam towards the optical resonator, - A step of focusing the laser beam reflected from the optical resonator using a detection lens. - A step of determining an error signal proportional to the beam ellipticity of the focused laser beam using a detector. - A step of 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, where this interval includes error signal values ​​corresponding to the determined beam ellipticity of a substantially circular laser beam. Includes.

[0094] Furthermore, this is achieved by a device that frequency-locks the laser to an optical resonator (see Embodiment 27), which, - A beam shaper for shaping a 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 a laser beam focused by a detection lens. This includes the following: Here, the axial distance between the detection lens and the detector is such that, when the device, in particular the beam shaper, is supplied with a laser beam at the resonant frequency of the optical resonator, the zero-crossing of the error signal falls in the 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 laser beam.

[0095] Furthermore, this disclosure relates to a method for manufacturing an apparatus for frequency-locking a laser to an optical resonator, the method comprising the following steps: - A step of providing a beam shaper for shaping a laser beam so that the laser beam acquires ellipticity. - A step of providing an optical resonator for receiving a laser beam after beam shaping, - A step of providing a detection lens for focusing a laser beam reflected from an optical resonator, - A step of providing a detector for determining an error signal proportional to the beam ellipticity of a laser beam focused by a detection lens, - Steps include providing a laser beam at the resonant frequency of an optical resonator. - A step of shaping the laser beam with a beam shaper so that the laser beam acquires ellipticity, - The step of directing the laser beam towards the optical resonator, - A step of focusing the laser beam reflected from the optical resonator using a detection lens. - A step of determining an error signal proportional to the beam ellipticity of the focused laser beam using a detector. - A step of 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, the interval containing error signal values ​​corresponding to the determined beam ellipticity of a substantially circular laser beam.

[0096] To understand the primary stabilization methods, we should recall the Hermitian Gaussian (HG) spatial modes supported by optical resonators. These cavities can support a set of transverse electromagnetic modes (TEM). mn ) 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 perfectly aligned optical modes entering the optical resonator, odd modes are induced by shifting the input beam from its aligned state, either by tilting or shifting the input beam. Even modes, on the other hand, are induced by focusing / mode mismatch and are independent of beam tilt or shift.

[0097] Previous implementations of spatial mode interference involved tilting / shifting the input beam to achieve first-order interference (TEM). 01 and TEM 10 It relied on inducing the mode of ) . What we are interested in here is the basic HG mode TEM which is displayed as "00". 00These are specific second-order HG modes and second-order TEM11-like modes, indicated by a "+", and these are all even-order modes. Even-order modes are induced by mode mismatch, and this method is free from the effects of misalignment drift that generates first-order odd-order modes. A slightly elliptical beam with horizontal / vertical orientation can be mathematically decomposed into a major "00" component and a small "+" component. In such a laser beam, the phase difference between these two modes encodes information about ellipticity. When incident on an optical resonator near the "00" resonance, the two modes undergo a differential phase shift due to reflection. This is because modes of different orders are generally non-degenerate, so only one of them resonates. This mechanism causes the reflected signal to have opposite ellipticity on the opposite side of the optical resonator's resonance. To utilize this effect, the ellipticity of the laser beam reflected from the optical resonator is measured after focusing.

[0098] These modes generally have different resonant frequencies. Regarding cavity modes, the spatial resolution of a slightly elliptical beam oriented diagonally is largely "TEM 00 This is the "TEM" mode. 11The contribution of the second-order mode is small. In such a beam, the phase difference between these two modes encodes information about the deviation from circularity. When incident on the resonator at a frequency close to one of its fundamental frequencies, the two modes naturally undergo a differential phase shift because only one of them resonates. As a result, a change in the shape of the reflected beam occurs. This mechanism causes the reflected signal to acquire an inverse ellipticity on the opposite side of the resonance, thereby generating a determinable error signal. To utilize this property, an error signal proportional to the beam ellipticity of the reflected laser beam is used, and the axial distance between the detector and the detection lens must be appropriately adjusted. By setting the axial distance and supplying the laser beam to the beam shaper at the resonant frequency of the optical resonator, the zero-crossing of the error signal lies in an interval between the minimum and maximum values ​​of the error signal, and that interval contains error signal values ​​corresponding to the determined beam ellipticity of a substantially circular beam. By utilizing the reflected signal that acquires an inverse ellipticity on the opposite side of the resonance, the deviation and / or drift of the laser beam can be determined, and this can be used to lock and stabilize the laser beam. The determined error signal can be considered as interference between the resonant "00" mode leaking from the resonator and the second-order "+" mode that directly reflects from the optical resonator and forms the phase reference. This detection method is insensitive to alignment drift and fluctuations because small misalignments simply generate first-order modes in the mode decomposition.

[0099] The optical resonator may be a linear resonator or a ring resonator. For example, a ring resonator in which the incident beam and the reflected beam travel through separate paths is available. The optical resonator may be, for example, a two-mirror resonator or a three-mirror resonator. However, any number of mirrors and resonators of any shape are also available. When a two-mirror (standing wave) resonator is used as the optical resonator, the reflected beam can be separated from the incident path by, for example, 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, in which light circulates around a sphere or ring (inside the material). The optical resonator may also be a fiber ring in which light can be evanescently coupled. This coupling can be done, for example, using a prism, in which case the evanescent tail during total internal reflection from the prism couples with the light inside and outside the optical resonator. In this way, the prism effectively interfaces a waveguide-like structure within the free-space optical system. While the prism might resemble an input mirror for a free-space resonator, it possesses a highly elliptical mode structure. In this case, input mode shaping requires preparing a beam profile that is further compressed for the eigenmodes of the micro-resonator-prism assembly.

[0100] This method and apparatus can be used with a wide range of laser beam wavelengths. For example, the wavelengths may be in the ultraviolet region between 190 nm and 400 nm, the visible region between 400 nm and 800 nm, the near-infrared region between 800 nm and 1800 nm, and / or the infrared region between 1800 nm 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 an optical resonator based on the error signal by feeding the error signal back into the laser current. Optionally, the apparatus includes a laser. Locking is based on monitoring changes in the ellipticity of the laser beam reflected from the optical resonator.

[0101] Laser beam focusing refers to focusing a laser beam onto at least one plane. Providing the laser beam at the resonant frequency of an optical resonator preferably involves a step of adjusting the laser (particularly finding the resonant frequency). Optionally, the method of adjusting the apparatus (without beam shaping, i.e., before inserting a beam shaper into the apparatus) - The step of (completely) coupling the laser beam into the resonator and measuring the characteristics of the reflected beam, particularly with a CCD camera, - The step of inserting the beam shaping device into the apparatus, This includes, optionally, a method for adjusting the apparatus, which includes measuring the in-plane and out-of-plane beam waists of the laser beam reflected from the optical resonator at a distance of 2 or more.

[0102] The axial distance between the detection lens and the detector refers to their distance along the path of the laser beam. For methods and apparatus for frequency-locking a laser to an optical resonator, the positions of the detector and detection lens are fixed. For methods of adjusting and manufacturing the apparatus, the detector and / or detection lens are movable. Zero crossing means that the error signal has a value of zero. By choice, the minimum is the global minimum, and / or the maximum is the global maximum. By choice, there are no further extreme values ​​between the minimum and maximum. By choice, the interval spans from the minimum to the maximum. Between the minimum and maximum values ​​of the error signal, defined by the slope of the error signal, the interval containing the error signal corresponding to a circular laser beam (providing the laser beam with a frequency at its resonant frequency) is available for laser locking. By choice, the resonant frequency of the optical resonator is TEM 00 Resonance frequency, or TEM 02 or TEM 20 or TEM 11 This is the resonant frequency of the resonance. Optionally, the axial distance may be adjusted within the adjustment interval to determine the error signal curve and set accordingly.

[0103] It is preferable that the detector is a quad-segment photodiode and the error signal is determined by the diagonal signals of the quad-segment photodiode. This diagonal signal is composed of the difference in the sum of the signals from the diagonal sensor regions of the quad-segment photodiode. In this way, an error signal corresponding to the beam ellipticity can be achieved particularly easily. Specifically, a quad-segment photodiode is a photodiode having four divided sensor regions. The gaps between the sensor regions can be optimized to be as small as possible. Specifically, a quad-segment photodiode generates a current proportional to the input light from each of the four sensor regions. The signals from each sensor region are denoted as A, B, C, and D, and on the quad-segment photodiode the sensor regions 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, meaning the laser beam is circular. A non-zero DIAG signal indicates an elliptical beam on the quad-segment photodiode. The quad-segment photodiode may be an InGaAs-based or silicon-based detector. The signal can be converted to a voltage (e.g., using standard analog electronic equipment with an operational amplifier).

[0104] Stability constraints still exist, for example, due to residual variations in the incident beam shape caused by changes in the second-mode component affecting the phase reference. To help alleviate these remaining constraints, pre-selection and post-selection procedures for polarization are available, which are described using the weak-value concept. For this purpose, optionally, the laser beam is polarized before being directed to the optical resonator, so that the laser beam reflected from the optical resonator is polarized. Optionally, the optical resonator is birefringent. In this way, technical noise that constrains system performance can be coherently suppressed. The apparatus preferably comprises a first polarizing beam splitter (or first polarizer) for polarizing the laser beam before it is directed to the beam shaper, and a second polarizing beam splitter (or second polarizer) for polarizing the laser beam after emission from the optical resonator, particularly between the detection lens and the detector. Polarization filtering can be used as a method for frequency locking the laser and / or as a method for adjusting the apparatus for frequency locking the laser.

[0105] When polarization degrees of freedom are included, the cavity reflection coefficient is the reflection operator acting on the input polarization state |ψ1>. Replaced with JPEG0007933876000005.jpg97. When performing a post-selection of the state, the effective reflection coefficient r obtained is w This is given by the weak value of the reflection operator. JPEG0007933876000006.jpg2069((Formula 1) Here, δ is the frequency detuning of the laser cavity, κ is the total linewidth of the cavity, and γ'=Aγ. γ is a dimensionless parameter characterizing the round-trip loss within the cavity. The "amplification" parameter A depends on a specific post-selection and is real for linearly polarized states |ψ1> and |ψ2>. It is a function of the polarizer angle, but operationally, it is the residual power ratio after post-selection when the light is out of resonance |<ψ2|ψ1>|. 2 ≈ 1 / (2A+1) 2It is possible to relate this to the following. The right-hand side of Equation 1 has the same form as the reflection from a normal high-finesse resonator, except for the variable loss parameter meter γ' (at a fixed value κ). Note that γ<1, γ=1, γ>1, and γ>2 are identified as the under-coupling, impedance matching, over-coupling, 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 Equation 1. By adjusting the post-selection to an impedance matching configuration, noise is reduced for frequency stabilization. In this configuration, the attenuation of the signal-generating portion is small, while the non-signal-generating beam component incident on the detector is strongly attenuated, so the slope of the error signal is effectively increased and sensitivity to beam shape and intensity variations is reduced.

[0106] Next, referring to the method of adjusting the device, it is preferable to adjust the axial distance between the detection lens and the detector along the optical axis of the laser beam so that the transmission peak of the laser beam reflected from the resonator coincides with the zero-crossing of the error signal, which is to adjust the axial distance between the detection lens and the detector along the optical axis of the laser beam so 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 minimum reflection value, that is, the SUM channel of the four-segment photodiode shows a dip (the SUM channel of the four-segment 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. Therefore, the axial position of the QPD can be adjusted until the transmission peak and the zero-crossing of the error signal coincide. If this coincidence does not occur, cross-coupling of intensity noise to frequency noise occurs. The zero-crossing or 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 positions of the detector and / or detection lens. When the laser provides a laser beam at the resonant frequency of the optical resonator, the axial distance between the detection lens and the detector along the optical axis of the laser beam is also preferred in the context of methods and apparatus for frequency-locking a laser if the transmission peak of the laser beam reflected from the resonator substantially coincides with the zero-crossing of the error signal, or within ±5%, ±2%, or ±1%. This can also be replaced by the feature that "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 lies in the 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 laser beam." A baseline offset of the error signal is also possible.Without experience and knowledge of astigmatism, it might seem that this method would not yield the best results. However, this evaluation 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 detector along the optical axis of the laser beam so that the zero-crossing of the error signal falls within the interval between the minimum and maximum values ​​of the error signal (this interval includes error signal values ​​corresponding to the determined beam ellipticity of a substantially circular laser beam) includes adjusting the axial distance so that the in-plane and out-of-plane signal waists of the laser beam reflected from the resonator coincide with the beam waist of the resonant cavity mode of the optical resonator (in particular, the average beam waist of the cavity mode, 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 detector along the optical axis of the laser beam so that the transmission peak of the laser beam reflected from the resonator coincides with the zero-crossing of the error signal. Optionally, additionally or alternatively, the axial distance is adjusted so that the in-plane and out-of-plane beam waists of the laser beam reflected from the resonator coincide (with each other). The baseline offset of the error signal may be due to the displacement of the detector position from the position where the beam takes a circular profile (ignoring astigmatism). The axial distance can be selected such that this baseline offset is eliminated. This beam waist match can be used as a first approximation before the matching of the transmission peak and zero crossing, or it can be used on its own. If there is no astigmatism, these two will yield the same result. In this case, since no astigmatism remains in the mode, setting the baseline to zero automatically leads to a plateau in transmission. In this method and apparatus for frequency locking of a laser, it is also preferable 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 match the beam waist of the resonant cavity mode of the optical resonator, or within ±5%, ±2%, or ±1% of the beam waist of the resonant cavity mode of the optical resonator.This can be replaced by the characteristic that "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 lies in the 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 laser beam."

[0108] By choice, the detector is a four-segment photodiode, and the error signal is determined by the diagonal signals of the four-segment photodiode. This diagonal signal is composed of the difference between the sum of the signals from the diagonal sensor regions of the four-segment photodiode.

[0109] Optionally, the method further includes the step of centering the laser beam onto the quadrilateral photodiode by translating the detection lens and / or quadrilateral photodiode in a direction perpendicular to the optical axis of the laser beam such that the left / right (LR) and / or up / down (UD) signals of the quadrilateral photodiode are minimized. Here, the left / right (LR) signal and the up / down (UD) signal are each composed of the difference of the sum of adjacent sensor regions of the quadrilateral photodiode, and the left / right signal and the up / down signal use the sum of different halves of adjacent sensor regions. Optionally, this is performed iteratively. In this way, the laser beam can be perfectly aligned with the quadrilateral photodiode, achieving optimal beam fit on the active region of the quadrilateral photodiode and generating the most accurate error signal. Optionally, translation is performed in two directions, which are mutual and perpendicular to the optical axis of the laser beam.

[0110] Optionally, the following steps are performed iteratively (i.e., alternately, i.e., sequentially): adjusting the axial distance between the detection lens and the four-segment photodiode along the optical axis of the laser beam so that the transmission peak of the laser beam reflected from the resonator coincides with the zero-crossing of the error signal; and centering the laser beam on the four-segment photodiode. All axial distance adjustments minimize the left-right and up-down signals. Repeating this process optimizes the zero-crossing so that it precisely coincides with the transmission plateau.

[0111] In all methods and apparatus disclosed herein, it is preferable that the detection lens be a convex spherical lens.

[0112] Furthermore, it is preferable that the laser beam is propagated through a biased optical fiber and optionally emitted from a fiber collimator before the laser beam is shaped to acquire ellipticity. The fiber collimator may have an adjustable focus.

[0113] It is preferable that shaping the laser beam so that it acquires ellipticity is performed using at least one pair of cylindrical lenses (which, by choice, have non-parallel, and in particular, vertically oriented focusing axes). By choice, the two cylindrical lenses have the same focal length, and further optionally, one is oriented horizontally and the other vertically. By choice, the focal lengths are such that the two lenses are coaxially arranged (as if forming a single spherical lens) and the incident beam is TEM 00 The cavity mode is selected to perfectly match the mode. The distance between the pair of cylindrical lenses determines the amount of light in the "+" mode, which can be up to, for example, 10% of the total output (e.g., 400 μW). One cylindrical lens can be oriented in-plane and the other out-of-plane. By adjusting the position of the cylindrical lenses and, optionally, the fiber collimator, the input mode can be manipulated to induce up to the second-order mode. TEM 00 TEM 02 TEM 20All modes other than the specified mode are suppressed, and the latter two modes can be aligned so that their transparency levels are equal. TEM 02 and TEM 20 To balance these forces, the cylindrical lens position can be set / adjusted so that the in-plane and out-of-plane beam waists converge and diverge. The average resonator waist is precisely positioned where the in-plane and out-of-plane beam waists coincide. If the beam diverges out of the cavity, its average waist may also coincide with the cavity-mode average waist. TEM 00 TEM 02 and TEM 20 The balance of the quantities can be adjusted by the distance between the two cylindrical lenses while maintaining the center of gravity. Each of the paired components is translated symmetrically in the opposite direction from this reference configuration, so TEM 02 Mode and TEM 20 The modes become equiamplitude, as observed in the resonator transmission spectrum. Physically, this operation causes the beam waist in the horizontal and vertical planes to move forward and backward from the cavity waist position. Alternatively, shaping the laser beam so that it acquires ellipticity is done by a combination of lenses, wedge prisms, and / or anamorphic prism pairs, which optionally produce an elliptic or astigmatic beam. Next, with respect to the apparatus for frequency-locking the laser to the optical resonator, it is preferable that the detector is a quad-segment photodiode. Optionally, the detector is configured to determine an error signal by the diagonal signals of the quad-segment photodiode, which consists of the difference between the sum of the signals from the diagonal sensor regions of the quad-segment photodiode.

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

[0115] For example, embodiments 1 to 29 that have selected the present disclosure will be further described below and shown in the drawings. However, these embodiments should not be considered limiting to the present disclosure.

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

[0117] 2. A method according to Embodiment 1, wherein at least one operating parameter of the laser is -Temperature of the laser resonator, - Laser resonator path length, - Pumping of laser resonators, - Phase of 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 the laser pumping is performed by an electric current passing through a 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 the method includes determining the geometric beam shape of the incident signal by deriving an error signal.

[0120] 5. A method according to any one of Embodiments 1 to 3, wherein the detector is a four-segment photodiode, and the derivation of the error signal involves determining the diagonal signals of the four-segment photodiode, which are composed of the difference between the sum of signals from the diagonal sensor regions of the four-segment photodiode.

[0121] 6. A method according to any one of Embodiments 1 to 5, wherein the propagation of the incident signal includes assigning an incident angle to the detector.

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

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

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

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

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

[0127] 12. A method for continuously shifting the wavelength of the output signal of an injection-locked laser using a method according to any one of Embodiments 1 to 12, - A step to shift the wavelength of the optical seed signal. Furthermore, it includes a method for changing the error signal by adjusting the wavelength of the optical seed signal (10 wavelengths).

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

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

[0130] 15. A method for frequency-locking a laser to an optical resonator, - Steps of supplying a laser beam from a laser, - A step of shaping the laser beam so that it acquires ellipticity, - The step of directing the laser beam towards the optical resonator, - A step of focusing the laser beam reflected from the optical resonator using a detection lens. - A step of determining an error signal proportional to the beam ellipticity of the focused laser beam using a detector. - A step of locking the laser into the optical resonator based on the error signal, A frequency-locking method comprising the following, wherein 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 lies in an interval between the minimum and maximum values ​​of the error signal, that interval includes error signal values ​​corresponding to the determined beam ellipticity of a substantially circular laser beam.

[0131] 16. A method according to Embodiment 15, wherein the detector is a four-segment photodiode, and the error signal is determined by the diagonal signals of the four-segment photodiode, and this diagonal signal is composed of the difference between the sum of signals from the diagonal sensor regions of the four-segment photodiode.

[0132] 17. A method for adjusting a device for frequency-locking a laser to an optical resonator, the device being: - A beam shaper for shaping a 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 a laser beam focused by a detection lens. This method includes, - Steps include providing a laser beam at the resonant frequency of an optical resonator. - A step of shaping the laser beam with a beam shaper so that the laser beam acquires ellipticity, - The step of directing the laser beam towards the optical resonator, - A step of focusing the laser beam reflected from the optical resonator using a detection lens. - A step of determining an error signal proportional to the beam ellipticity of the focused laser beam using a detector. - A step of adjusting the axial distance between the detection lens and the detector, along the optical axis of the laser beam, so that the zero-crossing of the error signal is in the interval between the minimum and maximum values ​​of the error signal. A method that includes an error signal value in which this interval corresponds to the determined beam ellipticity of a substantially circular laser beam.

[0133] 18. A method according to Embodiment 17, comprising adjusting the axial distance between the detection lens and the detector such that the transmission peak of the laser beam reflected from the optical resonator coincides with the 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 between the detection lens and the detector so that the in-plane and out-of-plane beam waists of the laser beam reflected from the optical resonator coincide with the beam waist of the resonant cavity mode of the optical resonator, before adjusting the axial distance between the detection lens and the detector along the optical axis of the laser beam so that the transmission peak of the laser beam reflected from the optical resonator coincides with the 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-segment photodiode, and the error signal is determined by the diagonal signals of the four-segment photodiode, and this diagonal signal is composed of the difference of the sum of signals from the diagonal sensor regions of the four-segment photodiode.

[0136] 21. A method according to Embodiment 20, - A step of centering the laser beam onto the four-segment photodiode by translating the detection lens and / or the four-segment photodiode in a direction perpendicular to the optical axis of the laser beam, such that the left and right signals and / or up and down signals of the four-segment photodiode are minimized. The method further includes the fact that the left-right signal and the up-down signal are each composed of the difference in the sum of adjacent sensor regions of a four-part photodiode, and the left-right signal and the up-down signal use the sum of different halves of adjacent sensor regions.

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

[0138] 23. A 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 through a biased optical fiber and optionally emitted from a fiber collimator before the laser beam is shaped to acquire 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 non-parallel, particularly 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 pairs of anamorphic prisms, which produce an elliptic beam or an astigmatic beam.

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

[0142] 27. A device for frequency-locking a laser to an optical resonator, - A beam shaper for shaping a 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 a laser beam focused by a detection lens. The apparatus is equipped such that, when a laser beam is supplied to the beam shaper at the resonant frequency of the optical resonator, the zero-crossing of the error signal falls within the interval between the minimum and maximum values ​​of the error signal, this interval containing error signal values ​​corresponding to the determined beam ellipticity of a substantially circular laser beam.

[0143] 28. An apparatus according to Embodiment 27, wherein the detector is a four-segment photodiode, and optionally the detector is configured to determine an error signal by the diagonal signals of the four-segment photodiode, and this diagonal signal is composed of the difference between the sum of signals from the diagonal sensor regions of the four-segment 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 that are oriented non-parallel to each other, particularly perpendicularly.

[0145] For illustrative purposes, this disclosure will be further described in relation to several selected embodiments shown in the drawings. However, these embodiments should not be considered limiting to this disclosure. Figures 1 to 6 relate to embodiments 1 to 14, and Figures 7 to 15c relate to embodiments 15 to 29. [Brief explanation of the drawing]

[0146] [Figure 1] This is a schematic diagram of beam ellipticity detection using a four-segment photodiode. [Figure 2] This shows the error signal as a function of the laser's detuning relative to the optical seed signal. [Figure 3] This 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] This is a schematic diagram of an example of a closed-loop control system for controlling a self-injection-synchronous laser, which uses a portion of the laser's output signal as an optical seed signal. [Figure 5] This figure shows an example of a closed-loop control system for controlling a self-injection synchronous laser equipped with an external resonator that provides a reflected signal for self-injection. [Figure 6] This figure shows an example of a closed-loop control system for controlling self-injection synchronization with an external resonator that provides a reflected signal for self-injection, having a second detector between the laser and the external resonator. [Figure 7] This is a schematic diagram of one embodiment of a device for frequency-locking a laser to an optical resonator. [Figure 8a] This is a schematic diagram of the geometric configuration of a triangular ring resonator and its associated Hermitian-Gaussian mode resonance. [Figure 8b] This figure shows beam ellipticity detection using a four-segment photodiode. [Figure 8c] This figure shows the frequency dependence of the error signal. [Figure 9] This figure shows the positions of the relevant elements in the beam path and the resulting beam characteristics in the embodiment shown in Figure 7. [Figure 10] This figure shows the error signal and resonator transmission in the embodiment shown in Figure 7. [Figure 11] This diagram shows the error signal and transmission. [Figure 12] This is a schematic diagram of the axial distance between the detection lens and the detector. [Figure 13a] This figure shows the use of the first polarizer and the second polarizer in the embodiments shown in Figures 7 and 14. [Figure 13b] This figure shows the normalized reflected power measured for different post-selection configurations. [Figure 13c] This figure shows the real and imaginary parts of the obtained weak value. [Figure 14] This figure shows two copies of the lock-up configuration used to demonstrate the performance of the stabilization method. [Figure 15a] Figure 14 shows the lock instability over the averaging time of the experimental configuration, compared to other stabilization methods. [Figure 15b] This figure shows the instability achieved with or without post-selection, and the time trace of the corresponding beat note at the bottom. [Figure 15c] This figure shows the estimated limits for stability. [Modes for carrying out the invention]

[0147] Figure 1 shows the detection of the beam ellipticity of the incident signal 1 using detector 2, and the mode decomposition of the slightly elliptical beam. Detector 2 is a quad-segment photodiode (QPD) 3 having four sensor regions. The quad-segment photodiode 3 generates a current proportional to the incident light from each of the four sensor regions 4. The derivation of the error signal 9 (not shown in Figure 1; see Figure 2) involves determining the diagonal signals of the quad-segment photodiode 3. These diagonal signals are composed of the difference between the sums of the signals from the diagonal sensor regions 4 of the quad-segment photodiode 3. The signals from each sensor region 4 are labeled A, B, C, and D, and on the quad-segment photodiode 3, the sensor regions 4 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), and DIAG=(A+B)-(C+D). The beam ellipticity (or a parameter directly proportional to the beam ellipticity; this depends on the rotational alignment of the detector with the resonator that determines the associated modes) can be measured using DIAG calculations, as can be seen in Figure 2. A DIAG signal of zero 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 elliptic incident signal 1 on the quad-segment photodiode 3. If the sign of the DIAG signal is reversed, it indicates the opposite ellipticity. The quad-segment photodiode 3 may be an InGaAs-based or silicon-based detector. The signal is further converted into a voltage. Furthermore, an error signal 9 is generated using a transimpedance filter (AMP) 5, for example, using standard analog electronic equipment with an operational amplifier, and a controller 6 (not shown in Figure 1). The incident angle, i.e., the incident angles P7 and Y8 of the incident light signal 1 to 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 unfolding surface of the detector 2 (for example, by tilting the mirror upstream of the quad-segment photodiode 3). The relative change in the geometric beam shape of the incident signal 1 is most pronounced when the laser light resonates with the optical seed signal 10 (not shown in Figure 1, see Figure 3).Therefore, this method achieves the highest sensitivity at the transmission peak of the incident signal 1, which is also the transmission peak of the optical seed signal 10. The distance between the laser 11 and the detector 2 is TEM. 00 This affects the Guoy phase between the mode and the second-order spatial mode. Therefore, the image obtained on the detector 2 can be adjusted by adjusting the position of the detector 2 relative to the laser 11. The positions of the detector 2 and the laser 11 are fixed during calibration and maintained constant during operation. There is an axial distance at which the error signal 9 is purely dispersive, and other axial distances at which the error signal 9 is purely absorptive, or has any signal shape between absorptive and divergent.

[0148] Figure 2 shows the frequency dependence of the error signal 9 with respect to the optical seed signal 10 (not shown in Figure 2) near 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 cross at zero detuning of the laser 11 with respect to the optical seed signal 10. This also corresponds to the substantially circular incident signal 1 on the detector 2. As detuning occurs, the geometric shape of the incident signal 1 changes from circular to elliptic, resulting in a non-zero error signal 9. The sign of the error signal 9 indicates the direction of the elliptic 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 value 13 and the maximum value 14 of the error signal 9. The interval has values ​​of the error signal 9 corresponding to the determined ellipticity of the incident signal 1. The incident signal 1 exhibits inverse ellipticity on the opposite side of the peak of the transmitted signal 12. This is used to determine the error signal 9, which is then used to control the laser 11 and stabilize the laser output signal 15. The incident signal 1 can be considered as the interference between the resonant "00" mode leaking from the laser 11 and the second-order "+" mode that is directly reflected from the laser 11 and forms a phase reference. This detection modality makes the error signal 9 insensitive to alignment drift and variation, because small alignment deviations generate (virtually only) the first-order mode in the mode decomposition of the incident signal 1. The slope of the error signal 9 is used to control the laser 11 and achieve laser locking to the optical seed signal 10.

[0149] Figure 3 schematically shows an 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 that 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 being re-injected 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 towards the detector 2 and used as the incident signal 1 propagated on the detector 2. The axial distance between the laser 11 and the detector 2 is set so that the transmission peak 12 of the incident signal 1 coincides with the 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 QPD3, as described in relation to Figure 1. The controller 6 is configured to receive a signal from the sensor region 4 of the QPD3 and derive an error signal based on its geometric beam shape, in this case its ellipticity. The derivation of the error signal 9 involves applying an offset to the error signal 9 so that the error signal 9 is zero at a setpoint. The setpoint is determined by the target state of the laser resonator 22 of the laser 11. For example, the target state is when the optical seed signal 10 is TEM 00 This is achieved when the laser mode is perfectly matched and / or when the center frequency of the optical seed signal 10 resonates with the laser. The controller 6 is configured to stabilize the laser 11 to a set value by controlling one or more operating parameters of the laser 11 based on the error signal 9. The operating parameters of the laser 11 with the laser resonator 22 include the following: -Temperature of laser resonator 22, -Path length of laser resonator 22, - Pumping of laser resonator 22, - Phase of optical seed signal 10.

[0151] 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. By adjusting the wavelength of the optical seed signal 10, the error signal 9 is changed. The change in the error signal 9 causes the controller 6 to control at least one operating parameter of the laser 11, thereby restoring the relationship between the laser 11 and the optical seed signal 10, in other words, bringing it to 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 through a closed control loop. The laser 11, beam splitter 21 and detector 2 can be integrated on a general micro-optics or silicon photonics platform (not shown).

[0152] The closed-loop control system 16 performs a method for monitoring the optical signal 15 extracted from the optical resonator 42 (see, for example, Figure 7), or in this embodiment, the laser resonator 22. This method is - A step of receiving at least a portion of the optical signal (15) as the incident signal 1, - A step of converting the incident signal 1 to an error signal 9 (see Figure 2) by applying a conversion configuration that includes propagating the incident signal 1 to a detector (2) and deriving an error signal 9 based on the geometric beam shape of the incident signal 1 propagating to the detector 2 (see Figures 1 and 2), particularly the beam shape ellipticity, for example, in proportion to the beam shape ellipticity. This includes the following: Here, the error signal 9 has a local minimum, in particular a minimum value 13 (see Figure 2) and a local maximum, in particular a maximum value 14 (see Figure 2), where the local minimum and local maximum demarcate an interval 59 of the error signal value (see, for example, Figure 2 or Figure 10), and the transformation configuration configures the interval 59 such that the error signal 9 includes a zero crossing 58 (see Figure 10) when the optical resonator 42 is in the target state.

[0153] Figure 4 shows another closed-loop control system 16 for controlling a self-injection synchronous laser 11, which is used to inject a portion of the output signal 15 of the laser 11 back into the laser 11. Compared to the configuration in Figure 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, and a 50 / 50 splitting ratio is particularly beneficial in the illustrated configuration. The frequency-selective back reflector 24 may be any optical element that performs self-injection by back-reflecting at least a portion of the laser output back into the laser 11. This may be, for example, a mirror (in which case it may not be frequency-selective), an optical grating, a resonant optical cavity, or any optical element that back-reflects at least a portion of the input optical power. The back reflector 24 preferably back-reflects light of a specific wavelength in order to achieve a narrow linewidth for the optical seed signal 10, thereby narrowing the 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 of the detector 2 has a detector angle 7,8 that is different from 0° with respect to the normal vector of the main extension surface of the effective area of ​​the detector 2. This minimizes the reflection of the detector back into the laser 11. In addition, the applied angles 7,8 can be used to adjust the error signal 9 during the calibration of the system 16. Alternatively, the detector surface can be used as a back reflector 24 for self-injection, and parameters other than the applied angle can be used to calibrate the error signal 9.

[0154] Figure 5 shows another embodiment of a closed-loop control system 16 for controlling a self-injection synchronous laser 11. An 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. Therefore, by controlling the optical path length of the optical resonator 25, the optical seed signal 10 can be controlled, thereby shifting the output signal 15 of the laser 11.

[0155] The laser 11 has an output signal 15, which is split by the beam splitter 21. In this embodiment, the transmitted portion is further shaped by the lens 39 and propagates through the 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 the optical resonator 25. Before the optical resonator 25, the optical output signal 15 is mode-matched to the optical resonator 25 using the lens 29. The optical resonator 25 consists of a second beam splitter 30, a second mirror 31, and a second piezo-mounted mirror 32. The spectral transmittance through the optical resonator 25 is determined using the second piezo-mounted mirror 32. The optical resonator 25 is configured such that the reflected intensity at the second beam splitter 30 is propagated onto the detector 2, which is a detection lens 33 and a QPD 3. The detection lens 33 is a convex spherical lens 34. The transmission intensity that has passed through the second beam splitter 30 and optical resonator 25 is reflected back towards the laser 11. This provides an optical seed signal 10 for self-injecting the laser 11.

[0156] Controller 6 (not shown in Figure 5; see Figure 3) derives an error signal 9 and controls one or more operating parameters of the laser 11, which is sensitive to the error signal 9. These parameters include: -Temperature of laser resonator 22, -Path length of laser resonator 22, - Pumping of laser resonator 22, - Phase of optical seed signal 10. The phase and injection of the optical seed signal 10 can be controlled by the piezo-controlled mirror 28.

[0157] In this example, laser 11 is a standard Eagleyard DFB 780nm 100mW laser. The external optical resonator 25 assembly is formed of three mirrors: a convex [r=10cm], a planar, and a concave mirror. The resonator has a linewidth of 10MHz. The L-shape of the optical resonator 25 facilitates mode matching and allows the reflected beam to be accessed as the incident signal 1 to the detector 2. However, many different resonator shapes are available, including a confocal two-mirror resonator, and only light circulating within the resonator can be self-injected into the laser, as long as the directly reflected beam is not directly reflected back into the laser.

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

[0159] In this embodiment, the error signal 9 from detector 2 is used by controller 6 to stabilize the phase of the optical seed signal 10. The incident signals to detectors 2 and 36 are used to resonate an optical resonator such as a laser resonator. 00 This can be understood as interference between the mode and the second-order "+" mode, which is reflected directly from the optical resonator and generates a phase reference. In the case of detector 2, the signal is TEM of optical resonator 25. 00This is based on the interference between the mode and the output signal 15 of the laser 11. However, the signal in the second detector 36 is TEM of the laser resonator 22. 00 This is based on the interference between the mode and the transmitted signal passing through the optical resonator 25. This is reflected back to the laser 11 and acts as the optical seed signal 10, and is also reflected from the laser resonator 22 to form a phase reference. Therefore, it is beneficial to use the error signal 9 based on the signal from the (first) detector 2 to adjust the phase between the laser resonator 22 and the optical resonator 25 (which is also the phase of the optical seed signal) using the piezo-mounted mirror 28, and to use the second error signal 38 based on the signal from the second detector 36 to control one or more of the remaining operating parameters of the laser 11. By using two detectors 2 and 36 in this embodiment, more information about injection lock is obtained and used to control the operating parameters, resulting in a more stable configuration.

[0160] Figure 7 schematically shows one embodiment of a device 40 for frequency-locking a laser 41 to an optical resonator 42. In this embodiment, the device 40 includes a laser 41, and the optical resonator 42 is formed as a three-mirror optical resonator. The laser beam from the laser 42 is coupled to a polarized optical fiber 43 by 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 "useful" light. Furthermore, a polarized optical fiber (PZF) 46 and a fiber collimator 47 are provided for emitting the laser beam from the polarized optical fiber 43. 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 (CLPs) 49 whose focusing axes are oriented perpendicular to each other. The laser beam is then irradiated onto a rotatable first polarized beam splitter (PBS) 50.

[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 focuses the laser beam reflected from the optical resonator 42. The laser beam then passes through a second polarizing beam splitter 53. The apparatus 40 further includes 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 quad-segment photodiode 55, and the detector 54 is configured to determine the error signal based on the diagonal signals of the quad-segment photodiode 55. The diagonal signals consist of the difference between the sums of the signals from the diagonal sensor regions of the quad-segment photodiode 55.

[0162] The apparatus 40 according to the embodiment shown in Figure 7 performs a method of 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. This method includes the following steps: - A step of receiving at least a portion of the optical signal 15 as the incident signal 1 (in this case, the signal incident on the four-segment photodiode 55); - A step of converting the incident signal 1 into an error signal 9 (see Figure 10) by applying a conversion configuration that includes propagating the incident signal 1 to the detector 2 (in this case, a four-segment photodiode 55) and deriving an error signal 9 based on the geometric beam shape of the incident signal 1 propagating to the detector 2 (see Figure 8C), particularly the beam shape ellipticity, for example, in proportion to the beam shape ellipticity. Here, the error signal 9 has a local minimum, in particular a minimum value 13 (see Figure 2 or Figure 10), and a local maximum, in particular a maximum value 14 (see Figure 2 or Figure 10), where the local minimum and local maximum demarcate intervals 59 of the error signal value (see, for example, Figure 10). The conversion configuration sets interval 59 such that the error signal 9 crosses zero 58 (see Figure 10) when the optical resonator 42 is in the target state. In this case, the target state refers to the optimal frequency lock of the laser 41 to the optical resonator 42.

[0163] Figures 8a, 8b, and 8c conceptually illustrate the generation of the error signal that forms the basis for laser locking (the dotted line in Figure 7 shows that the error signal is used as feedback to lock the laser, for example, by adjusting the laser current).

[0164] Figure 8a schematically shows the geometric configuration of a triangular ring resonator and its associated Hermitian-Gaussian mode resonance. In this embodiment, the optical resonator 42 consists of two planar mirrors and one convex mirror (e.g., with a radius of curvature of 5 cm). Furthermore, the inset shows the spatial amplitude structure of the "00" and "+" modes. In the intrinsic cavity modes, the "+" mode appears as a superposition of the "02" and "20" modes. Alignment drift belongs to the "01" and "10" modes. Here ν FSR ν represents the free spectral range, and n is an integer. In this example, the resonator parameters are as follows: round-trip length 7.01 cm, ν FSR =4.28GHz, '00' mode finesse 195.

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

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

[0167] The arrangement of the detection lens 52 and the detector 54 will be described later in relation to Figures 9 and 11.

[0168] Figure 9 shows the positions of the relevant elements in the beam path and the resulting beam characteristics, particularly the in-plane and out-of-plane beam waists and the cavity mode beam waist, in the embodiment of Figure 7. 00TEM 02 and TEM 20 The configuration of the optical elements that excite the modes is shown. The in-plane and out-of-plane beam waists of the laser beam coincide before reaching the beam shaper 48, which has a pair of cylindrical lenses 49. As can be seen from the figure, the beam shaper 48 imparts ellipticity to the laser beam. The laser beam is focused at the detection lens 52. The detector 54 is positioned so that the in-plane and out-of-plane beam waists coincide with the beam waists of the resonant modes. 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 adjustable interval 56 is shown in which the axial distance 57 between the detector lens 52 and the detector 54 can be adjusted for the use of an error signal for laser frequency locking. The dimensions in the figure (e.g., positions in cm in Figure 9) are selected for illustrative purposes and may take different values.

[0169] Figure 10 shows the error signal and resonator transmission in the embodiment of Figure 7. The axial distance 57 is adjusted so that the zero-crossing 58 of the error signal falls within the interval 59 between the minimum value 60 and the maximum value 61 of the error signal, and so that the error signal value corresponding to a predetermined beam ellipticity of a substantially circular laser beam falls 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 to 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] The second laser system can also be locked to the opposite side of the optical resonator 42, mirroring the left side. In principle, this three-mirror resonator can lock up to three lasers simultaneously. This can be scaled up by increasing the number of mirrors. Alternatively, if the laser wavelengths are sufficiently different, multiple lasers can be locked using the same input mirror, provided that each wavelength is separated later by a filter.

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

[0172] Figure 12 schematically shows the axial distance 57 between the detection lens 52 and the detector 57. It is also possible to center the laser beam to the detector 57. This is achieved by translating the detection lens 52 (and / or detector 54) in directions 62 perpendicular to the optical axis 63 ("LR" i.e., left-right direction and "UD" i.e., up-down direction) such that the left-right and up-down signals of the four-segment 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 regions of the four-segment photodiode 55, and the left-right signal and up-down signal use the sum of different halves of adjacent sensor regions. When the axial distance 57 is adjusted, it is desirable that the laser beam be recentered. Therefore, it is desirable that these steps be performed in a reproducible and iterative manner. That is, each time the axial distance 57 is adjusted, the LR signal and UD signal should be minimized again. By repeating this process, the zero-crossing 58 is optimized to precisely match the transmission plateau, as shown in the "optimized" error signal in Figure 11.

[0173] Figures 13a to 13c show the effective resonator under weak-value emphasis, i.e., pre-selection and post-selection of polarization. In particular, Figure 13a shows the use of the first and second polarizers in the embodiment of Figure 7 and Figure 14 described below. Reflection operator using input polarizer 11 and output polarizer 53 The weak value r of JPEG0007933876000007.jpg86 w As a result, an effective resonator with adjustable reflectivity is obtained. Both polarizers 50 and 53 are linear polarizers. Therefore, the laser beam is polarized before it reaches the optical resonator 42, and the laser beam reflected from the optical resonator is also polarized. Figure 13b shows the normalized reflected power measured for different post-selection configurations. The curve represents its off-resonance residual power fraction |<ψ2|ψ1>| 2 Labeled by [labeling]. The 1 / 2 curve is identical to the original resonator reflection curve, and the 1 / 60 curve is for an impedance-matched (γ'=1) configuration. The dashed-dotted line represents the intensity contribution level from the non-"00" mode component. Figure 13c shows the (theoretical) r when matched to the "00" mode. w The real and imaginary parts are shown, and the advantages of post-selection are demonstrated: an effective increase in the slope of the error signal and 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. These two copies share the same optical resonator 42 and otherwise each has (uniquely) the same elements as the apparatus 40 described in the context of Figure 7. Therefore, refer to the description in Figure 7, and identical elements are indicated by the same reference numbers. Two separate lasers 41 were stabilized in two degenerate opposing propagation "00" modes in the triangular ring resonator 42. The distance between a pair of cylindrical lenses 49 determined the amount of light in the "+" mode, which accounted for about 10% of the total power output (about 400 μW) used in the experiment.

[0175] Comparisons between different optical frequency stabilization systems can be achieved by evaluating their performance as fractional instability with respect to the cavity linewidth. Typically, absolute instability increases proportionally with the cavity linewidth. This is because the typical cause of the instability in question stems from variations in the error signal shape. In this experiment, a 22-MHz full-linewidth resonator and two 780nm external resonator diode lasers with natural linewidths of 500kHz were used.

[0176] Figure 15a shows the lock instability over the averaging time of the experimental configuration shown in Figure 14, compared to other stabilization methods. It shows the performance of the implementation method while locked to the "00" mode resonance. The averaging time ranges from 0.25 seconds to 100 seconds. -7 It reached the lock instability region, surpassing previous cutting-edge technologies, and achieved a 5 × 10 averaging time of 10 seconds. -7 The instability was reached. The Allan deviation of beat notes recorded for 10 hours was taken, and it was assumed that the instability was equal between the two rocks. The comparison was made with respect to the following method. -Salomon (Salomon, C., Hills, D. & Hall, J.L. Laser stabilization at the millihertz level. J.Opt.Soc.Am.B5, 1576 (1988)): Transmission modulation method; -Slagmolen (Slagmolen, BJJ, Shaddock, DA, Gray, MB & McClelland, DEFrequency stability of spatial mode interference (tilt) locking). IEEE J. Quantum Electron. 38, 1521-1528 (2002).): Double-pass "tilt-lock" 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). The data was extracted and expressed as relative values ​​of the cavity line widths used in each study.

[0177] Figure 15b shows a comparison of the achieved instability with and without post-selection on the upper side, and the corresponding beat note traces on the lower side.

[0178] Figure 15c shows the estimated limits to stability in post-selection and post-selection operations. The primary contribution from beam shape noise is shown by a solid line, and secondary contributions are shown by a dashed line. The secondary contributions are as follows: measured in-loop error signal noise (X), detector dark noise (◇), and laser intensity drift noise (〇).

[0179] A more detailed description of the experiment conducted with the configuration shown in Figure 14 follows. However, it should be noted that different parameters may be used, but this should not be considered limiting to the invention.

[0180] The optical resonator 42 consists of one high-reflectivity curved mirror (radius of curvature 5 cm) and two low-reflectivity (-98.93 for p-polarized light at an incident angle of 30°) planar mirrors. The reflectivity is chosen so that the linewidth of the resonator is greater than that of the laser 41 (500 kHz) for simplicity. The mirrors are mounted in epoxy resin to an aluminum block with three perforations forming a nearly equilateral triangle. A fiber collimator 47 and a pair of cylindrical lenses 49, which form the beam shaper 48, are mounted in a cage assembly located on a 5-axis tip-tilting translation stage. This assembly facilitates mode matching to the optical resonator 42, along with the axial translation function of the fiber collimator lenses. The optical resonator 42, along with the other optics, is all placed directly on an optical table and enclosed in a metal box for stability against temperature and air fluctuations. An aluminum cavity spacer is stabilized within 1 mK to control the cavity length.

[0181] To experimentally identify the cavity mode reference profile (see also Figure 9), light is backcoupled to the "00" mode from one of the mirrors, and the mode profile emitted from the resonator 42 (along the path to the quad-segment photodiode (QPD) 55) is analyzed with a CMOS camera. The incident beam profile is adjusted by positioning the optical elements to match the reference mode upon reflection from the resonator. Once aligned, all modes except the "00" mode and the "02"-"20" modes are suppressed, and the transmission levels of the latter two become equal.

[0182] Following the alignment of the input light, the position of the QPD55 is aligned. To center the beam on the QPD55, the detection lens (DL) 52 is translated while monitoring the left-right and up-down signals from the QPD55. Once the beam is centered, the baseline offset on the error signal is due to the axial displacement of the QPD55's position from the position where the beam has a circular profile. The position of the QPD55 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 transmitted peak from the error signal zero-crossing 58. Therefore, it is possible to adjust the axial position of the QPD55 until the transmitted peak and the error signal zero-crossing 58 coincide. If this coincidence does not occur, the intercoupling of intensity noise with frequency noise appears. Electronic backend:

[0183] The QPD55 and feedback circuit are homemade. All quadrants of the QPD55 are independently amplified with a transimpedance gain of 380kΩ. Then, calculations are performed in the operational amplifier circuit to simultaneously access the diagonal, left / right, up / down, and sum signals. These, respectively, yield the resonator lock error signal, the beam centering signal, and the overall light intensity reflected from the resonator.

[0184] Signals from the diagonal channels go to an analog feedback circuit and supply laser current to maintain resonance with the resonator of laser 41 when locked. The feedback loop includes two integrators (1 / f² response) from DC to 20 kHz and one integrator (1 / f response) from 20 kHz to a lock bandwidth of 100 kHz, providing lock tightness that does not impose significant limitations on stability (marked with an "x" in Figure 15c). For ground loop elimination, which is important for performance, all input and output signals to the feedback circuit are differential.

[0185] The QPD55 and its implemented amplification circuit are limited by shot noise above 300 Hz. Below this frequency, amplification noise becomes dominant. However, the lock stability observed on timescales exceeding 10 ms is independent of both shot noise and amplification noise. This is consistent with other experiments operating on such timescales. System performance, comparison, and noise characteristics evaluation:

[0186] Assuming that two copies of the laser resonator-locked 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 are added orthogonally. This correction has already been incorporated in the explanation of Figure 15 and is a standard procedure. This generally underestimates the achievable instability, because if we assume that one of the two configurations has higher performance, we would need to divide by an even larger number to extract the performance of that configuration. In this case, since the analyzed beam shape noise, which limits performance, turned out to be equal for both copies, the performance of both copies is approximately equal.

[0187] To compare with previous state-of-the-art technologies, instability data was extracted from the studies cited in Figure 15 (see below) using a web plot digitizer, and data representing the ratio to the cavity linewidth used in each study were measured. In selecting specific publications for comparison, particular attention was paid to studies containing two or more similar configurations to obtain actual measured stability via laser beat notes. Note that lock stability can be inferred from a single copy system by recording the in-loop error signal and using an appropriate transformation factor. This method is not necessarily reliable because the feedback loop suppresses the error signal regardless of its physical origin. For example, the in-loop error signal in this study is continuously 10 -8 Although it is averaged out on an instability scale, the actual stability limit is brought about by beam-shape noise causing the error signal to deviate from the actual frequency value. In the comparative data, the curves for "Slagmolen" and "Salomon," along with the results presented in this study, were obtained by locking two lasers onto the same resonator, while the curve for "Matei" was obtained by comparing two lasers locked onto separate resonators, and is limited by the thermal noise of the resonators used.

[0188] The estimated limits to stability shown in Figure 15c (see below) were performed out of situ. Beam shape noise was measured with the cavity modes physically blocked. Therefore, there is no accumulation of light within the resonator. In this configuration, the input mirror directs the incident light directly to the QPD, making it possible to identify variations in beam ellipticity within the incident beam. No further modeling was performed to account for the shape noise filtering effect that the resonator would perform when the modes are not blocked. QPD dark noise was measured by blocking the incident light and monitoring the error signal. Intensity drift noise was measured by simultaneously monitoring the laser beat note frequency and the total channels of the QPD 55 while the laser 41 was locked into the resonator 42. Except for intensity noise, the measured values ​​were converted to frequencies using the slope of the error signal at resonance. For intensity drift noise, the measured laser intensity dependence of the beat note during lock operation was used. Note that no active intensity stabilization was performed in this study.

[0189] It was verified that the stability limit actually stems from variations in beam ellipticity, and not indirectly from beam directivity variations combined with the non-ideal nature of the QPD detection system. By translating the QPD position perpendicular to the beam axis, 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 an order of magnitude larger than the voltage noise in the ellipticity signal, thus ruling out beam directivity crosstalk as the primary noise source. A narrower laser linewidth, assuming no other dominant defect sources in the system, theoretically improves stability on a short-time scale. In this study, there was no performance degradation even when the laser linewidth was artificially widened from 500 kHz to 1 MHz (by injecting current noise). This suggests that linewidth is not a limiting factor. It should also be noted that in this parameter domain, since the lock bandwidth (100 kHz) is smaller than the initial laser linewidth, no reduction in laser linewidth occurred. The advantage of the locking method examined in this study compared to the second-best "transmission modulation" method (Figure 15a) is the use of a reflected beam, a property common to the PDH method as well. In this operating mode, the lock bandwidth is not directly limited by the cavity linewidth. This is important in the context of narrow-linewidth, high-finesse resonators when attempting to narrow the laser linewidth in addition to stabilizing the center frequency.

[0190] As a further point, it has been shown that the fundamental limitations of many frequency-locking techniques using optical shot noise are the same within a numerical range of one order of magnitude, and this 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 sensitivity to beam directivity variations. These variations only secondarily affect the signal in the diagonal QPD channel and do not appear to cause limitations in this study, but are nevertheless further reduced. Another non-ideal aspect suppressed by post-selection is the adverse effect of astigmatism in the triangular cavity modes. Astigmatism causes the zero-crossing of the error signal to be shifted from the maximum transmitted intensity. As a result, performance is reduced through intensity-frequency noise conversion. As discussed in the alignment procedure, this effect can be mitigated by fine-tuning the alignment, but even so, post-selection significantly suppresses these problems and facilitates alignment.

[0192] Qualitatively, the improvement brought about by post-selection can be thought to stem from differential measurements performed on two orthogonally polarized beams: one entering the resonator and another directly reflected 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, it is also observed that post-selection introduces a small sensitivity of the frequency lock point to changes in cavity length. Preliminary characterization suggests that this increased sensitivity may be due to a change in the polarization of light reflected from the resonator with changes in optical frequency, effectively changing the post-selection angle. Although cavity length is a parameter that is inevitably well controlled in frequency stabilization applications, it is not a dominant limitation in this implementation. Application to different cavity shapes:

[0194] In Figures 7 and 14, ring resonators are used in which the incident and reflected beams travel along separate paths. Different applications may require different resonator designs. For example, ultra-low vibration sensitivity generally requires the use of a standing wave resonator with two mirrors. In this configuration, nothing changes in terms of the primary method. The reflected beam travels along the same path as the incident beam, but can be easily separated by using a quarter-wave plate with a polarizing beam splitter, for example (as is commonly done in the PDH method). Also note that the specific mode spectrum of the resonator is irrelevant to the implementation. The only requirement for proper operation is TEM 00 The mode is not spectrally degenerate with the second mode, and therefore the second mode can have any common resonant frequency. This primary method can also be used for resonators such as waveguides that require evanescent coupling, such as 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 the light in and out of the resonator. In this way, the prism effectively connects the structure on the waveguide to the free-space optical system and looks like an input mirror for the free-space resonator in applying the method developed in this study. Post-selected beam deployment and weak value of the reflection coefficient

[0195] Here, we outline the derivation of the effective reflection coefficient and related equations used above. First, we model cavity reflection without the degree of freedom of polarization in the quantum mechanical language used to describe weak values, and then we take polarization into account.

[0196] In the first case, the problem has two degrees of freedom: "mode shape" and "channel". The state that spreads in the first degree of freedom is various TEM modes, and the state that spreads in the second degree of freedom is "incident". JPEG0007933876000008.jpg613, "Reflection" JPEG0007933876000009.jpg613, "Transparent" JPEG0007933876000010.jpg613 is displayed. The initial state is defined as a product state of two degrees of freedom: JPEG0007933876000011.jpg738. Here, the state "Φ" represents an arbitrary input mode shape. To phenomenologically describe the reflection near the "00" resonance from the cavity, a unitary operator JPEG0007933876000012.jpg797 is defined. If the "mode shape" is in the state |0>, this evolves the "channel" state with another unitary operator JPEG0007933876000013.jpg89, otherwise, it directly evolves the incident state JPEG0007933876000014.jpg58 to the reflection state JPEG0007933876000015.jpg69. Here, JPEG0007933876000016.jpg86 is the identity operator for the "mode shape" degree of freedom, and |0><0| is the projection operator for the "00" mode. When considering only the reflected portion of the beam, the effective evolution operator R̂ that acts only on the input "mode shape" state |Φ> can be defined using the projection onto the reflected portion after the state evolution: JPEG0007933876000017.jpg748. This procedure yields the effective evolution operator JPEG0007933876000018.jpg751. Here JPEG0007933876000019.jpg839 is the amplitude reflection coefficient, which is quantified experimentally.

[0197] Next, when the polarization degree of freedom is included, the initial state JPEG0007933876000020.jpg752 A more general unitary operator that evolves is introduced. Here, |ψ1> is the initial state of the polarization degree of freedom. Using the above procedure, the effective evolution operator R̂=R̂ that acts only on the initial "mode shape" and "polarization" states JPEG0007933876000021.jpg727 H +R̂ V is derived, where JPEG0007933876000022.jpg897 and JPEG0007933876000023.jpg890 are obtained. In these expressions, r H and r V are the amplitude reflection coefficients for the horizontal polarization state |H> and the vertical polarization state |V> relative to the principal axis of the cavity. Here, a potential relative birefringence phase shift Φ is explicitly included for direct reflection from the tilted cavity input mirror. Using the definition R^ post |Φ>=<ψ'2|R^|Φ>|ψ1> to further project onto the polarization state |ψ'2>, the final post-selection evolution operator R^ post =<ψ2|ψ1>[r w |0><0|+(I-|0><0|)] is obtained. This acts purely on the input mode state |Φ>. Here, r w =<ψ2|r^|ψ1> / <ψ2|ψ1> is the weak value of the reflection operator r^=r H |H><H|+r V |V><V|, and the overlap coefficient <ψ2|ψ1> of R^ post further represents the loss property 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: |ψ2>=(|H><H|+|V><V|e -iΦ )|ψ'2>. Although the small birefringence phase is neglected above, the asymmetry remaining in the curve of FIG. 15b may be caused by this.

[0198] Next, r related to this experimental configuration w is evaluated. By using linear polarizers, the pre-selection state and the post-selection state can be expressed as |ψ1>=cosθ1|H>+sinθ1|V> and |ψ2>=cosθ2|H>+e -iΦ sinθ2|V>. Accordingly, the weak value becomes r w =r H ((1 / 2)+B)+r V ((1 / 2)-B), and the overlap coefficient is <ψ2|ψ1>=cos(θ2+θ1) / (B+(1 / 2)+e -iΦ (B-(1 / 2))). Here, the complex number B=(1+e -iΦtanθ2tanθ1) -1 -(1 / 2) is defined. In this study, the cavity is of polarization splitting type, and the vertical polarization component is regarded as a largely detuned resonance, so r H =1-γ / (1-i δ / (κ / 2) ), and r v =1 can be used. Here, γ, δ, and κ are as defined in the main text. Thereby, the weak value becomes r w =1-γ(B+(1 / 2)) / (1-i δ / (κ / 2) ). Finally, the small birefringence phase approximation Φ<<1 is applied, and performing a power series expansion of Φ leads to the formula used in the main text. As a result, JPEG0007933876000024.jpg1446(2) JPEG0007933876000025.jpg14118(3) is obtained. Here, γ'=Aγe -iΦ(A-1) , and A=(1 / 2)+B| Φ=0 . The successful post-selection probability can be expressed as follows. JPEG0007933876000026.jpg970 Here, the associated post-selection angle is assumed to satisfy cos(θ2+θ1) 2 ~1. The expression used above further sets Φ=0.

Claims

1. A method for monitoring an optical signal (15) extracted from an optical resonator (42), The steps include receiving at least a portion of the optical signal (15) as an incident signal (1), The steps include converting the incident signal (1) to an error signal (9) by applying a conversion configuration that includes propagating the incident signal (1) to a detector (2) and deriving an error signal (9) based on the geometric beam shape of the incident signal (1) propagated on the detector (2), Includes, The error signal (9) has a minimum value and a maximum value that depend on the transverse spatial mode configuration of the optical resonator (42). The minimum value and the maximum value demarcate the interval (59) of the error signal (9). The conversion configuration is a method of configuring the interval (59) such that the error signal (9) crosses zero (58) when the optical resonator (42) is in the target state.

2. The method according to claim 1, wherein the conversion configuration is configured such that the interval (59) includes the value of an error signal (9) corresponding to a circular image of the incident signal (1) propagated on the detector (2).

3. The method according to claim 1 or 2, wherein the detector (2) is a four-segment photodiode (3), and the derivation of the error signal (9) includes determining the diagonal signals of the four-segment photodiode (3), the diagonal signals being composed of the difference between the sum of signals from the diagonal sensor regions of the four-segment photodiode (3).

4. The method according to claim 1, wherein the propagation of the incident signal (1) includes adjusting the incident angle (7, 8) of the incident signal (1) onto the detector (2).

5. The method according to claim 1, wherein the propagation of the incident signal (1) includes shaping the incident signal (1).

6. The method according to claim 1, wherein the derivation of the error signal (9) includes giving an offset to the error signal (9) such that the error signal (9) is set to zero when the optical resonator (42) is in the target state.

7. The method according to claim 1, wherein propagating the incident signal (1) includes using a detection lens (33) to focus the incident signal (1) onto the detector (2).

8. The method according to claim 7, wherein propagating the incident signal (1) includes adjusting the axial distance (57) between the detection lens (33) and the detector (2) such that the transmitted peak of the incident signal (1) coincides with the zero-crossing of the error signal (9).

9. The method according to claim 8, wherein adjusting the axial distance (57) between the detection lens (33) and the detector (2) includes adjusting the axial distance (57) such that the beam waist of the incident signal (1) in a first plane containing the optical axis of the incident signal (1) and the beam waist of the incident signal (1) in a second plane that is not parallel to the first plane and contains the optical axis of the incident signal (1) coincide with the beam waist of the resonant cavity mode of the optical resonator (42).

10. The detector (2) is a four-segment photodiode (3), and the incident signal (1) is focused onto the four-segment photodiode (3) by a detection lens (33). The method according to claim 1, further comprising the step of centering the incident signal (1) on the four-segment photodiode (3) by translating the detection lens (33) and / or the four-segment photodiode (3) in a direction perpendicular to the optical axis of the incident signal (1), wherein the left-right signals and the up-down signals are each composed of the difference in the sum of adjacent sensor regions (4) of the four-segment photodiode (3), and the left-right signals and the up-down signals use the sum of different halves of adjacent sensor regions (4), the left-right signals and / or the up-down signals of the four-segment photodiode (3) are minimized.

11. The method according to claim 10, wherein the steps of adjusting the axial distance (57) between the detection lens (33) and the four-segment photodiode (3) along the optical axis of the incident signal (1) and centering the incident signal (1) on the four-segment photodiode (3) are performed iteratively.

12. A method for frequency locking a laser (41) to an optical resonator (42) according to claim 7, The steps include providing a laser beam from the laser (41), The steps include shaping the laser beam so that it acquires ellipticity, A step of directing the laser beam towards the optical resonator (42), wherein the laser beam reflected from the optical resonator (42) is the incident signal (1) propagated to the detector (2), A step of locking the laser (41) to the optical resonator (42) based on the error signal (9), Methods that further include this.

13. Shaping the laser beam so that it acquires ellipticity is performed by a pair of cylindrical lenses (49) whose focal axes are oriented non-parallel to each other, Shaping the laser beam so that it acquires ellipticity is done by a combination of lenses, wedge prisms, and / or pairs of anamorphic prisms, which generate an elliptic or astigmatic beam. The method according to claim 12, wherein at least one of the above.

14. The method according to claim 12, wherein the laser beam is subjected to a polarization filter before being directed towards the optical resonator (42), and the incident signal (1) is polarization filtered.

15. The method according to claim 1, 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 as follows: The steps include adjusting the temperature of the optical resonator (42), A step of adjusting the optical path length of the optical resonator (42), The method according to claim 15, comprising one or more of the above.

17. The method according to claim 1, wherein the optical resonator (42) is a laser resonator (22) of a laser (11).

18. The method according to claim 16, wherein the adjustment of the optical resonator (42) includes the step of adjusting the current of the semiconductor laser (17).

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