Optical device and method
The optical device and method utilize a multi-mode optical parametric oscillator with an optical feedback loop to achieve sub-Hertz linewidth and broad tunability, addressing the limitations of existing narrow-linewidth lasers.
Patent Information
- Application Number
- PCT/EP2024/078420
- Authority / Receiving Office
- WO · WO
- Patent Type
- Applications
- Current Assignee / Owner
- Priority Date
- 2023-10-10
- Filing Date
- 2024-10-09
- Publication Date
- 2025-04-17
AI Technical Summary
Existing narrow-linewidth lasers face challenges in achieving extended wavelength regions and tunability due to limitations in gain medium optical transitions and stringent phase matching conditions.
A multi-mode optical parametric oscillator with a main optical resonator cavity made of nonlinear resonator medium, optically coupled with a continuous-wave laser, and an optical feedback loop with a band-pass filter to select and re-inject a frequency mode, thereby reducing the linewidth of the reinjected line.
The solution achieves a coarse tuning of up to 20 nm and an intrinsic linewidth down to sub-Hertz level, three orders of magnitude lower than the pump, enabling simultaneous narrow-linewidth emission and continuous tuning across a broad wavelength range.
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Figure EP2024078420_17042025_PF_FP_ABST
Abstract
Description
[0001] Optical device and method
[0002] TECHNICAL FIELD
[0003] The invention relates to a multi-mode optical parametric oscillator having a main optical resonator cavity made of nonlinear resonator medium, wherein the main optical resonator cavity is configured to receive a continuous-wave laser light being optically coupled therewith. The invention also relates to a method for producing laser light with a narrow linewidth.
[0004] BACKGROUND ART
[0005] Narrow-linewidth and tunable lasers are essential for applications that require low-phase-noise and wavelength versatility. Applications include precision spectroscopy, quantum optics, optical atomic clocks, sensing, high performance communications and LIDAR, to name a few. The most widely used approaches to implement narrow-linewidth lasers have relied on spectral purification of a free-running diode, solid-state or fibre laser, for example, active frequency locking the laser to a high-Q cavity with an electronic servo element or optical feedback of a laser with an external optical element such as grating, mirror, cavity or waveguide. Recent advances in photonic heterogeneous integration have enabled coherent lasers with a linewidth at the 1 Hz level on a chip. The gain of most conventional narrow-linewidth lasers is mainly provided by stimulated emission, thus their operating wavelength bands, i.e., maximum tuning ranges are fundamentally limited by the optical transitions allowed in the gain medium.
[0006] In principle, narrow-linewidth and tunable coherent oscillators can be attained at extended wavelength regions by making use of nonlinear optical frequency conversion processes, such as harmonic generation, sum / difference frequency generation or four-wave mixing. However, this remains a challenging endeavour because one or two narrow-linewidth pump lasers are required, and the phase matching condition needs to be stringently satisfied over a large frequency range. Stimulated Brillouin scattering is another well-known mechanism for implementing narrow-linewidth lasers; however, the emission wavelength is fundamentally constrained by the pump.
[0007] There is thus a need for improvements within this field.
[0008] SUMMARY OF THE INVENTION An object of the present disclosure is to provide an optical device and a method for producing laser light with a narrow linewidth that seeks to mitigate, alleviate, or eliminate one or more of the above-identified deficiencies in the art and disadvantages singly or in any combination.
[0009] This object is obtained by an optical device comprising:
[0010] - a multi-mode optical parametric oscillator having a main optical resonator cavity made of nonlinear resonator medium, wherein the main optical resonator cavity is configured to receive a continuous-wave laser light being optically coupled therewith,
[0011] - an optical feedback loop connected to the output of the optical parametric oscillator, wherein the optical feedback loop comprises a band-pass filter arranged to select a frequency mode of the parametric oscillator, wherein the optical feedback loop is configured to re-inject a selected line from the multi-mode parametric oscillator into the main optical resonator cavity, thereby reducing the linewidth of the reinjected comb line.
[0012] The invention provides an optical device in the form of a tunable coherent oscillator based on a multimode coherent optical parametric oscillator (OPO) in a high-Q microresonator, i.e., a microcomb. Single-mode coherent oscillation is realized through re-injection of a selected comb line into a main optical resonator cavity. This is also known as self-injection locking (SIL) of one selected comb line. The optical device achieves a coarse tuning of up to 20 nm, and an intrinsic linewidth down to sub-Hertz level, which is three orders of magnitude lower than the pump.
[0013] Besides the spectral measurement of comb lines, the stability of the repetition rate has also been measured. For this purpose, the repetition rate is electro-optic down-converted into low frequency range (~ 470 MHz) with a 25 GHz intensity modulation, and then its phase noise is measured with a phase-noise analyser. It has been shown that single-sideband phase noise is reduced at low-offset frequencies while increased at higher frequency offset (>100 kHz) after SIL. This phenomenon can be understood as the repetition rate phase noise is mainly determined by the relative frequency stability between the pump and the narrow-linewidth oscillator. At lower offset frequencies, the relative frequency noise is mainly limited by the linewidth of the comb line due to the thermorefractive noise; therefore, comb linewidth narrowing can lead to lower repetition rate phase noise. While, at higher offset frequency region, the linewidth of the pump becomes to be a limiting factor as comb line -33 features a much lower frequency noise (or higher at the spike frequencies). Hence, in order to exploit the narrow-linewidth oscillator for extremely pure microwave generation, a narrow-linewidth pump laser is required.
[0014] The band-pass filter may be tunable. This allows for an effective selection of the comb line that is to be re-injected into the main optical resonator cavity.
[0015] The nonlinear resonator medium may be a second or third order non-linear medium.
[0016] The optical feedback loop may be one of a waveguide, optical fibre, integrated optical waveguide or a free space optical cavity. The optical feedback loop may be longer than the main optical resonator cavity. These alternatives have been shown to produce a well-functioning feedback loop.
[0017] The optical feedback loop may comprise an optical amplifier. The optical feedback loop may comprise a phase tuner. The optical feedback loop may comprise at least one polarization controller and / or at least one attenuator. Each one of these additional components may add to the controllability of the optical device.
[0018] The feedback power ratio (q) of the selected frequency mode into the main optical resonator cavity may be less than one. It has been shown that the invention works well with such a feedback power ratio (q).
[0019] This object is further obtained by a method for producing laser light with a narrow linewidth, wherein the method comprises:
[0020] - providing a multimode optical parametric oscillator having a main optical resonator cavity made of nonlinear resonator medium,
[0021] - providing an optical feedback loop connected to the optical parametric oscillator,
[0022] - receiving, into the main optical resonator cavity, a continuous-wave laser light being optically coupled therewith,
[0023] - selecting a frequency mode of the parametric oscillator, - selecting a line to re-inject into the main optical resonator cavity via the optical feedback loop, wherein the optical feedback loop comprises a band-pass filter arranged to select the line,
[0024] - re-injecting the selected line from the multi-mode parametric oscillator via the optical feedback loop into the main optical resonator cavity, thereby reducing the linewidth of the reinjected line.
[0025] The method may further comprise tunably selecting the line of the multi-mode parametric oscillator using a tunable band-pass filter.
[0026] The method may further comprise arranging the optical feedback loop to be longer than the main optical resonator cavity.
[0027] The method may further comprise feeding back the selected line into the main optical resonator cavity with a feedback power ratio (q) of less than one.
[0028] In this application, a narrow-linewidth and tunable coherent oscillator based on self-injection locking (SIL) of a multimode continuous-wave OPO. In the examples of the application, a microcomb with a single-soliton state is employed here. However, other coherent states such as Turing rolls or other soliton states can also be leveraged. Like standard SIL configuration, the system is composed of a microcomb and an external optical feedback loop.
[0029] In one example, to ensure a single-frequency narrow-linewidth oscillator generation, a tunable band-pass filter (TBPF) is inserted into the feedback loop for selecting one comb line that can be arbitrary except for the pump.
[0030] Although optical gain can be involved into the feedback loop to compensate the loss, however, in principle the feedback loop can be purely passive. This configuration is thus fundamentally different from laser cavity microcombs, where the gain plays a crucial role for the comb generation. It is worth noting that, the terminology of SIL in microcombs have been utilized in different contexts, including mode-locked state generation and pump diode laser SIL. OPO SIL has been discussed in the scenario of optical frequency division where the signal and the idler are subharmonics of the pump. The optical device of the application demonstrates the feasibility of dramatic spectral narrowing of a continuous wave (CW) OPO with SIL, and realize the first sub-Hertz intrinsic linewidth oscillator based on parametric gain. The invention shows that there exists a dynamic regime in which the frequency noise of the comb line can be consistently reduced to be three orders lower than the pump, regardless of the feedback phase. The invention lays the foundation for understanding SIL dynamics in OPOs.
[0031] In an alternative example of the invention, an arbitrary comb line (except for the pump) for SIL can be selected by simply setting the TBPF. Therefore, the invention enables simultaneous narrow-linewidth emission and continuous tuning in extremely broad wavelength range. Furthermore, as the oscillator is one comb line of the soliton microcomb, its spectral purification can spontaneously facilitate for microcomb stabilization beyond the limits imposed by the thermorefractive noise in the cavity.
[0032] BRIEF DESCRIPTION OF THE DRAWINGS
[0033] Figure 1 schematically shows an optical device according to a first embodiment of the invention,
[0034] Figure 2 schematically shows an optical device according to a second embodiment of the invention,
[0035] Figure 3 schematically shows a graph over the power spectral density for one comb line with and without re-injection of a selected comb line into a main optical resonator cavity,
[0036] Figure 4 shows a flowchart of a method for producing laser light with a narrow linewidth.
[0037] DETAILED DESCRIPTION
[0038] Aspects of the present disclosure will be described more fully hereinafter with reference to the accompanying drawings. The apparatus disclosed herein can, however, be realized in many different forms and should not be construed as being limited to the aspects set forth herein. Like numbers in the drawings refer to like elements throughout.
[0039] The terminology used herein is for the purpose of describing particular aspects of the disclosure only, and is not intended to limit the disclosure. As used herein, the singular forms "a", "an" and "the" are intended to include the plural forms as well, unless the context clearly indicates otherwise.
[0040] Unlike ordinary lasers, the gain of optical parametric oscillators (OPOs) originates from an optically driven nonlinear polarization, thus the operating wavelength band is only limited by the absorption characteristics of the nonlinear medium used and the ability to engineer the phase matching among the waves involved in the process. Nowadays, OPOs are widely exploited for tunable light generation ranging from UV to terahertz, particular in the wavelength regions where ordinary lasers hardly reach. The fundamental parametric process can be described as a new pair of photons, referred to as the signal photon(s) (a>s) and the idler photon(s) (c ^), created from one (or two) pump cp) photon(s) mediated by a (or ) nonlinear medium.
[0041] According to energy conservation, the sum of the frequencies of signal and idler waves follows the pump frequency (or its double in the case), while their relative frequency could undergo a random diffusion process. Over the past decades, several approaches have been pursued in view of generating narrow-linewidth signal (or idler) via OPOs. The most commonly applied approaches include utilizing frequency selective elements such as intracavity etalons or gratings.
[0042] Self-injection seeding is an alternative technique for spectral narrowing in pulse pumped OPOs. These all-optical approaches permit signal spectral narrowing for several orders with compatibility of broadband wavelength tuning, however, the narrowing effect is usually insufficient to reach a signal wave with a linewidth smaller than the pump. Another strategy is to stabilize the signal or idler into a frequency reference, such as a Fabry-Perot (FP) cavity, a frequency comb, atomic resonance, or narrow-linewidth laser. These approaches could give superior spectral coherence but suffers from using a sophisticated servo system.
[0043] Figure 1 schematically shows an optical device 1 according to a first embodiment of the invention. Figure 1 is a more general description of the optical device 1 according to the invention. The optical device 1 comprises a multi-mode optical parametric oscillator 2 having a main optical resonator cavity 3 made of nonlinear resonator medium 4. The main optical resonator cavity 3 is configured to receive a continuous-wave laser light 5 being optically coupled therewith.
[0044] An optical feedback loop 6 is connected to an output of the optical parametric oscillator 2. The optical feedback loop 6 comprises a band-pass filter 7 arranged to select a frequency mode of the optical parametric oscillator 2. The optical feedback loop is configured to re-inject a selected line 8 into the main optical resonator cavity 3, thereby reducing the linewidth of the reinjected line.
[0045] To be able to choose selectively which line that is to be fed back to the main optical resonator cavity 3, the band-pass filter 7 may be a tunable band pass filter.
[0046] The nonlinear resonator medium 4 of the main optical resonator cavity 3 may be a second or third order non-linear resonator medium.
[0047] The optical feedback loop 6 may be one of a waveguide, optical fibre, integrated optical waveguide or a free space optical cavity and may be longer than the main optical resonator cavity. The optical feedback loop may further comprise an optical amplifier. The optical feedback loop may further comprise one or more of a phase tuner, at least one polarization controller and at least one attenuator.
[0048] A feedback power ratio (q) of the selected line into the main optical resonator cavity is less than one. Similar to semiconductor lasers with optical feedback, this system contains a wealth of nonlinear dynamical behaviours. It has been found that the dynamics of this system can be classified as four distinct feedback regimes according to the comb line's noise performance. Regime I corresponds to very weak feedback strength. The spectral width can be slightly broadened or narrowed from time to time, inferred as the variation of the feedback phase. With increasing feedback strength, spectral line narrowing is observed for most of time.
[0049] However, mode hopping starts to occur frequently with a given feedback strength, reflected as a burst of spikes in the beating spectrum. This regime is termed as regime II. When the feedback reaches to a certain level (regime III), the mode hopping ceases altogether and the line is persistently narrowed. This means the spectral narrowing is no longer sensitive to the feedback phase. Therefore, this regime allows obtaining an extremely narrow-linewidth oscillator under relaxed conditions. In the following, the SIL state refers to this regime unless otherwise specified. If the feedback strength is increased further (regime IV), a so-called coherence- collapsed state is obtained. Not only the original beating note spectrum gets broadened, but also new frequency components emerge. This regime corresponds to chaos generation. If the feedback ratio approaches one, the feedback power is too strong and the soliton would vanish. The optical device 1 produces a coherent single-frequency narrow-linewidth oscillator output from the optical parametric oscillator 2.
[0050] Figure 2 schematically shows an optical device 1 according to a second embodiment of the invention. Figure 2 shows a more detailed setup of an optical device 1 of the invention. In figure 2, a continuous wave laser 5 provides laser light into the main optical resonator cavity 3, where the laser light is optically coupled with the main optical resonator cavity 3. A first amplifier 9, in this example an erbium doped fibre amplifier (EDFA), amplifies the laser light before it enters the main optical resonator cavity 3. Alternatives to the EDFA can be a semiconductor optical amplifier, a Brillouin amplifier or a Raman amplifier, among others.
[0051] The optical device 1 of figure 2 comprises a main optical resonator cavity 3 in the form of an integrated silicon nitride (SisIX ) waveguide in the shape of a microresonator used for microcomb generation. The height and the width of the SisIX waveguide are in one non-limiting example 740 nm and 1800 nm, respectively, which result in a group velocity dispersion coefficient of [32 = -70 ps2 / km for TEOO mode. The radius of the microresonator is 227 pm, so the corresponding free spectral range (FSR) is approximately 100 GHz. The gap between ring and bus is 450 nm. Different cross-sections than the one mentioned above work as well. Further, different materials such as lithium niobate, tantalum pentoxide and aluminum oxide can be used instead of silicon nitride.
[0052] The feedback loop 6 of the example of figure 2 comprises an optical band pass filter 7 to tunably select the desired comb line, a feedback amplifier 10 (in this example an EDFA (small-signal gain ~35 dB, saturation power ~3.2 pW)) and an electronic variable optical attenuator (EVOA) 11 for feedback strength control. To make the feedback efficient, a fibre polarization controller may be inserted into the feedback loop 6. The optical loss of the feedback loop 6 is around 10 dB when the EDFA 10 is off. The total length of the feedback fibre loop is approximately 50 m, including approximately 27 m fibre in the EDFA 10.
[0053] A feedback strength can be quantified as the on-chip power ratio of the feedback field for reentering the microresonator and emitting one. In the example of figure 2, the power feedback ratio q can be measured from the comb line power ratio at nodes A and B when the feedback loop 6 is closed. It is important to emphasize that q < 1 is satisfied all the time in this work, that is to say, the EDFA 10 here is not employed as the 'gain source' for the compound cavity laser but instead it is served as the loss compensator of the feedback loop 6. Therefore, the feedback loop 6 could be purely passive. The whole system is operated in atmospheric environment without any servo control loop.
[0054] A measurement arrangement 12 is connected to the optical device 1. In this way, the coherent single-frequency narrow-linewidth oscillator output from the optical parametric oscillator 2 can be used for very accurate measurements. Examples of applications, not limited to measurements, for the coherent single-frequency narrow-linewidth oscillator output from the optical parametric oscillator 2 include precision spectroscopy, quantum optics, optical atomic clocks, sensing, high performance communications and LIDAR.
[0055] Figure 3 schematically shows a graph over the measured frequency noise for a selected comb line with re-injection of the selected comb line into a main optical resonator cavity 20 and without re-injection of the selected comb line into a main optical resonator cavity 21. For comparison, the frequency noise for the pump 22 is also included.
[0056] As can be seen from figure 3, re-injecting the selected comb line via the optical feedback loop into the main optical resonator cavity, thereby producing a coherent single-frequency narrowlinewidth oscillator output from the optical parametric oscillator, provides a great decrease in the frequency noise compared to the source laser light from the continuous wave laser and to the selected comb line without the re-injection of the selected comb line.
[0057] For the case without re-injection through the feedback loop, the selected comb line features a much higher frequency noise than the pump mainly due to the thermorefractive noise of the microcavity. However, the selected comb line's frequency noise gets reduced by three to four orders of magnitude with re-injection through the feedback loop, resulting in a sub-Hertz intrinsic linewidth, which is two orders lower than the pump. The results indicate that both the pump noise and thermorefractive noise can be greatly suppressed. The noise reduction from re-injection through the feedback loop is due to large cavity induced noise dilution.
[0058] It is further possible to obtain continuous wavelength tuning via simultaneous tuning of the pump frequency and the cavity resonance over one FSR of the microresonator.
[0059] Besides the spectral measurement of comb lines, the stability of the repetition rate has also been measured. For this purpose, the repetition rate is electro-optic down-converted into low frequency range (~ 470 MHz) with a 25 GHz intensity modulation, and then its phase noise is measured with a phase-noise analyser. It has been shown that single-sideband phase noise is reduced at low-offset frequencies while increased at higher frequency offset (>100 kHz) after SIL. This phenomenon can be understood as the repetition rate phase noise is mainly determined by the relative frequency stability between the pump and the narrow-linewidth oscillator. At lower offset frequencies, the relative frequency noise is mainly limited by the linewidth of the comb line due to the thermorefractive noise; therefore, comb linewidth narrowing can lead to lower repetition rate phase noise. While, at higher offset frequency region, the linewidth of the pump becomes to be a limiting factor as comb line -33 features a much lower frequency noise (or higher at the spike frequencies). Hence, in order to exploit the narrow-linewidth oscillator for extremely pure microwave generation, a narrow-linewidth pump laser is required.
[0060] Figure 4 shows a flowchart of a method 100 for producing laser light with a narrow linewidth. In step 110, a multimode optical parametric oscillator having a main optical resonator cavity made of nonlinear resonator medium is provided. In step 120, an optical feedback loop connected to the optical parametric oscillator is provided. In step 130, a continuous-wave laser light is received into the main optical resonator cavity, where the continuous-wave laser light is optically coupled with the main optical resonator cavity. In step 140, a frequency mode of the parametric oscillator is selected. In step 150, a line of the optical parametric oscillator is selected to be re-injected into the main optical resonator cavity via the optical feedback loop, wherein the optical feedback loop comprises a band-pass filter arranged to select the line. In step 160, the selected line is re-injected via the optical feedback loop into the main optical resonator cavity, thereby reducing the linewidth of the reinjected line.
[0061] The description of the example embodiments provided herein have been presented for purposes of illustration. The description is not intended to be exhaustive or to limit example embodiments to the precise form disclosed, and modifications and variations are possible in light of the above teachings or may be acquired from practice of various alternatives to the provided embodiments. The examples discussed herein were chosen and described in order to explain the principles and the nature of various example embodiments and its practical application to enable one skilled in the art to utilize the example embodiments in various manners and with various modifications as are suited to the particular use contemplated. The features of the embodiments described herein may be combined in all possible combinations of methods, apparatus, modules, systems, and computer program products. It should be appreciated that the example embodiments presented herein may be practiced in any combination with each other.
[0062] It should be noted that the word "comprising" does not necessarily exclude the presence of other elements or steps than those listed and the words "a" or "an" preceding an element do not exclude the presence of a plurality of such elements. It should further be noted that any reference signs do not limit the scope of the claims, that the example embodiments may be implemented at least in part by means of both hardware and software, and that several "means", "units" or "devices" may be represented by the same item of hardware.
[0063] The various example embodiments described herein are described in the general context of method steps or processes, which may be implemented in one aspect by a computer program product, embodied in a computer-readable medium, including computer-executable instructions, such as program code, executed by computers in networked environments. A computer-readable medium may include removable and non-removable storage devices including, but not limited to, Read Only Memory (ROM), Random Access Memory (RAM), compact discs (CDs), digital versatile discs (DVD), etc. Generally, program modules may include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Computer-executable instructions, associated data structures, and program modules represent examples of program code for executing steps of the methods disclosed herein. The particular sequence of such executable instructions or associated data structures represents examples of corresponding acts for implementing the functions described in such steps or processes.
[0064] In the drawings and specification, there have been disclosed exemplary embodiments. However, many variations and modifications can be made to these embodiments. Accordingly, although specific terms are employed, they are used in a generic and descriptive sense only and not for purposes of limitation, the scope of the embodiments being defined by the following claims.
Claims
CLAIMS1. Optical device (1) comprising: a multi-mode optical parametric oscillator (2) having a main optical resonator cavity(3) made of nonlinear resonator medium (4), wherein the main optical resonator cavity (3) is configured to receive a continuous-wave laser light (5) being optically coupled therewith, an optical feedback loop (6) connected to the output of the optical parametric oscillator (2), wherein the optical feedback loop (6) comprises a band-pass filter (7) arranged to select a frequency mode of the optical parametric oscillator (2) , wherein the optical feedback loop (6) is configured to re-inject a selected line (8) from the multi-mode parametric oscillator (2) into the main optical resonator cavity (3), thereby reducing the linewidth of the reinjected line (8).
2. Optical device (1) according to claim 1, wherein the band-pass filter (7) is tunable.
3. Optical device (1) according to claim 1 or 2, wherein the nonlinear resonator medium (4) is a second or third order non-linear medium.
4. Optical device (1) according to any one of the preceding claims, wherein the optical feedback loop (6) is one of a waveguide, optical fibre, integrated optical waveguide or a free space optical cavity.
5. Optical device (1) according to any one of the preceding claims, wherein the optical feedback loop (6) is longer than the main optical resonator cavity (3).
6. Optical device (1) according to any one of the preceding claims, wherein the optical feedback loop (6) comprises an optical amplifier (10).
7. Optical device (1) according to any one of the preceding claims, wherein the optical feedback loop (6) comprises a phase tuner.
8. Optical device (1) according to any one of the preceding claims, wherein the optical feedback loop (6) further comprises at least one polarization controller and / or at least one attenuator (11).
9. Optical device (1) according to any one of the preceding claims, wherein the feedback power ratio (q) of the selected frequency mode into the main optical resonator cavity (3) is less than one.
10. Method for producing laser light with a narrow linewidth, wherein the method comprises:- providing a multimode optical parametric oscillator (2) having a main optical resonator cavity (3) made of nonlinear resonator medium (4),- providing an optical feedback loop (6) connected to the optical parametric oscillator (2),- receiving, into the main optical resonator cavity (3), a continuous-wave laser light (5) being optically coupled therewith,- selecting a frequency mode of the parametric oscillator,- selecting a line (8) from the multi-mode parametric oscillator (2) to re-inject into the main optical resonator cavity (3) via the optical feedback loop (6) , wherein the optical feedback loop (6) comprises a band-pass filter (7) arranged to select the line (8),- re-injecting the selected line (8) via the optical feedback loop (6) into the main optical resonator cavity (3), thereby reducing the linewidth of the reinjected line (8).
11. Method according to claim 9, wherein the method comprises:- tunably selecting the line (8) from the multi-mode parametric oscillator (2) using a tunable band-pass filter (7).
12. Method according to claim 9 or 10, wherein the method comprises:- arranging the optical feedback loop (6) to be longer than the main optical resonator cavity (3).
13. Method according to any one of claims 9-11, wherein the method comprises:- feeding back the selected line (8) into the main optical resonator cavity (3) with a feedback power ratio (q) of less than one.