Systems and methods for high-power multimode fiber laser amplifiers with controlled output beam

Abstract

Provided herein is a multimode fiber laser amplification system. The multimode fiber laser amplification system includes a multimode fiber (MMF) amplifier and a first polarization-resolved wavefront shaping unit positioned before the MMF amplifier, where the amplifier is arranged and disposed to receive a shaped seed from the first polarization-resolved wavefront shaping unit and generate an amplified output beam. Also provided herein are methods of multimode laser amplification using the system.

Description

Attorney Docket No. 047162-7524WO1 (02775)SYSTEMS AND METHODS FOR HTGH-POWER MULTIMODE FIBER LASER AMPLIFIERS WITH CONTROLLED OUTPUT BEAMCROSS-REFERENCE TO RELATED APPLICATIONS

[0001] The present application claims priority under 35 U. S. C. § 119(e) to U. S.Provisional Patent Application No. 63 / 729,693, filed December 9, 2024, which application is incorporated herein by reference in its entirety.STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

[0002] This invention was made with government support under FA9550-20-1-0129, FA9550-24-1-0182, and FA9550-24-1-0033 awarded by Air Force Office of Scientific Research. The government has certain rights in the invention.BACKGROUND OF THE INVENTION

[0003] To achieve further power scaling of fiber laser amplifiers, various techniques have been developed to mitigate harmful nonlinear optical effects such as stimulated Brillouin scattering (SBS) and transverse mode instability (TMI), but mostly restrict themselves to singlemode amplification to ensure high beam quality. However, strong light confinement in conventional single-mode fibers leads to deleterious nonlinear optical effects and instabilities, which limit the achievable output power of fiber laser amplifiers. Also, while the output beam of a single-mode fiber amplifier has a smooth spatial profile, its polarization state is hard to control at very high power where vector modulation instability (MVI) prevails.

[0004] Recently, several works have demonstrated that multimode fiber amplifiers can strongly suppress these effects, making them promising candidates for high-power fiber laser systems. However, degraded output beam quality is often an issue for employing multimode fiber amplifiers. For example, single-frequency multimode fiber laser amplifiers produce speckled output beams with spatial-varying polarizations.

[0005] Accordingly, there remains a need in the art for systems and methods that improve upon existing systems and methods for multimode fiber laser amplifiers. The present disclosure meets this need.Attorney Docket No. 047162-7524WO1 (02775)SUMMARY

[0006] In one aspect, a multimode fiber laser amplification system includes a multimode fiber (MMF) amplifier and a first polarization-resolved wavefront shaping unit positioned before the MMF amplifier; wherein the amplifier is arranged and disposed to receive a shaped seed from the first polarization-resolved wavefront shaping unit and generate an amplified output beam. In some embodiments, the first polarization-resolved spatial wavefront shaping unit is designed to simultaneously reduce detrimental nonlinear effects and spatio-temporal instabilities in the MMF amplifier and to create selective spatial field distribution and polarization state of the amplifier output beam.

[0007] In some embodiments, the system further includes a second polarization-resolved wavefront shaping unit positioned after the MMF amplifier. In some embodiments, the first polarization-resolved spatial wavefront shaping unit is designed to shape the spatial field distributions of two orthogonal linear polarizations of the seed into the MMF amplifier for maximal suppression of detrimental nonlinear effects and spatio-temporal instabilities in the MMF amplifier; and the second polarization-resolved spatial wavefront shaping unit is designed to transform the amplifier output to a coherent beam of target spatial field distribution and polarization state.

[0008] In some embodiments, the system further includes a laser source, the laser source configured to provide a coherent seed to the MMF laser amplification system. In some embodiments, the laser source comprises a fiber laser, a semiconductor laser, or a solid-state laser. In some embodiments, the laser source produces spatially coherent light in a single transverse mode.

[0009] In some embodiments, the system further includes one or more preamplifiers positioned sequentially after the laser source and before the MMF amplifier. In some embodiments, the one or more preamplifiers comprise fiber preamplifiers, semiconductor optical amplifiers, solid-state preamplifiers, or combinations thereof. In some embodiments, the one or more preamplifiers operate in a single transverse mode and the laser is a coherent laser.

[0010] In some embodiments, the system further includes at least one lens; at least one isolator; at least one attenuator; at least one beam expander; at least one spatial filter; at least one beam splitter; at least one objective; or combinations thereof.Attorney Docket No. 047162-7524WO1 (02775)

[0011] In some embodiments, the MMF amplifier comprises a signal MMF; an active MMF; at least one pump laser; and a signal -pump combiner coupling the at least one pump laser to the active MMF.

[0012] In some embodiments, the system suppresses stimulated Brillouin scattering (SBS) in fiber laser amplifiers. In some embodiments, the system suppresses transverse modal instability (TMI) in fiber laser amplifiers. In some embodiments, the system suppresses stimulated Raman scattering (SRS) in fiber laser amplifiers. In some embodiments, the system suppresses modulation instability (MI) in fiber laser amplifiers. In some embodiments, the system suppresses vector modulation instability (VMI) in fiber laser amplifiers.

[0013] In some embodiments, the system operates in the state of continuous wave (CW), quasi-CW, or pulsed.

[0014] In another aspect, a method of multimode laser amplification includes providing the system according to any of the embodiments described herein, designing spatial light modulation patterns to create desired vector field distributions with the first polarization-resolved spatial wavefront shaping unit; and amplifying the shaped multimodal laser waveform with the MMF amplifier. In some embodiments, the amplification of the shaped multimodal laser waveform suffers minimal detrimental nonlinear effects and spatio-temporal instabilities in the MMF amplifier. In some embodiments, the amplification of the shaped multimodal laser waveform generates a coherent output beam with desired spatial field distribution and polarization state, and simultaneously reduces detrimental nonlinear effects and spatio-temporal instabilities in the MMF amplifier. In some embodiments, the method increases the lowest power threshold of SBS, SRS, TMI, MI and VMI. In some embodiments, the MMF amplifier output has spectral (20 dB) linewidth narrower than 20 kHz.

[0015] In another aspect, a method of multimode laser amplification includes providing the system according to any of the embodiments disclosed herein, designing spatial light modulation patterns to create desired vector field distributions with the second polarization-resolved spatial wavefront shaping unit; and transforming the amplifier output beam to a coherent beam with desired spatial field distribution and polarization state. In some embodiments, the MMF amplifier output has spectral (20 dB) linewidth narrower than 20 kHz.Attorney Docket No. 047162-7524WO1 (02775)BRIEF DESCRIPTION OF THE DRAWINGS

[0016] For a fuller understanding of the nature and desired objects of the present invention, reference is made to the following detailed description taken in conjunction with the accompanying drawing figures wherein like reference characters denote corresponding parts throughout the several views.

[0017] FIGS. 1A-B show schematically two embodiments of multimode fiber (MMF) amplification system. (A) Input light field E(1)passes through the first polarization-resolved spatial wavefront shaping unit before entering a MMF amplifier. Output light of the amplifier passes through the second polarization-resolved spatial wavefront shaping unit to generate the target field E(o). The first polarization-resolved spatial wavefront shaping unit shapes the spatial field distributions of two orthogonal linear polarizations to maximize power threshold of detrimental nonlinear effects and spatio-temporal instabilities in the MMF amplifier. The second polarization-resolved spatial wavefront shaping unit transforms the amplifier output to a coherent beam of target spatial profile and polarization state. (B) Only one polarization-resolved wavefront shaping unit is positioned before a MMF amplifier to simultaneously reduce detrimental nonlinear effects and spatio-temporal instabilities in the MMF amplifier and to create selective spatial profiles and polarization state of the output beam.

[0018] FIGS. 2A-E show schematics of some configurations for polarization-resolved spatial wavefront shaping unit. (A) An input light is split by a polarizing beam-splitter (PBS) to two beams with orthogonal linear polarizations, EH and Ev. Each beam passes through several spatial light modulators (SLMs) designed to shape spatial distributions of field amplitude and phase. The two shaped wavefronts of orthogonal polarizations then recombine at a second PBS. M denotes mirror. (B) EH and Ev are reflected separately between a SLM and a planar mirror (M) multiple times, each time the spatial wavefront of EH or Ev is modulated by a different part of the SLM. (C) Spatial wavefront for one polarization component EH is modulated by multiple reflections from different parts of the first SLM, its polarization state is changed by a half-wave-plate (HWP) and then the spatial wavefront of the orthogonal linear polarization component Ev is modulated by a second SLM. (D) A quarter-wave-plate (QWP) is inserted between a SLM and a mirror to modify the polarization state. The polarization is rotated to the orthogonal direction by passing the QWP twice. The reflections of light from the SLM before the QWP shape the spatial wavefront of EH, and the reflections after the QWP shape the spatial wavefrontAttorney Docket No. 047162-7524WO1 (02775)of Ev. (E) A phase hologram is written to a SLM to generate two spatial wavefronts and diffract them to different directions. Both beams are linearly polarized in the same direction. A half-wave-plate (HWP) is placed in the path of one beam to rotate its polarization to the orthogonal direction. The two wavefronts are then combined by a beam displacer to generate a coherent beam of target spatial profile and polarization state.

[0019] FIG. 3 shows optimal multimode excitations in a multimode fiber (MMF) amplifier for suppressing stimulated Brillouin scattering (SBS) (panels a and b) and transverse mode instability (TMI) (panels c and d). Panel (a) shows optimal mode content for SBS suppression in a MMF with 160 modes excites a few clusters of fiber modes with small difference in propagation constants in order to create a significantly broadened SBS spectrum (inset) with much lower peak value compared to exciting only the lowest-order mode (fundamental mode FM). The fiber modes are ordered by their propagation constant (wavevector component parallel to the fiber axis). Higher order modes have larger propagation constants. Panel (b) shows scaling of SBS threshold enhancement over FM-only excitation with the number of fiber modes for equal mode excitation (blue) and optimal excitation (green), in comparison with best single mode excitation (orange). The optimal multimode excitation results in the highest SBS threshold enhancement. Panel (c) shows optimal mode content for the maximal TMI suppression in a MMF amplifier with 80 modes avoids exciting fiber modes of intermediate propagation constant. Such modes couple strongly with modes of both larger and small propagation constants, and not exciting them reduces the thermo-optical coupling between fiber modes and lower the TMI threshold. Panel (d) shows scaling of TMI threshold enhancement over FM-only excitation with the number of fiber modes for equal mode excitation (blue) and optimal excitation (green), in comparison with best single HOM excitation (orange). The optimal multimode excitation and equal-mode excitation lead to a linear increase in TMI threshold enhancement with number of fiber modes.

[0020] FIGS. 4A-B show optimal mode content for joint suppression of SBS and TMI for different ratios of SBS threshold (PSBS) and TMI threshold (PTMI) for FM-only excitation. (A) For PSBS / PTMI=L optimal mode content is similar to the SBS-only suppression. Both SBS and TMI thresholds are enhanced 9.4 times. (B) For PSBS / PTMI=1-5, optimal mode content enhances SBS threshold 8.8 times and TMI threshold 13.2 times, so that the two thresholds are equal.Attorney Docket No. 047162-7524WO1 (02775)

[0021] FTG. 5 shows polarization-resolved wavefront shaping of a single-frequency multimode fiber amplifier output beam. The input wavefront of a coherent seed is modulated by a SLM to suppress detrimental nonlinear effects in the MMF amplifier. The amplifier output beam is split by a polarizing beam-splitter (PBS) to two beams with orthogonal linear polarizations, EH and Ev. Both are speckled beams with distinct spatial distributions of intensity and phase, shown in panels (a) and (f). Their beam propagation factors are M2= 6.18 and 6.22, respectively. Each beam passes through three lossless SLMs with free-space propagation in between. The optimized phase modulation patterns on SLMs, shown in panels (a-d) and (g-i), transform both EH and Ev to smooth beams with same spatial intensity distribution, shown in panels (e) and (j). The beam propagation factors are 1.24 and 1.37, respectively. EH and Ev recombine at another PBS, and their relative (global) phase is adjusted with the last SLMs in their paths to control the polarization state of the combined Gaussian beam, e.g., making a righthand circular polarized a beam, shown in panels (k-n), with a beam propagation factor M2= 1.15.

[0022] FIG. 6 shows input wavefront shaping of a MMF amplifier with mode-dependent gain, strong mode coupling and polarization mixing. Panel (a) shows a Gaussian beam E(1)at wavelength 1064 nm is split by a PBS to EH and Ev. Panel (b) shows each beam passes through three SLMs and then recombine at a second PBS, before entering the MMF amplifier. Panel (c) shows the modulation patterns on the SLMs are optimized to create spatial intensity and phase distributions for EH and Ev, so that the amplifier output beam is in the LP01 mode with M2= 1.08. Panel (d) shows the global phase of the third SLM for EH relative to that of Ev is further tuned to set the output polarization state to right circular polarization (RCP). Both near-field and far-field intensity distributions for RCP are smooth, while the intensities for left circular polarization (LCP) vanish.

[0023] FIG. 7 shows input wavefront shaping of a nonlinear multimode fiber amplifier. The step-index fiber with a 42 pm core and 0.06 NA supports 15 modes per polarization at wavelength 1064 nm. The input signal power is 10 W and pump power is 500 W at 976 nm wavelength. The output signal power is 444 W and residual pump power 3 W. The saturated gain coefficient at the fiber output end is roughly 50 times smaller than the linear gain coefficient, which is nearly mode independent. A linearly-polarized Gaussian beam (panel (a)) passes through three SLMs (panel (b)) before entering the multimode fiber amplifier. The SLMs areAttorney Docket No. 047162-7524WO1 (02775)designed to have the amplifier output in LP01 mode. Panel (c) shows spatial intensity and phase distributions of input light to the MMF amplifier. Panel (d) shows output field intensity distribution in near-field resembles the LP01 mode, and far-field intensity profile is smooth.

[0024] FIGS. 8A-B show schematics illustrating optical setup of a multimode fiber amplifier. (A) Coherent narrowband laser of 5W from a preamplifier is expanded, wavefrontshaped by a SLM, before seeding a MMF amplifier that comprises a Yb-doped step-index fiber with 76 guided modes at wavelength 1064 nm. Amplified light output of the MMF amplifier is characterized by a near-field camera, power meter, spectral linewidth measurement setup, and spectrometer. On the fiber input side, time trace of backscattered light is monitored for detecting the onset of SBS. (B) Detailed schematic of an MMF amplifier and characterization setup.

[0025] FIGS. 9A-C show graphs illustrating stimulated Brillouin scattering (SBS) in fiber amplifiers. (A) Output signal power at the SBS instability threshold is estimated for a 15-pm-core single-mode fiber (SMF) amplifier, and compared to a 42-pm-core multimode mode fiber (MMF) amplifier under fundamental-mode (FM)-only excitation. Calculated intensity profile of the FM is shown as inset. SBS instability thresholds for the MMF amplifier are measured with few-mode and multimode excitations, and output intensity distributions are shown as inset. Both SMF and MMF are 18 m long. (B) Brillouin gain spectrum calculated with equal excitation of all modes (purple) is broader than FM-only excitation (green) in the MMF amplifier. (C) SBS instability threshold as a function of effective fiber length 1 / Leff for the 42-pm-core MMF with multimode excitation. Blue crosses are experimental data, and purple solid curve shows the 1 / Leff scaling. All data points are taken with output beam focused to a diffraction-limited spot. Green circles represent the estimated SBS instability thresholds under FM-only excitation at corresponding lengths. Inset shows intensity and phase distributions across the focal spot at output signal power of 173 W. 76% of total output power is concentrated inside the focus. Color represents phase and brightness the intensity.

[0026] FIG. 10 shows a graph illustrating heterodyne spectrum for measuring the spectral linewidth of the seed laser. The full width at -20 dB of maximum is 34 kHz.

[0027] FIG. 11 shows images illustrating output focusing at varying power. Images of output focal spot by input wavefront shaping at amplified signal powers of 233 W and 455 W. The focal spots are at the MMF output facet. The power ratio (PR), i.e., the ratio of power within the focal area and the total power, is given in the image.Attorney Docket No. 047162-7524WO1 (02775)

[0028] FTGS. 12A-D show images and a graph illustrating beam propagation factor of focal spot at MMF amplifier output. (A) Intensity profile of a diffraction-limited spot created at 300 pm from the MMF distal facet by shaping phase front of input signal. 76% of total output power is concentrated inside the focal area. (B) Output beam profile recorded at axial distance z ~ 200 pm from the focal plane. (C) Output beam profile recorded at axial distance z ~ 400 pm from the focal plane. (D) Focal spot radii along x and y axes (perpendicular to fiber axis z), wxand wy, increase with axial distance z. Squares and circles mark experimental data, solid curves represent the fit with Eq. 3 that gives Mx« 1.05 and My « 1.35.

[0029] FIGS. 13A-C show graphs illustrating high-power MMF amplifier with narrow linewidth. The fiber core diameter is 42 pm and effective length Leff= 3.7 m. (A) Amplified signal power as a function of absorbed pump power gives the slope efficiency of 82%. Red circles are experimental data, black dashed line is a linear fit of data, and blue curve from our theoretical model. (B) Optical spectrum of amplifier output at 453 W power shows the amplified signal peak at 1064 nm on top of a broad amplified ASE band. The signal-to-ASE peak ratio is 52 dB. The small peak centered at 971 nm is from residual pump. (C) Heterodyne spectra for input (blue) and output (red) signals of the MMF amplifier at the output signal power of 503 W. The sweep time is 41 ms. Full width at -20 dB of maximum is 35 kHz for both input and output signals, indicating no detectable spectral broadening by the MMF amplifier. After subtracting the reference linewidth, the 20-dB width of input / output signal is 18 kHz.

[0030] FIGS. 14A-B show graphs illustrating relative power noise (RPN) and phase noise of an MMF amplifier. (A) Measured RPN spectra of the MMF amplifier with input wavefront shaping (WFS) at output power of 455 W (blue dotted), in comparison to that without WFS and pumping of MMF amplifier (orange solid). (B) Measured phase noise spectra of the MMF amplifier with input WFS at output power of 455 W (blue dotted), in comparison to that without WFS and pumping of MMF amplifier (orange solid). The measurement time is 20 ms in A and 75 ms in B.

[0031] FIGS. 15A-B show graphs illustrating noise induced by wavefront shaping. (A) Measured RPN spectra of the preamplifier output beam before (orange solid) and after (blue dotted) the SLM, showing a couple of peaks between 1 kHz and 3 kHz which are induced by wavefront shaping (WFS). The measurement time is 20 ms. (B) Measured heterodyne peak of the output signal of theMMF amplifier at 80W without input wavefront shaping (orange), showingAttorney Docket No. 047162-7524WO1 (02775)similar linewidth to that of 503 W amplified signal which is focused to a spot by input wavefront shaping (blue). The sweep time is 41 ms.

[0032] FIGS. 16A-C show graphs illustrating various power measurement. (A) Numerically calculated pump and signal power evolution in the MMF amplifier with parameters given in Table 1. Vertical dashed line marks the shortest active fiber length 5.5 m of our MMF amplifier, where the signal power (blue curve) is 490Wand the residual pump power (orange curve) is 98W. (B) Signal power evolution in individual fiber modes shows weak modedependent gain. (C) Relative modal power, given by the ratio of signal power in each fiber mode and that averaged over all fiber modes, varies with z in the first half of the active fiber and becomes nearly z independent in the second half. The brightness of each curve is proportional to the total signal power, highlighting the region where SBS is significant.

[0033] FIG. 17 shows a schematic of an MMF amplifier with full-field control. The Yb-doped step-index fiber with a core diameter of 42-pm and NA of 0.1 supports a total of 80 modes. It is 4.8 meter long, and co-pumped by laser diodes. Passive MMF with matching parameters is used in the signal-pump combiner. The abbreviations are: X / 2, half-wave plate; P, polarizer; SLM, spatial light modulator; and BS, beam sampler.

[0034] FIGS. 18A-C show images illustrating an MMF amplifier output beam with M2= 1.17. (A) Optimized phase pattern on the spatial light modulator. (B) Field amplitude and phase of input light in horizontal polarization (H pol) and vertical polarization (V pol), generated by first-order diffraction from the SLM. (C) Measured output beam profile in near-field and far-field. The fiber core (42 pm) and NA (0.1) boundaries are marked by while circles. The amplified signal power is 210 W.DETAILED DESCRIPTIONDefinitions

[0035] As used herein, each of the following terms has the meaning associated with it in this section. Unless defined otherwise, all technical and scientific terms used herein generally have the same meaning as commonly understood by one of ordinary skill in the art to which this disclosure belongs. Generally, the nomenclature used herein and the laboratory procedures are those well-known and commonly employed in the art. It should be understood that the order ofAttorney Docket No. 047162-7524WO1 (02775)steps or order for performing certain actions is immaterial, so long as the present teachings remain operable. Any use of section headings is intended to aid reading of the document and is not to be interpreted as limiting; information that is relevant to a section heading may occur within or outside of that particular section. All publications, patents, and patent documents referred to in this document are incorporated by reference herein in their entirety, as though individually incorporated by reference.

[0036] In the application, where an element or component is said to be included in and / or selected from a list of recited elements or components, it should be understood that the element or component can be any one of the recited elements or components and can be selected from a group consisting of two or more of the recited elements or components.

[0037] In the methods described herein, the acts can be carried out in any order, except when a temporal or operational sequence is explicitly recited. Furthermore, specified acts can be carried out concurrently unless explicit claim language recites that they be carried out separately. For example, a claimed act of doing X and a claimed act of doing Y can be conducted simultaneously within a single operation, and the resulting process will fall within the literal scope of the claimed process.

[0038] As used herein, the singular form “a,” “an,” and “the” include plural references unless the context clearly dictates otherwise.

[0039] Unless specifically stated or obvious from context, as used herein, the term “about” is understood as within a range of normal tolerance in the art, for example within 2 standard deviations of the mean. “About” can be understood as within 10%, 9%, 8%, 7%, 6%, 5%, 4%, 3%, 2%, 1%, 0.5%, 0.1%, 0.05%, or 0.01% of the stated value. Unless otherwise clear from context, all numerical values provided herein are modified by the term about.

[0040] As used herein, the terms “comprises,” “comprising,” “containing,” “having,” and the like can have the meaning ascribed to them in U. S. patent law and can mean “includes,” “including,” and the like.

[0041] Unless specifically stated or obvious from context, the term “or,” as used herein, is understood to be inclusive.

[0042] Ranges provided herein are understood to be shorthand for all of the values within the range. For example, a range of 1 to 50 is understood to include any number, combination of numbers, or sub-range from the group consisting 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,Attorney Docket No. 047162-7524WO1 (02775)16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, or 50 (as well as fractions thereof unless the context clearly dictates otherwise).

[0043] As used herein, the term “ratio” refers to a relationship between two numbers (e.g., scores, summations, and the like). Although, ratios can be expressed in a particular order (e.g., a to b or a:b), one of ordinary skill in the art will recognize that the underlying relationship between the numbers can be expressed in any order without losing the significance of the underlying relationship, although observation and correlation of trends based on the ration may need to be reversed. For example, if the values of a over time are (4, 10) and the values of b over time are (2, 4), the ratio a:b will equal (2, 2.5), while the ratio b:a will be (0.5, 0.4).Although the values of a and b are the same in both ratios, the ratios a:b and b:a are inverse and increase and decrease, respectively, over the time period.Detailed Description

[0044] Provided herein is a multimode fiber (MMF) laser amplification system with controlled output beam profile and polarization state. In some embodiments, as illustrated in FIG. 1A, a narrowband, spatially coherent laser beam provides the seed 101 to a MMF laser amplification system 100. It passes through a first polarization-resolved spatial wavefront shaping unit 110, before entering a MMF amplifier 120. The first polarization-resolved spatial wavefront shaping unit 110 is arranged and disposed to receive the seed 101 and shape its spatial field distributions for two orthogonal polarizations to generate a target wavefront. The amplifier 120 is arranged and disposed to receive the seed with shaped wavefront 103 and amplify it. The output beam 105 of the MMF amplifier 120 enters a second polarization-resolved wavefront shaping unit 130 positioned after the MMF amplifier 120 to transform the spatial field distributions for two orthogonal polarizations. The first polarization-resolved wavefront shaping unit 110 before the MMF amplifier 120 is designed for optimal multimode excitation in the MMF amplifier 120 to minimize detrimental nonlinear effects and spatio-temporal instabilities. The second polarization-resolved wavefront shaping unit 130 after the MMF amplifier is designed to convert the amplifier output to a coherent beam with target spatial profile and polarization state 107.Attorney Docket No. 047162-7524WO1 (02775)

[0045] In some embodiments, as illustrated in FIG. 1B, a MMF laser amplification system 200 only includes one polarization-resolved wavefront shaping unit 110, which is positioned before the MMF amplifier 120 (i.e., no second polarization-resolved wavefront shaping unit 130 after the amplifier 120). This unit 110 is designed to simultaneously reduce detrimental nonlinear effects and spatio-temporal instabilities in the MMF amplifier 120 and to create selective spatial profiles and polarization state of the output beam.

[0046] The seed to the amplifier is generated by a laser source 140 with a single transverse (spatial) mode. The input beam can be a continuous wave or pulsed in time. In some embodiments, the laser source has a narrow-linewidth, for example less than 1 GHz. In some embodiments, the seed is amplified by one or multiple preamplifiers 831 (FIG. 8A) positioned sequentially after the laser 140 before entering the MMF amplifier 120. The preamplifiers 831 also operate with single transverse (spatial) mode. Suitable laser sources 140 and preamplifiers 831 include, but are not limited to, fiber lasers and amplifiers, semiconductor lasers and amplifiers, solid-state lasers and amplifiers.

[0047] The polarization-resolved wavefront shaping unit 110 is designed to transform an input beam with arbitrary spatial wavefront and polarization to the desired spatial field distributions for two orthogonal polarizations EH and Ev. It includes any suitable spatial light modulators (SLMs) 810, such as optical phase masks and amplitude masks, programmable spatial light modulators based on liquid crystal technologies, or optical microelectromechanical systems (MEMS). The unit also includes, but not limited to, optical mirrors, polarizing and / or non-polarizing beam-splitters, quarter-waveplates, and / or beam displacers.

[0048] FIGS. 2A-E show examples using liquid-crystal -based SLMs 810 that modulate only the phasefront of a specific linear polarization. In FIG. 2A, the input light to the polarization-resolved wavefront shaping unit is split by a polarizing beam-splitter (PBS) 201 to two beams with orthogonal linear polarizations EH (202) and Ev (204). Each beam passes through several SLMs (810a-810f) designed to shape spatial distributions of field amplitude and phase. The two shaped wavefronts of orthogonal polarizations recombine at a second PBS 203. Alternatively, in FIG. 2B, the input light is reflected between a single SLM (810a, 810b) and a planar or curved mirror (21 la, 21 lb) multiple times, each time the spatial wavefront of light is modulated by a different part of the SLM. In FIG. 2C, the two polarizations of light are not separated spatially. After the spatial wavefront of one linear polarization component of an inputAttorney Docket No. 047162-7524WO1 (02775)light is modulated by multiple reflections from different parts of the first SLM 810a, its polarization state is changed by a half-wave-plate (HWP) 220 and then the spatial wavefront of orthogonal linear polarization component is modulated by a second SLM 810b. Alternatively, in FIG. 2D, two SLMs can be replaced by a single SLM, and a quarter-wave-plate (QWP) 230 is inserted between the SLM and the mirror to modify the polarization state. The reflections of light from the SLM before the QWP shape the spatial wavefront of one linear polarization, and the reflections after the QWP shape the spatial wavefront of orthogonal linear polarization component. In some embodiments, as illustrated in FIG. 2E, a phase hologram is designed and written to a SLM in order to generate two spatial wavefronts and diffract them to different directions. Both beams are linearly polarized in the same direction. A half-wave-plate (HWP) 220 is placed in the beam path to rotate its polarization to the orthogonal direction. The two wavefronts are then combined by a beam displacer 240 or polarizing beam splitter to generate a coherent beam of target spatial profile and polarization state.

[0049] The amplifier 120 includes any suitable multimode fiber amplifier for use in connection with the shaped wavefront generated by the SLM. For example, in some embodiments, the amplifier 120 includes one or more active multimode fibers 121 doped with Ytterbium (Yb), Erbium (Er), Thulium (Tm), Holmium (Ho), Neodymium (Nd), Praseodymium (Pr), or combinations thereof. The MMF amplifier 120 also includes one or more pump lasers 821 and a pump combiner 823 that couples light from the pump laser(s) 821 to the active fiber. For example, in one embodiment of the multimode fiber amplifier 120, a passive multimode fiber 825 is coupled to the pump diode lasers 821 through a pump combiner 823, and spliced 829 to the active fiber 827. As will be appreciated by those skilled in the art, any suitable number of pump lasers 821 having any suitable power may be selected to provide a desired pump power. The pump lasers 821 can be diode lasers, fiber lasers, solid-state lasers or a combination of some of them. For example, in one embodiment, the MMF amplifier includes five diode lasers with an optical power of approximately 150 W each, for a total optical pump power of approximately 600 W. Additional diode lasers can be added to reach higher pump power. The pumping configurations include co-pumping, where the signal and pump light are coupled to the doped multimode fiber from the same end; counter-pumping, where the signal and pump light are coupled to the doped fiber from the opposite ends; and bi-directional pumping, where pump light is coupled to the doped fiber from both sides.Attorney Docket No. 047162-7524WO1 (02775)

[0050] In some embodiments, the amplifier includes one or more additional elements positioned along the beam path. Suitable additional elements include, but are not limited to, preamplifier(s) 831, lens(es) 832, isolator(s) 835, attenuator(s) 836, beam expander(s) 837, spatial filter(s) 838, beam splitter(s) 833; objective(s) 834, and / or combinations thereof.

[0051] Also provided herein is a method of multimode laser amplification. In some embodiments, the method includes generating a multimodal excitation in a MMF amplifier with the polarization-resolved spatial wavefront shaping unit before the amplifier to maximize the power threshold for detrimental nonlinear effects including but not limited to stimulated Brillouin scattering (SBS), and simulated Raman scattering (SRS). In some embodiments, the method includes generating and optimizing a multimodal excitation in a MMF amplifier with the polarization-resolved spatial wavefront shaping unit before the amplifier to maximize the power threshold for spatio-temporal instabilities, including but not limited to transverse mode instability (TMI), modulation instability (MI), and vector modulation instability (VMI). In some embodiments, the method includes generating and optimizing a multimodal excitation in a MMF amplifier with the polarization-resolved spatial wavefront shaping unit before the amplifier to simultaneously suppress detrimental nonlinear effects and spatio-temporal instabilities. In some embodiments, the method includes generating and optimizing a multimodal excitation in a MMF amplifier with the polarization-resolved spatial wavefront shaping unit to reduce detrimental nonlinear effects and spatio-temporal instabilities and simultaneously control the output beam pattern and polarization state, for example, to create a circularly polarized output beam with the beam propagation factor M2close to 1. Additionally or alternatively, the method includes tailoring the amplifier output beam with the polarization-resolved spatial wavefront shaping unit after the amplifier to transform it to the target beam profile and polarization state, for example, a focused beam or collimated beam with linear or circular polarization and the beam propagation factor M2close to 1. Finally, one or more of the embodiments disclosed herein enables real-time control of the output beam profile and polarization state of the multimode fiber amplifier.

[0052] Additionally or alternatively, in some embodiments, the system includes a high-beam-quality single-frequency multimode-fiber amplifiers with full-field control. In some such embodiments, the system is configured to modulate both field amplitude and phase for two orthogonal polarizations of an input signal to an MMF amplifier. For example, in some embodiments, the system is configured to realize full-field shaping by using the first-orderAttorney Docket No. 047162-7524WO1 (02775)diffraction of a phase-only spatial light modulator (SLM), which enables complete control of the spatial profile and polarization state of the amplified beam. Accordingly, the system is capable of generating an output Gaussian beam e.g., M2< 1.2) from a single-frequency MMF amplifier having a plurality of modes (e.g., 80 modes).

[0053] Referring to FIG. 17, in some embodiments, the system 1700 includes a laser 140, a preamplifier 831, a beam expander 837, a beam shaper 1701, an SLM 810, a beam displacer 1703, and an MMF 120. The laser 140 includes any suitable laser source, such as, but not limited to, a single-frequency, linearly-polarized laser at wavelength 1064 nm. The preamplifier 831 is configured to provide any suitable preamplification of the laser, such as, but not limited to, to 45 W. The beam expander 837 is configured to expand the Gaussian beam from the preamplifier 831 and the beam shaper 1701 is configured to convert the expanded beam to any suitable shape, such as, but not limited to, a flat-top beam. A first order diffraction of the shaped beam from the SLM 810 is composed of two sub-orders, which produce two distinct field patterns. The system 1700 is configured to convert one of the field patterns to an orthogonal polarization, then combine the orthogonal polarization with the other field pattern through the beam displacer 1703. The resulting combined beam is launched to the MMF amplifier 120 as the seed. In some embodiments, the MMF amplifier 120 is co-pumped by laser diodes 821 at any suitable wavelength (e.g., 976 nm). The amplified signal and residual pump are separated at the output. Both near-field and far-field intensity distributions of the amplified signal are recorded simultaneously. In some embodiments, the system 1700 also includes one or more beam isolators 835, an attenuator 836, one or more lenses 832, one or more mirrors 1705, or a combination thereof. For example, in some embodiments, the system 1700 includes an isolator 835 and a variable attenuator 836 positioned between the preamplifier 831 and the beam expander 837 and / or an isolator 835 between the SLM 810 and the beam displacer 1703.

[0054] In some embodiments, the system includes an iterative algorithm configured to improve the output beam quality of the MMF amplifier. In some embodiments, the iterative algorithm is configured to determine a SLM pattern that delivers a desired amplitude and phase modulations of both polarizations into the MMF with minimal diffraction loss. For example, in some embodiments, at each iteration, the algorithm updates the SLM pattern to minimize the deviation between the measured intensity distributions and the ideal Gaussian profiles in bothAttorney Docket No. 047162-7524WO1 (02775)near-field and far-field. The beam propagation factor (M2) can be directly calculated from the second moment of the measured near- and far-field intensity distributions.

[0055] The systems and methods according to one or more of the embodiments disclosed herein results in a high-power narrow-band (single-frequency) MMF laser amplifier that provides output light with high spatial and temporal coherence, which facilitate optical interferometry and coherent beam combining. The coherent excitation of many modes in a highly multimode fiber amplifier, as described herein, greatly suppresses detrimental nonlinear optical effects and spatiotemporal instabilities that have limited further power scaling of single-mode fiber amplifiers. Accordingly, the MMF amplifier with input and / or output wavefield shaping, as described herein, can simultaneously achieve high power, high coherence, and high beam quality, even in the presence of thermo-optical nonlinearity, Kerr nonlinearity, and gain saturation. Additionally or alternatively, the systems and methods according to one or more of the embodiments disclosed herein enhance the power thresholds of SBS, SRS, TMI, MI, and / or VMI. More specifically, and without wishing to be bound by theory, it is believed that the power thresholds of detrimental nonlinear effects and spatio-temporal instabilities increase with the fiber core dimeter and the number of modes excited in the fiber. The methods described herein are applicable to step-index fibers, graded-index fibers, microstructured fibers and specialty fibers.

[0056] Those skilled in the art will recognize, or be able to ascertain using no more than routine experimentation, numerous equivalents to the specific procedures, embodiments, claims, and examples described herein. Such equivalents are considered to be within the scope of this invention and covered by the claims appended hereto.

[0057] It is to be understood that wherever values and ranges are provided herein, all values and ranges encompassed by these values and ranges, are meant to be encompassed within the scope of the present invention. Moreover, all values that fall within these ranges, as well as the upper or lower limits of a range of values, are also contemplated by the present application.

[0058] The following examples further illustrate aspects of the present invention.However, they are in no way a limitation of the teachings or disclosure of the present invention as set forth herein.Attorney Docket No. 047162-7524WO1 (02775)EXAMPLESEXAMPLE 1 -Multimode excitation for SBS and TMI suppression in multimode fiber amplifiersIntroduction

[0059] By wavefront shaping of the input light to a multimode fiber (MMF) amplifier (as shown in FIGS. 1A-B), a complete control of the modal excitation in the MMF can be achieved. As such, optimal modal excitations can be realized to maximize the power threshold for detrimental nonlinear effects and spatio-temporal instabilities. Theoretically, globally optimal excitation can be achieved for suppressing SBS and TMI individually as well as jointly, using a linear programming-based approach. In a single-frequency step-index MMF amplifier, this can lead to an order of magnitude higher SBS and TMI thresholds than a single-mode fiber (SMF) amplifier.Methods

[0060] Our method in finding the optimal multimode content for SBS or TMI suppression involves mapping the optimization problem to a linear programming problem, which can be solved efficiently with standard solvers. For instance, multimode SBS suppression problem can be transformed as described below. Analogous method works for TMI suppression as well as joint suppression of SBS and TMI. The effective Brillouin gain in the m-th guided optical mode at frequency £1 is given by G_m(Ω) = ∑_l G_B^(m,l)(Ω) P_l, a weighted sum of Brillouin gain coefficients Gj? ' (fl) with signal powers Ptin various mode / as weights. The maximum Brillouin gain across all the fiber modes and frequencies fl determines the reflected Stokes power and need to be minimized, while keeping the total signal power constant. This leads to the following optimization problem: min[max G_B^(m,l)(Ω_i)P_l], along with the{pq m.ri,-constraints: Pi > 0,Pt= Pout, where Poutis the total output signal power. Both constraints in this problem are linear functions of the variables Pi, while the maximum value of the weighted Brillouin coefficients is not. We use a standard transformation to linearize the problem: introducing a slack variable t that is constrained to be larger than all possible values of G_B^(m,l)(Ω_i)P_l, and minimizing this variable. The resulting problem is a standard linearAttorney Docket No. 047162-7524WO1 (02775)program, solvable with standard solvers. The discontinuities in the original objective are replaced by the intersection of MX linear constraints, where M is the number of fiber modes and N is the number of the frequency steps fl;.Results(a) SBS Suppression

[0061] FIG. 3, panel (a), shows the optimal mode content obtained using this method for maximal suppression of SBS in a MMF with 160 guided modes. The fiber modes are ordered by their propagation constant, which is the wavevector component parallel to fiber axis. Higher order modes have smaller propagation constants. The maximal SBS threshold is 9.6 times of that with only the fundamental mode (FM) excited in the same fiber. For maximal SBS suppression, higher order modes are excited more than lower order modes, since the Brillouin gain coefficients typically decrease with increasing mode order. Multiple modes are excited instead of a single high-order mode (HOM), to take advantage of relatively lower intermodal gain coefficients compared to the intramodal gain. Finally, we observe that the optimal mode content involves a few clusters of modes with significant separation in the propagation constants. This is because the Brillouin gain coefficients for mode pairs with well separated propagation constants are peaked at very different frequencies. As a result, the overall Brillouin gain spectrum is broadened and has a much lower peak value. This is illustrated in the inset of FIG. 3, panel (a), where the Brillouin gain spectrum for FM-only excitation is compared to the optimal multimode excitation. In FIG. 3, panel (b), we show the scaling of the SBS threshold enhancement over FM-only excitation with the number of fiber modes for the optimal multimode excitation and compare it with equal-mode excitation and best single mode excitation. Exciting a single HOM shows smallest threshold enhancement. The equal-mode excitation does lead to a significantly higher threshold enhancement, which increases with the number of fiber modes, reaching a maximum of 6.5 times enhancement of SBS threshold with equal excitation of 160 modes in the MMF. The enhancement of the SBS threshold upon optimal multimode excitation also increases with the number of fiber modes and is consistently higher than both the equal-mode and single-HOM excitations, reaching a maximum value of 9.6 in a MMF with 160 modes.Attorney Docket No. 047162-7524WO1 (02775)(b) TMI Suppression

[0062] FIG. 3, panel (c), shows the optimal mode content obtained using the linear programming method for maximal suppression of TMI. Generically most guided modes in the fiber are excited, with relatively higher weight given to higher order modes. The optimal mode content reveals a new strategy to further increase the threshold by not exciting a group of modes with intermediate propagation constants. These modes have a significant number of ‘neighboring modes’ with slightly larger and smaller propagation constants. As a result, they experience the strongest thermo-optical coupling, and not exciting these modes increases the TMI threshold. In FIG. 3, panel (d), we show the scaling of the TMI threshold enhancement over FM-only excitation with the number of fiber modes for optimal excitation and compare with equal-mode excitation and single-HOM excitation. Both optimal and equal-mode excitation have notably higher TMI thresholds than both FM-only and single-HOM excitations, and they exhibit a linear increase in the TMI threshold with the number of excited modes. This is consistent with the sparse nature of thermo-optical coupling between fiber modes. The slope of the linear scaling is significantly higher for the optimal excitation (0.20) compared to the equal-mode excitation (0.15). When 80 fiber modes are excited, the optimal excitation and equal-mode excitation lead to 17 times and 13 times enhancement of TMI threshold over that of the FM-only excitation, respectively.(c) Joint SBS and TMI Suppression

[0063] Using the linear programming approach, an optimal multimode excitation can also be obtained which suppresses both SBS and TMI simultaneously. This is achieved by minimizing the maximum Brillouin and thermo-optical gain across all the fiber modes and frequencies. The optimal mode content depends on the relative value of SBS and TMI threshold for the FM-only excitation in a given fiber. In FIGS. 4A-B, we show the results for optimal mode content for different ratios of SBS threshold PSBS and TMI threshold PTMI with FM-only excitation. In FIG 4A where PSBS / PTMI = 1 for the FM-only excitation, the optimal mode content is similar to the SBS-only optimization. This is because it is relatively easier to increase the TMI threshold with generic multimode excitations. As such, if SBS and TMI threshold are equal for FM-only excitation, the optimal multimode excitation is similar to that for SBS only suppression, which also causes a sufficient TMI suppression. In FIG. 4B, PSBS / PTMI = 1.5 for the FM-onlyAttorney Docket No. 047162-7524WO1 (02775)excitation, and the optimal mode content is between SBS-only and TMI-only suppressions. The increase in TMI threshold is higher than the SBS threshold, making final thresholds for SBS and TMI equal.

[0064] Overall, with sufficient control of the modal excitation by input wavefront shaping, we can optimally suppress SBS and TMI as well as both simultaneously.EXAMPLE 2 - Polarization-resolved wavefront shaping of a MMF amplifier output beam Introduction

[0065] As described in Example 1, coherent multimode excitation in a MMF amplifier can be optimized to minimize detrimental nonlinear effects and spatio-temporal instabilities. This is realized by shaping the input beam to the MMF amplifier with a polarization-resolved wavefront shaping unit. However, the amplifier output beam typically has low beam quality. An optical beam quality is usually described by the beam propagation factor M2. Its value is given by the normalized product of the second moments of time-averaged intensities in the beam waist and the far field. A large value of M2means low beam quality, which limits how tightly an optical beam can be focused by a lens, but also how well it can be collimated to the far field. A diffraction-limited Gaussian beam has the minimum value of M2= 1, which corresponds to the highest beam quality. Due to uncontrolled mode coupling in the MMF, the amplifier output beam typically has M2» 1, and thus difficult to focus or collimate without any further beam shaping. We design a polarization-resolved wavefront shaping unit, positioned after the multimode fiber amplifier, to convert the output beam with M2» 1 to a smooth beam with M2~ 1. This conversion process, in principle, does not suffer any power loss. We have performed extensive numerical simulations to verify our idea.Methods and Results

[0066] We consider a single-frequency MMF amplifier. The output beam displays a speckled pattern, formed by interference of light in the multiple modes of the fiber. The spatial field distributions for two linear polarizations are different, thus the polarization state of output light varies spatially. FIG. 5 is a schematic illustrating our design of a polarization-resolved wavefront shaping unit, positioned after the MMF amplifier, which transforms the speckled output beam to a linearly polarized beam with M2« 1. In our numerical simulation, theAttorney Docket No. 047162-7524WO1 (02775)multimode fiber core diameter is 42 μm and NA is 0.1. At the optical wavelength λ = 1064 nm, the number of guided modes is N = 80.

[0067] The amplifier output beam is split by a polarizing beam-splitter (PBS) to two beams with orthogonal linear polarizations. The two beams have same power, but distinct speckle patterns, as illustrated in FIG. 5. Their beam propagation factors are M2= 6.18 and 6.22, respectively.

[0068] Each beam passes through three lossless SLMs with free-space propagation in between. Each SLM modulates only the phase of light field and does not cause any amplitude modulation. The distance between two adjacent SLMs is chosen such that free-space propagation from one SLM to the next results in sufficient mixing of spatiallymodulated light via diffraction. We simulate light propagation and phase modulation in a forward numerical model. After the optical beam passes the last SLM, we compute the beam propagation factor M2from the near-field and far-field intensity distributions. The M2value is used as the cost function for optimizing phase modulations on three SLMs. We iteratively search for the optimal phase modulation patterns using error back-propagation and gradient descent method.

[0069] FIG. 5 shows the optimized phase modulation patterns that transform both polarization components of the multimode fiber amplifier output to Gaussian beams with same spatial intensity distribution. Their far-field intensity patterns are smooth. The beam propagation factors are 1.24 and 1.37 for the two linearly polarized beams. We find numerically that adding the fourth SLM to each beam path makes M2value even closer to 1. Finally, the two orthogonally polarized beams are recombined by another PBS. By adjusting the relative (global) phase between the two beams with the last SLMs in their paths, we can control the polarization state of the combined Gaussian beam, e.g., making a right-circularly polarized beam with M2= 1.15.

[0070] Our method works for arbitrary beam profiles emerging from a multimode fiber amplifier, with any number of guided modes and arbitrary modal compositions. To generate an output beam other than a Gaussian beam with M2~ 1, the cost function is changed to the difference between the shaped field distribution and the target one. Additional constraints may be added to the cost function to limit the transverse size of modulation patterns on the SLMs.Attorney Docket No. 047162-7524WO1 (02775)EXAMPLE 3 - Polarization-resolved wavefront shaping of input light to a MMF amplifier Introduction

[0071] The methods described in Examples 1 and 2 show that a polarization-resolved wavefront shaping unit placed before a MMF amplifier can optimize the input wavefront to the amplifier to maximize the power threshold for detrimental nonlinear effects and spatio-temporal instabilities, and a second polarization-resolved wavefront shaping unit placed after the amplifier can transform the output beam to target spatial profile and polarization state. This example shows a simpler implementation, where the polarization-resolved wavefront shaping unit after the multimode fiber amplifier is removed. There is only one polarization-resolved wavefront shaping unit before the amplifier. It is designed to simultaneously suppress detrimental nonlinear effects and spatio-temporal instabilities in the fiber amplifier and control the output beam profile and polarization state.Methods and Results

[0072] We first consider a multimode fiber amplifier with negligible nonlinearity. For such cases, the linear relation between the vector field distribution at the amplifier output facet and that at input facet is given by the field transmission matrix, which can be measured experimentally.

[0073] We have numerically simulated a linear MMF amplifier with mode-dependent gain, strong mode coupling and polarization mixing. The active fiber is a step-index fiber with 80 guided modes for both polarizations. We compute the field transmission matrix T that includes two orthogonal linear polarizations for input and output fields. Its inverse T-1provides the input field distribution to create a target output beam profile and polarization state. As an example, we consider the output beam is in the fundamental mode LPoi with right circular polarization. We separate its field components of orthogonal linear polarizations. They are multiplied by T'1to give the input fields for both polarizations, which are shown in FIG. 6.

[0074] We know from Example 2 that any vector field distribution can be created by designing a polarization-resolved wavefront shaping unit. Here, we apply the same method to design six SLMs for two orthogonal linear polarization components of the seed at the MMF amplifier input end, in order to create the desired input wavefront to the amplifier. The costAttorney Docket No. 047162-7524WO1 (02775)function for our gradient descent algorithm is the difference between the shaped field distribution and the target input.

[0075] In the polarization-resolved wavefront shaping unit, the input light is split by a PBS, and then launched to separate SLMs for parallel wavefront shaping of two orthogonal linear polarizations. We further tuned the relative global phase between the last SLMs for the two beams to control the output polarization state. At the fiber output end, the field is in LPoi mode with right circular polarization, as targeted. We computed the M2=1.08, in agreement with the M2value for LPoi mode of the step-index fiber.

[0076] While the amplifier output is in a single mode, all guided modes in the fiber are excited due to strong mode coupling and polarization mixing. Such multimode excitation suppresses nonlinear effects such as SBS and TMI.

[0077] Next we consider a nonlinear multimode fiber amplifier with strong gain saturation and pump depletion. As the mapping from input to output fields is nonlinear, the field transmission matrix no longer exists. For a nonlinear MMF with gain saturation and thermo-optical nonlinearity, we have proved theoretically and confirm numerically that input wavefront shaping can create target output beam profile, as long as the amplifier operates below the instability threshold and reaches a steady state. The spatial wavefront of an input beam to create a target output beam profile can be found by considering a complementary multimode fiber with absorption. FIG. 7 shows our simulation results of a MMF amplifier with 15 guided modes. We target the amplifier output beam in LPoi mode. For simplicity, polarization mixing in the fiber is neglected, while random mode coupling is included in our simulation.

[0078] To find the input wavefront to the nonlinear MMF amplifier, we numerically launch the amplified signal in LPoi mode together with the residual pump into the distal end of an equivalent multimode fiber absorber. The transmitted field at the fiber proximal end is phase conjugated and defines the input field pattern to the MMF amplifier. It is then generated by three SLMs placed at the amplifier input end. FIG. 7, panel (b), shows phase modulation patterns on three SLMs, obtained with the same optimization procedure and cost function as in the previous example. For confirmation, we numerically simulate an input Gaussian beam passes through the designed SLMs and then couples into the nonlinear MMF amplifier in FIG. 7, panel (c). FIG. 7, panel (d), confirms the output beam in LPoi mode, as targeted. If polarization mixing in the multimode fiber amplifier is not negligible, input wavefronts for two orthogonal linearAttorney Docket No. 047162-7524WO1 (02775)polarizations are shaped by a polarization-resolved wavefront shaping unit to generate the target spatial profile and polarization state of the amplifier output beam.

[0079] While the mapping from a nonlinear MMF amplifier to the complementary absorber confirms the existence of an input wavefront that creates a target output beam, it is hard to implement experimentally. With programmable SLMs, the required phase modulations can be found by minimizing the difference between measured output beam profile and the target one. Alternatively, the inverse mapping from output to input of a nonlinear MMF amplifier may be obtained by training an artificial neural network. Experimentally, SLMs can create different input wavefronts and output beam profiles are recorded. These data are used for training a digital twin for the backward model that maps the amplifier output to input. After the training, the digital twin can predict the input field pattern for a target output.EXAMPLE 4 - Suppression of Stimulated Brillouin Scattering in a MMF Amplifier Introduction

[0080] A major bottleneck for further power scaling of single-frequency fiber amplifiers is the stimulated Brillouin scattering (SBS). It scatters forward propagating signal to backward Stokes light, limiting the amplifier output power. The intense Stokes pulses can also damage upstream lasers. Various methods have been developed to suppress SBS in single-mode fiber amplifiers, e.g., by broadening the signal linewidth, which is detrimental to applications that require coherent beams, such as coherent beam combining.

[0081] In this Example, a highly multimode fiber (MMF) amplifier was built to successfully suppress SBS and attain high output beam quality with input wavefront shaping. The SBS is greatly suppressed due to reduced light confinement in a large-core fiber and broadened Brillouin spectrum by multimode excitation. Experimentally, a SBS-free singlefrequency MMF amplifier was realized with output power of 503 W and spectral (20-dB) linewidth of 20 kHz. The input spatial wavefront of a coherent seed was shaped into a MMF amplifier to control the output beam profile, e.g., focusing to a diffraction-limited spot. The superior performance of the MMF amplifier revealed its potential in further power scaling while maintaining high coherence and beam quality.Attorney Docket No. 047162-7524WO1 (02775)Results

[0082] FIG. 8A shows the experimental setup. In the master oscillator power amplifier (MOP A) configuration (FIG. 8A), the linearly polarized continuous-wave (CW) output of a single-frequency laser oscillator at wavelength λs= 1064 nm is preamplified by two-stage Yb-doped SMF preamplifiers. Using a liquid-crystal SLM, phase-only modulation was applied to the linearly-polarized signal before coupling it to a Yb-doped MMF amplifier that supports 76 spatial modes. The doped fiber is cladding-pumped by five diode lasers in the wavelength range of 966-971 nm to amplify the signal via stimulated emission of Yb ions (see Methods for more details).

[0083] In the fiber amplifier, the signal being amplified acts as a pump for SBS, generating a backwards propagating Stokes field by emitting forward-propagating acoustic waves. The Stokes field is amplified by both SBS and stimulated emission of excited Yb-ions as it propagates backward through the fiber. With increasing pump power, the time trace of the backscattered light intensity shows random spikes of duration ~10–100 ns (left inset of FIG. 8A). The sharp intensity fluctuations are attributed to an SBS-induced dynamic instability, and represents a precursor for the onset of SBS. Here we define the SBS instability threshold as the output signal power at which the maximum height of Stokes spikes is 1.5 times of the continuous background from Rayleigh scattering in backward intensity trace. This level is well below the conventional SBS threshold that is typically set by the condition that the reflected power equals a few percent of the transmitted signal power.

[0084] Compared to the standard SMF amplifier, the large cross-section of our MMF amplifier lowers the signal intensity within the fiber core, thus reducing SBS. Similar to the conventional SBS threshold, the SBS instability threshold is expected to scale quadratically with the fiber core diameter (see SM Sec. 2). If only the fundamental mode (FM) is excited in our MMF with a 42-μm core, the SBS threshold is approximately 8 times of that in a large-mode-area (LMA) SMF amplifier with a 15-μm core. From the previously measured SBS instability threshold in the SMF amplifier, we estimate the SBS instability threshold for FM-only excitation in our MMF amplifier to be 24 W, as shown in FIG. 9A.Attorney Docket No. 047162-7524WO1 (02775)Multimode SBS theory

[0085] The four-fold increase of SBS instability threshold (24 W to 97 W) by multimode excitation within the same MMF can be explained by our semi-analytic theory of SBS in the MMF amplifier. When the input light is distributed over multiple fiber modes, the forwardpropagating signal in each mode will be Brillouin-scattered to backward-propagating Stokes in all modes. This process is mediated by acoustic modes in the fiber, and the Stokes is frequency downshifted from the signal by the acoustic frequency Ω. The scattering coefficient for signal in the / -th mode and Stokes in the m-th mode, gB(m,l)(Ω), depends on the spatial overlap of the signal and Stokes mode profiles with various acoustic modes in the fiber. Each mode pair of I and m typically has the largest spatial overlap with a single acoustic mode, making gB(m,l)peaked at this acoustic mode frequency Ω. The total Brillouin gain at axial position z in the m-th mode due to signals in all fiber modes is:GB(m)(z) = Σl[equation image] HEIGHT="38" WIDTH="348" SRC="imgf000026_0001.tif" / > l where PS(l)is the signal power in the l-th mode. The growth rate of Stokes power in the m-th mode PB(m)is determined byy(2)Due to the exponential nature of growth, the total Stokes power PB= ΣmPB(m)is dominated by the mode m that has the maximum value of GB(m)HEIGHT="22" WIDTH="18" SRC="imgf000026_0005.tif" / > '.

[0086] GB(m)depends on the signal power distribution among the fiber modes. We consider three cases below. First, all the signal power is in the FM, l = 1. gB(1,1)HEIGHT="22" WIDTH="33" SRC="imgf000026_0006.tif" / > gB' is larger than gBn’1^ for any m > 1, due to better spatial overlap between optical and acoustic modes. Thus Brillouin scattering is strongest for the Stokes field in the FM (m = 1), mediated by the lowest-order acoustic mode in the fiber. Next, consider when a single HOM / > 1 is excited. Due to larger acousto-optic overlap, intramodal Brillouin scattering (m = / ) is stronger than intermodal scattering (m 1), g^’1^ > gBn'l)for any m A / . However, smaller acousto-optic overlap makesAttorney Docket No. 047162-7524WO1 (02775)(I D (1 1)9B 9B ■>anthe SBS threshold for Z > 1 is higher than that with FM-only excitation.Finally, if input power is distributed among all fiber modes, GB(m)is a sum of intramodal and intermodal scattering coefficients weighted by the signal power PS(l)in individual modes, which is much less than the total power. Because gB(m,l)for different mode pairs are peaked at varying frequencies Ω, GB(m)is spectrally broadened and has a notably lower peak value than single-mode excitation (FIG. 9B). As the peak value dictates the exponential growth of Stokes power, the SBS threshold is appreciably higher for multimode excitation than for the excitation of any single mode.Pump depletion and gain saturation

[0087] In a high-power MMF amplifier like ours, gain saturation and pump depletion are strong. These effects determine the signal power distribution throughout the fiber, which impacts the Stokes growth rate. To include these effects in our theory, we extend our earlier model as follows. We first calculate the signal power in each mode PS(l)(z) throughout the fiber, taking into account gain saturation and pump depletion in the MMF amplifier. At or below the SBS instability threshold, the Stokes power is much lower than the signal power, thus feedback due to SBS is neglected when computing the signal power and pump power distribution in the MMF amplifier (SM Sec. 2).

[0088] Numerical simulation of our MMF amplifier using the improved model (with parameters listed in Table 1) reveals that the pump power (at wavelength λP= 971 nm) is mostly absorbed in the first 6 m of the Yb-doped fiber. Beyond this distance, the signal power does not grow appreciably (FIG. 10). To investigate the scaling of SBS instability threshold with fiber length, we started with a long fiber and gradually cut it back in FIG. 9C. From the physical fiber length L and total signal power PS(z) = ΣlPS(l)(z) HEIGHT="22" WIDTH="77" SRC="imgf000027_0001.tif" / > = Szwecompute the effective fiber length for SBS: Leff= ∫0LHEIGHT="24" WIDTH="17" SRC="imgf000027_0002.tif" / > Psz)dz / Ps(L). FIG. 9C shows that the measured SBS instability threshold increases rapidly as Leffdecreases (blue crosses). The blue curve shows that SBS threshold scales inversely with Leff, as expected from our theory (SM Sec. 2). To illustrate the enhancement by multimode excitation, the green circles show the estimated thresholds with FM-only excitation at fiber lengths corresponding to experimental values. At the shortest lengthAttorney Docket No. 047162-7524WO1 (02775)Leff = 3.7 m, the measured SBS instability threshold reaches 503 W, about 5 times higher than the FM-only threshold.Table 1: Table ofMMF amplifier parametersParameter ValueFiber core diameter [pm] 42Fiber cladding diameter [pm] 570Core numerical aperture 0.1Signal wavelength λs[nm] 1064Pump wavelength λP[nm] 971Pump absorption cross-section <jp ’ [nr] 1.2 × 10−24Pump emission cross-section σP(e)[m2] 1.02 × 10−24Signal absorption cross-section σS(a)[m2] 5.8 × 10−27Signal emission cross-section σS(e)[m2] 2.71 × 10−25Yb concentration ρt[ions m−3] 6.5 × 1025Output beam shaping

[0089] All data points in FIG. 9C are taken with the output beam focused to a chosen spot outside the distal facet of the MMF by optimizing the input phase front with the SLM (see Methods). The inset in FIG. 9C is an optical image of focal spot taken at 173 W output signal power. With phase-only modulation of the input signal, 76% of total output power is concentrated within the focal area. We have obtained similar focusing efficiencies at higher powers (FIG. 11). The near-field focusing of the output beam is achieved through interference of signal in many fiber modes, and hence the SBS instability threshold is enhanced by multimode excitation in the amplifier.

[0090] To control the output beam profile by input wavefront shaping, the signal bandwidth must be narrower than the spectral correlation width of a MMF amplifier. The latter characterizes how fast the output field pattern decorrelates with frequency detuning for a fixed input wavefront. Despite gain saturation and pump depletion in our MMF amplifier, its spectral correlation width barely changes from that of the passive MMF. With increasing fiber length, the spectral correlation width decreases; but even for the longest MMF of our amplifier, theAttorney Docket No. 047162-7524WO1 (02775)correlation width exceeds 1 GHz, which is well above the signal bandwidth (see next section). The amplified signal is spatially coherent; that is, the relative phase of output fields between any two positions is time-invariant. Consequently, the speckle pattern created by multimode interference at any frequency within the signal bandwidth is almost identical, leading to high intensity contrast as seen in the inset of FIG. 9A.

[0091] The spatial coherence of the amplified signal allows us to generate the same output pattern, e.g., focusing to a chosen location, for all frequency components, by imposing a single wavefront on the input beam. To measure the phase of the output field, we perform an interferometric experiment (see Methods). The inset of FIG. 9C shows the measured phase is constant across the focal spot, confirming diffraction-limited focusing.

[0092] To track axial evolution of a focal spot, we image the transverse intensity distribution at different axial planes. Three exemplary images are presented in FIGS. 12A-C, revealing the spot radius increases with axial distance z from the focal plane. We extract the intensity profiles in x and y axes, and fit with Gaussian envelop to obtain the spot radius wxand wy(see Methods). FIG. 3D shows the growth of wxand wywith z. We fit the z dependence withw(z) = w(0)√(1 + (M²λSz / πn[w(0)]²)²)where w(0) is the focused spot radius at the focal plane (z = 0), Asis the signal wavelength in vacuum, and n « 1.45 is the refractive index of the glass endcap in which the output beam is focused and further propagates. Equation 3 fits the data well and give the beam propagation factors Mx2≈ 1.05 and My2≈ 1.35.Efficiency and linewidth

[0093] We further characterize the performance of our high-power MMF amplifier with the shortest Leff = 3.7 m. First, we measure the output signal power and residual pump power. The power of the amplified signal is given by the difference between input and output signal power, PS(amp)= PS(out)− PS(in)HEIGHT="23" WIDTH="132" SRC="imgf000029_0002.tif" / >gamp) = Ps(°ut) — Ps^in\ The absorbed pump power is obtained by subtracting the residual pump power at the amplifier output from the pump power launched into the MMF, PP(abs)= PP(in)− PP(out). FIG. 13A shows PS(amp)increases almost linearly with PP(abs). A linearAttorney Docket No. 047162-7524WO1 (02775)fitting gives the slope efficiency of 82%. The experimental data agree with theoretical prediction of our model.

[0094] We also measure the output spectrum of the MMF amplifier with an optical spectrum analyzer. In FIG. 13B, the signal appears as a narrow peak at λS= 1064 nm. It sits on top of a broad amplified spontaneous emission (ASE) band. The peak ratio of amplified signal to ASE is 52 dB.

[0095] The signal linewidth, which is too narrow to be resolved by the optical spectrum analyzer, is measured by a heterodyne interferometer (see Methods). FIG. 13C shows the heterodyne peak resulting from beating of the amplified signal and a reference (red curve). The full width at −20 dB of the maximum is ΔνM= 35 kHz. Subtracting the reference width of ΔνR= 17 kHz (see FIG. 10), the 20-dB width of the output signal is ΔνS= ΔνM− ΔνR= 18 kHz. It corresponds to a Lorentzian line of 3-dB bandwidth (full width at half maximum) ~ 1 kHz.

[0096] For comparison, we measure the linewidth of input signal to the MMF amplifier using the same method. As shown by the blue curve in FIG. 13C, the input signal has nearly identical width as the output (red curve), thus optical wavefront shaping and coherent multimode amplification do not cause detectable spectral broadening of the signal. As mentioned earlier, a common approach to mitigating SBS in SMF amplifiers is by broadening the input signal linewidth to tens of GHz. In our scheme, the 3-dB signal linewidth is 6 orders of magnitude narrower, thus the temporal coherence length is 6 orders of magnitude longer in the MMF amplifier.Discussion and conclusion

[0097] We have shown that input wavefront shaping allows us to simultaneously mitigate SBS in the MMF amplifier and control the output beam profile. Although an SLM placed at the fiber amplifier output end could shape the amplified beam and allow another SLM at the input end to optimize multimode excitation for maximal SBS suppression, we choose here to shape the low-power seed to avoid high-power handling. In the current experiment, phase-only modulation of the input wavefront limits output focusing efficiency such that about 20-30% of power is outside the focal spot. Complete focusing can be achieved with both amplitude and phase modulation of the input signal. Full-field modulation will also generate an output beam with M2closer to unity.Attorney Docket No. 047162-7524WO1 (02775)

[0098] In addition to output focusing, input wavefront shaping can generate different output profiles of the MMF amplifier, which will be useful for laser welding and material processing. Even in the presence of strong nonlinearity, gain saturation and pump depletion, as long as the amplifier operates below the instability threshold, optimal input wavefront can be found by minimizing the difference between the measured output beam shape and the target. In this work, the spatial profile of a single linear polarization of the output beam is controlled by input wavefront shaping. Although the input signal is linearly polarized, polarization mixing in the MMF causes depolarization. However, it is possible to control output beam profiles for two orthogonal polarizations by shaping both input polarizations. This can be done by separating two orthogonal polarizations of the input signal and shaping their field patterns separately before combining them and coupling to the MMF.

[0099] An instability-free increase of the power in a MMF amplifier beyond the level demonstrated here is possible by enlarging the fiber core to further suppress SBS. Even if only the fundamental mode is excited, the SBS threshold scales quadratically with the core diameter (SM Sec. 2). With multimode excitation, Brillouin spectrum broadening is more pronounced, as additional modes can be excited in a MMF with larger core, leading to higher enhancement of the SBS threshold. While our scheme does not rely on specialty fibers, wavefront shaping can be applied to specialized MMFs with lower optical nonlinearity and to microstructured fibers with large cross-section for higher-power operation.

[0100] More generally, our method can be extended to mitigating other detrimental nonlinear effects in high-power fiber amplifiers, e.g., transverse mode instability, stimulated Raman scattering, modulation instability, and also to other types of high-power lasers such as solid-state and semiconductor optical amplifiers. Our scheme of spatial and modal spread of the signal may be combined with temporal stretch of optical pulses for high-peak-power amplification. In total, the SBS threshold in our MMF amplifier is an order of magnitude higher than that in a standard step-index SMF amplifier.Experiments and methodsMultimode fiber amplifier

[0101] The oscillator is a single-frequency fiber laser (NP Photonics RFLM-100-1-1064 Rock module) that produces a CW linearly-polarized beam of power up to 100 mW atAttorney Docket No. 047162-7524WO1 (02775)wavelength As= 1064 nm. The oscillator output is split into two: one serves as the seed for fiber amplifiers, the other as the reference for interferometric measurements. The 25 mW seed is launched into a two-stage preamplifier with Yb-doped SMFs and amplified to 10 W. The output beam is then wavefront-shaped by a phase-only liquid-crystal SLM (Santec SLM-300), which is placed at the focal plane of the input facet of a triple-clad step-index MMF without Yb doping (Coherent FUD-4693). This fiber, labeled signal fiber, has a core diameter 42 μm and numerical aperture (NA) 0.1. It supports 76 guided modes at the signal wavelength As= 1064 nm. The proximal facet of the signal fiber is angle-cleaved (~ 8°) to avoid spurious reflection. Its distal end is combined with the SMF pigtails of five laser diodes, each providing up to 125 W power. With increasing electric current, the pump wavelength APgradually shifts from 966 nm to 971 nm. The output MMF of signal-pump combiner is spliced to a Yb-doped MMF (Coherent FUD-4715) with same parameters as the signal fiber (core, first cladding, and second cladding diameters are 42, 570, and 650 μm, respectively). The distal facet of the doped MMF is spliced to a short undoped MMF that is terminated by an antireflection-coated endcap. An optical isolator is placed between the seed laser and the preamplifier, a second one between the preamplifier and the SLM, and a third one between the SLM and the MMF amplifier. The SLM is water-cooled to ensure that prolonged laser exposure does not cause heat-induced degradation. Wavefront shaping

[0102] The SLM area covering the NA of MMF core is divided into 256 macropixels, each consisting of 72 x 72 SLM pixels of area 7.8 x 7.8 pm2(8 x 8 pm2pitch). We adjust the phase of each macropixel sequentially to maximize the output intensity at the chosen location. We start from the center macropixel, scan its phase < > from 0 to 2 / r, and measure the power within the focal area of diameter given by twice the full width at half maximum (FWHM) of the maximum intensity. Since the focused power typically varies sinusoidally with the phase, the measured power is fitted by an objective function cos( — < >0) + const, with < >0representing the phase for maximum power. After setting the phase of that macropixel toO, we repeat the phaseoptimization procedure for neighboring macropixels, proceeding in a spiral pattern outward from the center, until the optimal phases for all macropixels are determined. The entire optimization process takes a few minutes, which is limited by the speed of our liquid-crystal SLM. SwitchingAttorney Docket No. 047162-7524WO1 (02775)to MEMS-based SLMs can greatly reduce the optimization time, as demonstrated for output focusing through passive MMFs.

[0103] To generate a smoother input wavefront for higher output focusing efficiency, the number of SLM macropixels exceeds the number of fiber modes. This is not necessary, and it increases the optimization time. Alternatively, the input wavefront may be decomposed into fiber modes and the optimization performed in modal basis. This will provide a smoother input wavefront with a smaller number of modes to be optimized.

[0104] In our MMF amplifier, heat dissipation causes temperature and refractive index changes that induce mode coupling at high power. Thus, we need to re-optimize the input wavefront when changing the power. But at a given power, once the temperature distribution is stabilized, the optimized input wavefront for output focusing can work for many hours without readjustment.

[0105] After the optimization, we characterize the focusing efficiency by the ratio of power within the focal area and the total power. To determine the radius wxyof focal spot, we fit the intensity distribution along x or y axis with a Gaussian envelope and set wxywhere the intensity drops to e−2of the maximum at the center.Optical characterizations

[0106] The output signal is divided into multiple beams for spatial, spectral and temporal characterization. The signal power is measured by a water-cooled power meter (Ophir Spiricon L1500W-BB-50). The optical spectrum is recorded by an optical spectrum analyzer (Yokogawa AQ6370D). The temporal variation of signal intensity is detected by a fast photodiode (Thorlabs PDA20CS) that is connected to a high-speed oscilloscope (Keysight DSOX3014T). Spatial intensity distribution of the output signal on a transverse plane at a distance of 50 μm beyond the fiber and endcap interface is imaged by a lens to a CCD camera (Allied Vision Manta 6419B-NIR). A linear polarizer (Thorlabs LPIREA100-C) is placed before the camera to select one polarization. To measure its phase, the linearly-polarized signal interferes with a reference beam from the seed laser. The spatial phase distribution of the signal field is extracted from the off-axis hologram.

[0107] The spectral linewidth of the seed laser is measured by a delayed self-heterodyne interferometer. Schematics are shown in FIGS. 8A-B. The output beam is split into two. One ofAttorney Docket No. 047162-7524WO1 (02775)them is modulated by an acousto-optic modulator (AOM) at frequency 110 MHz. The other beam is delayed by propagating through a 25-km SMF. The two incoherent beams are then recombined by a beam splitter and the total intensity is measured by a fast photodiode (Thorlabs DET08C) which is connected to an electrical spectrum analyzer (Keysight N9030B-PXA). The 20-dB width (full width at -20 dB of maximum) of the heterodyne peak at 110 MHz is 34 kHz (FIG. 10). Since it is equal to two times the laser linewidth, thus the 20-dB linewidth of seed laser is 17 kHz.

[0108] To measure the MMF amplifier linewidth, a fraction of the output signal is modulated by the AOM at a frequency of 110 MHz. It is then combined with a reference beam from the seed laser. To remove its coherence with the signal, the reference is delayed via a 25-km SMF. The sum of the modulated signal and delayed reference is measured by a fast photodiode and an electrical spectrum analyzer. The width of the heterodyne peak at 110 MHz is equal to the sum of the signal linewidth and reference linewidth.Focusing efficiency

[0109] We measure the output focusing efficiency at varying signal power and observe no significant change. However, the percentage of total output power being concentrated within the focal area varies slightly with the transverse location of the focal spot. FIG. 11 shows the focusing results at output powers of 233 W and 455 W. The power ratio (PR), defined as the ratio of power within the focal area and the total power, is 80% at 233 W, 76% at 455 W.Noise characterization

[0110] To investigate potential impact of wavefront shaping and multimode excitation on the amplifier noise, we measure the relative power noise (RPN) and phase noise of the MMF amplifier output. For RPN, we record the temporal fluctuation of the MMF amplifier output power with a fast photodetector (with a transimpedance amplifier of bandwidth 10 MHz) and an oscilloscope (of bandwidth 100 MHz), when shaping the input wavefront for output focusing. For comparison, we also measure the RPN without wavefront shaping or pumping of the MMF amplifier. FIG. 14A shows small changes in RPN spectrum by wavefront shaping and multimode amplification.

[0111] To measure the differential phase noise, we interfere the output beam from the MMF amplifier with the seed laser modulated by an AOM. The optical path length mismatchAttorney Docket No. 047162-7524WO1 (02775)between two arms of the interferometer is on the order of 10 meter, which is much shorter than the temporal coherence length of the laser. As shown in FIG. 14B, we do not detect any notable change in the phase noise spectrum by input wavefront shaping and multimode amplification.

[0112] FIG. 15A shows that the SLM introduces a couple of peaks between 1 and 3 kHz in both RPN and phase noise spectra, but peak heights are small. FIG. 15B shows the input wavefront shaping for output focusing does not cause any detectable change in the spectral linewidth.Theoretical Model

[0113] We have derived a semi-analytic theory of vector optical modes coupled via scalar acoustic waves to model SBS in highly multimode fiber (MMF) amplifiers, extending previous studies on passive multimode fibers and single-mode-fiber (SMF) amplifiers. Here we fully take into account optical gain saturation and pump depletion, which were ignored in our previous studies. We consider only Stokes amplification by SBS, and not stimulated emission of Yb ions, because the Brillouin gain is much higher than that by stimulated emission below the SBS threshold. Our theoretical model elucidates the underlying mechanism of the SBS suppression and can account for the observed scaling of SBS threshold in the MMF amplifier.

[0114] As mentioned in the main text, coherent multimode excitation in an MMF amplifier offers two key advantages over the SMF counterpart. First, the larger fiber core reduces light confinement, and weaker signal intensity within the core reduces optical nonlinearity.Quantitatively, the SBS threshold for FM-only excitation scales linearly with the fiber core area. Second, the SBS threshold is further increased by highly multimode excitation in the MMF amplifier. The Brillouin gain is reduced, due to (i) intermodal scattering being weaker than intramodal scattering, (ii) effective spectrum broadens due to differential frequency displacement of Brillouin scattering peaks for different mode pairs. Hence, the SBS threshold is higher when the signal excites more modes in the fiber. These effects were predicted and observed in passive fibers previously; our improved theory, which includes gain saturation and pump depletion, shows that the physical origin of the reduction of Brillouin gain is the same in MM amplifiers. This allows us to make a solid scaling prediction for the variation of the threshold with the effective length of the fiber.

[0115] The SBS threshold varies with the fiber length. Since the Stokes power grows exponentially in the backward propagation through the fiber amplifier, shortening the fiberAttorney Docket No. 047162-7524WO1 (02775)length will reduce SBS. However, the output signal power may be lower if the signal amplification is truncated. The optimal length of a fiber amplifier is given by the minimum length for nearly complete absorption of the pump light. We numerically find such length by simulating the coupled evolution of pump power and signal power in a MMF amplifier, taking into account pump depletion and gain saturation.

[0116] Consider a co-pumped Yb-doped MMF amplifier. The fiber core is doped with Yb ions that are pumped to amplify the signal via stimulated emission. Below the SBS instability threshold, the Stokes power is much lower than the signal power, allowing us to ignore SBS when computing signal power and pump power throughout the MMF amplifier. The field amplitude A® of the signal in the Z-th fiber mode satisfies the following equation:d / 1s”‘>fe)=y „<.■..»(„)=fdf (4)dz LIW 2 J ' 1 + / . / >!,wHere r±and z denote transverse and longitudinal position respectively,is the propagation constant of mode m(Z),gEis the gain coefficient for mode m provided by mode I, g0is unsaturated gain coefficient which is assumed to be mode independent, i m) is the transverse field profile of fiber mode l(m), / sis the signal intensity given by Zs(f±, Z) =A;(^) V;z(C.)eI^z|2,and Zsatis the gain saturation intensity, gEn'l)depends on the spatial overlap of i / jmand i i weighted by the intensity-dependent gain saturation over Yb doped area of the fiber. The cross modal gain m = I is non-zero because of spatial hole burning, i.e., transverse-varying gain saturation by / s(f±, z)~. As a result, the equations for different modal amplitudes are coupled, unlike the MMF amplifier with unsaturated or weakly saturated gain. Both Zsatand g0depend on the pump power FP, which evolves along the fiber as:dPP(z) = -gp^P^z), g^z} = ^jd^±[ffpa)Pi(4< *) (5)dz ^e)pu(4<^)] Here gPz is the absorption coefficient of pump light, ptis the density of Yb ions, pLl(pi) is the fraction of Yb ions in the upper (lower) energy level, <7p (dp ) is the absorption (emission) cross-section of Yb ions at the pump wavelength AP. The integral is over Yb doped area of the fiber core, and Acis the area of pump core (the first cladding of a double- or triple-clad fiber). Note that we consider a single equation for the sum of pump power in all modes, because theAttorney Docket No. 047162-7524WO1 (02775)pump light is incoherent and fills the pump core uniformly. Finally, the fraction of upper (u) and lower (p\ = 1 — pu) level density at steady state is obtained from the rate equations of Yb ions. We solve Eq. 4 and Eq. 5 together with the expression for puusing a finite-difference method such as Euler or Runge-Kutta method.

[0117] FIG. 16A shows the pump power Pp(z) decays as the total signal power Ps(z) = Z ' i P$l)(z) grows along the MMF amplifier with parameters listed in Table 1. When the pump power is nearly depleted, the signal power levels off. The vertical dashed line marks the shortest active fiber length of 5.5 m, where the signal power (blue curve) grows from 10 W to 490 W, and the pump power (orange curve) reduces from 725 W to 98 W. The pump wavelength AP= 971 nm deviates from the absorption peak of Yb ions at 976 nm. If the pump wavelength is shift to 976 nm, the residual pump power drops to 8 W at z = 5.5 m. The final optical-to-optical conversion efficiency of the MMF amplifier with L = 5.5 m will increase to 80%.

[0118] FIG. 16B shows the signal power P^ of individual mode grows at slightly different rate, as the gain saturation varies from mode to mode. The power fraction Plz) = P^lz~) I Ps(-z) initially varies with z, then remains almost constant after z > 3 m, where the signal power is high and Brillouin Scattering is strong (FIG. 16C). Equation (1) in the main text becomesG^n)(z) = Ps(z) £ = Ps(z)G™(6)iwhere g^’^P^ is approximately independent of z for z > 3 m where Brillouin gain is high. Hence,p£m\z) =. (7)where PB’nL) is the Stokes power in the m-th mode at the distal end of the fiber (z = L) from spontaneous Brillouin scattering. The Stokes power at the proximal end (z = 0), summed over all modes, is dominated by the mode with the largest GBmIt is approximated bypGn\&) PBCm)(L)eG'Bm)Jo'ps^dz= P^(L)e^niPsa)Len(8)Attorney Docket No. 047162-7524WO1 (02775)where Leff= Ps(z)dz / P (L) is the effective fiber length. It refers to the length of an amplifier with constant signal power PSL~) along z and longitudinally integrated power equal to JQ£Ps(z)dz. In an amplifier with Ps(.z) < Ps( l ^eff is shorter than the fiber length L.

[0119] Our theory allows us to calculate the conventional SBS threshold based on the condition that the steady-state backward Stokes power exceeds a fraction of r of the forward signal power Ps L'), (in the literature r varies from 0.1% and 10%, here we choose 1%). We use a good approximation that the total Stokes power is dominated by the mode with the highest Stokes gain, P^mo~). The amplified signal power at the SBS threshold is given by1 1 rPS(X) / ’sW = - - (9)Leff G^ D PBCm)(L) Ignoring its weak (logarithmic) dependence on the output power level, the SBS threshold is inversely proportional to the effective fiber length Leffas found experimentally in FIG. 9C. The threshold is also inversely proportional to the effective Brillouin gain coefficient G^\

[0120] Experimentally we measure the SBS instability threshold, at which significant spikes appear in the time trace of reflected intensity. Such spiking results from SBS and occurs at power below the conventional SBS threshold. It prevents further power scaling as large Stokes pulses could damage the upstream lasers. Since the instability is induced by SBS, it should depend on the average SBS gain calculated in our theory, and its threshold is expected to follow a similar scaling with Leffas the conventional SBS threshold our model predicts. This conjecture is confirmed in FIG. 9C. In order to compare to experimental data, we assume the Stokes spiking emerges above certain reflectivity r0and treat the logarithmic term in Eq. (8) as a single fitting parameter, which we set to match the lowest threshold, Leff = 15.4 m in FIG. 9C.

[0121] Experimentally, the fiber modes are excited either by the seed at the MMF input end or via linear and nonlinear mode coupling throughout the fiber. It is hard to measure the spatially-varying refractive index change in the fiber core due to pumping and heating, and the resulting nonlinear mode coupling is difficult to estimate. Thus we do not know the exact number of modes excited experimentally in the MMF amplifier, and we believe the number of excited modes actually varies along the fiber. Nevertheless, the measured SBS instability thresholds under multimode excitations are approximately 4-5 times of the estimated FM-onlyAttorney Docket No. 047162-7524WO1 (02775)thresholds (blue crosses vs. green circles in FIG. 9C). Such degree of enhancement is very similar to the theoretical estimation for equal excitation of all spatial modes in the fiber. Hence, we expect most modes are excited in the MMF amplifier. The maximum SBS threshold enhancement is estimated to be 6-7 times, which can be achieved by input wavefront optimization solely for SBS suppression but the output beam would not be focused. Placing an SLM at the MMF output end can shape the amplified beam, allowing another SLM at the input end to optimize multimode excitation for maximal SBS suppression.EXAMPLE 5 - High-beam-quality single-frequency multimode-fiber amplifiers with full-field control

[0122] High power, high beam quality, and narrow linewidth are required for many applications of fiber laser amplifiers. However, further power scaling is limited by optical nonlinearities and instabilities in single-mode or few-mode fiber amplifiers. As discussed above, the present inventors have revealed that coherent multimode excitations in highly multimode fiber (MMF) amplifiers can mitigate stimulated Brillouin scattering (SBS) and transverse mode instability (TMI). Through such approaches, an output power of 500 W can be achieved by suppressing SBS in an MMF amplifier with 1 kHz linewidth. The main concern of using multimode fibers for single-frequency laser amplifiers is the output speckles formed by modal interference. While the present inventors have also demonstrated that the output beam of a highly multimode fiber can be focused to a diffraction-limited spot by shaping the phase front of a coherent input light, phase-only modulation of the input wavefront limits the output focusing efficiency and the beam propagation factor M2.

[0123] This Example describes an efficient method of modulating both field amplitude and phase for two orthogonal polarizations of an input signal to an MMF amplifier. The full-field shaping is realized by using the first-order diffraction of a phase-only spatial light modulator (SLM). It enables complete control of the spatial profile and polarization state of the amplified beam. In particular, an output Gaussian beam with M2< 1.2 was generated from a singlefrequency MMF amplifier with 80 modes.

[0124] FIG. 17 shows the experimental setup. A single-frequency, linearly-polarized laser at wavelength 1064 nm was preamplified to 45 W. The Gaussian beam was then expanded and converted to a flat-top beam before incident to a SLM. First-order diffraction from the SLMAttorney Docket No. 047162-7524WO1 (02775)is composed of two sub-orders, which produce two distinct field patterns. One of them is converted to the orthogonal polarization and then combined with the other by the beam displacer. The combined beam is launched to the MMF amplifier as the seed. A specialized iterative algorithm was introduced to determine the optimal SLM pattern that delivers the desired amplitude and phase modulations of both polarizations into the MMF with minimal diffraction loss. The MMF amplifier is co-pumped by laser diodes at wavelength 976 nm. The amplified signal and residual pump are separated at the output. Both near-field and far-field intensity distributions of the amplified signal are recorded simultaneously.

[0125] An iterative algorithm was introduced to improve the output beam quality of our MMF amplifier. At each iteration, the SLM pattern is updated to minimize the deviation between the measured intensity distributions and the ideal Gaussian profiles in both near-field and far-field. The beam propagation factor (M2) was directly calculated from the second moment of the measured near- and far-field intensity distributions. FIG. 18C shows the amplified beam with M2= 1.17 at the output power of 210 W. The refined SLM phase pattern and the corresponding input field patterns for horizontal and vertical polarizations are presented in FIG. 18A and 18B.EQUIVALENTS

[0126] Although preferred embodiments of the invention have been described using specific terms, such description is for illustrative purposes only, and it is to be understood that changes and variations may be made without departing from the spirit or scope of the following claims.INCORPORATION BY REFERENCE

[0127] The entire contents of all patents, published patent applications, and other references cited herein are hereby expressly incorporated herein in their entireties by reference.

Claims

Attorney Docket No. 047162-7524WO1 (02775)CLAIMSWhat is claimed is:

1. A multimode fiber laser amplification system, comprising:a multimode fiber (MMF) amplifier; anda first polarization-resolved wavefront shaping unit positioned before the MMF amplifier; wherein the amplifier is arranged and disposed to receive a shaped seed from the first polarization-resolved wavefront shaping unit and generate an amplified output beam.

2. The system of claim 1, wherein the first polarization-resolved spatial wavefront shaping unit is designed to simultaneously reduce detrimental nonlinear effects and spatio-temporal instabilities in the MMF amplifier and to create selective spatial field distribution and polarization state of the amplifier output beam.

3. The system of claim 1, further comprising:a second polarization-resolved wavefront shaping unit positioned after the MMF amplifier.

4. The system of claim 3, wherein:the first polarization-resolved spatial wavefront shaping unit is designed to shape the spatial field distributions of two orthogonal linear polarizations of the seed into the MMF amplifier for maximal suppression of detrimental nonlinear effects and spatio-temporal instabilities in the MMF amplifier; andthe second polarization-resolved spatial wavefront shaping unit is designed to transform the amplifier output to a coherent beam of target spatial field distribution and polarization state.

5. The system according to any one of claims 1-4, further comprising a laser source, the laser source configured to provide a coherent seed to the MMF laser amplification system.

6. The system of claim 5, wherein the laser source comprises a fiber laser, a semiconductor laser, or a solid-state laser.Attorney Docket No. 047162-7524WO1 (02775)7. The system of claim 5, wherein the laser source produces spatially coherent light in a single transverse mode.

8. The system of claim 5, further comprising one or more preamplifiers positioned sequentially after the laser source and before the MMF amplifier.

9. The system of claim 8, wherein the one or more preamplifiers comprise fiber preamplifiers, semiconductor optical amplifiers, solid-state preamplifiers, or combinations thereof.

10. The system of claim 8, wherein the one or more preamplifiers operate in a single transverse mode and the laser is a coherent laser.

11. The system according to any one of claims 1-4, further comprising:at least one lens;at least one isolator;at least one attenuator;at least one beam expander;at least one spatial filter;at least one beam splitter;at least one objective; orcombinations thereof.

12. The system according to any one of claims 1-4, wherein the MMF amplifier comprises:a signal MMF;an active MMF;at least one pump laser; anda signal-pump combiner coupling the at least one pump laser to the active MMF.

13. The system according to any one of claims 1-4, wherein the system suppresses stimulated Brillouin scattering (SBS) in fiber laser amplifiers.Attorney Docket No. 047162-7524WO1 (02775)14. The system according to any one of claims 1-4, wherein the system suppresses transverse modal instability (TMI) in fiber laser amplifiers.

15. The system according to any one of claims 1-4, wherein the system suppresses stimulated Raman scattering (SRS) in fiber laser amplifiers.

16. The system according to any one of claims 1-4, wherein the system suppresses modulation instability (MI) in fiber laser amplifiers.

17. The system according to any one of claims 1-4, wherein the system suppresses vector modulation instability (VMI) in fiber laser amplifiers.

18. The system according to any one of claims 1-4, wherein the system operates in the state of continuous wave (CW), quasi-CW, or pulsed.

19. A method of multimode laser amplification, the method comprising:providing the system according to any one of claims 1-4;designing spatial light modulation patterns to create desired vector field distributions with the first polarization-resolved spatial wavefront shaping unit; andamplifying the shaped multimodal laser waveform with the MMF amplifier.

20. The method of claim 19, wherein the amplification of the shaped multimodal laser waveform suffers minimal detrimental nonlinear effects and spatio-temporal instabilities in the MMF amplifier.

21. The method of claim 19, wherein the amplification of the shaped multimodal laser waveform generates a coherent output beam with desired spatial field distribution and polarization state, and simultaneously reduces detrimental nonlinear effects and spatio-temporal instabilities in the MMF amplifier.

22. The method of claim 19, wherein the method increases the lowest power threshold of SBS, SRS, TMI, MI and VMI.Attorney Docket No. 047162-7524WO1 (02775)23. The method of claim 19, wherein the MMF amplifier output has spectral (20 dB) linewidth narrower than 20 kHz.

24. A method of multimode laser amplification, the method comprising:providing the system according to any one of claims 3-4;designing spatial light modulation patterns to create desired vector field distributions with the second polarization-resolved spatial wavefront shaping unit; andtransforming the amplifier output beam to a coherent beam with desired spatial field distribution and polarization state.

25. The method of claim 24, wherein the MMF amplifier output has spectral (20 dB) linewidth narrower than 20 kHz.

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