Cross-correlated (C2) imaging of fiber and waveguide modes

General error analysis of matrix-operation-mode decomposition technique in few-mode fiber laser

The mode decomposition based on matrix operation (MDMO) is one of the fastest mode decomposition (MD) techniques, which is important to the few-mode fiber laser characterization and its applications. In this paper, the general error of the MDMO technique was analyzed, where different influencing factors, such as position deviation of the optical imaging system, coordinate deviation of the image acquisition system, aberrations, and mode distortion were considered. It is found that the MDMO technique based on far-field intensity distribution is less affected by optical imaging system position deviation, coordinate deviation of the image acquisition system, and mode distortion than those based on direct near-field decomposition. But far-field decomposition is more affected by aberration than those based on near-field decomposition. In particular, the numerical results show that the deviation of the coordinate axis direction is an important factor limiting the accuracy of MD. In addition, replacing the ideal eigenmode basis with a distorted eigenmode basis can effectively suppress the decrease in mode decomposition accuracy caused by fiber bending. Moreover, based on detailed numerical analysis results, fitting formulas for estimating the accuracy of the MDMO technique with imperfections are also provided, which provides a comprehensive method for evaluating the accuracy of the MDMO technique in practical engineering operations.

Optical Encoding Model Based on Orbital Angular Momentum Powered by Machine Learning

Based on orbital angular momentum (OAM) properties of Laguerre-Gaussian beams LG(p,ℓ), a robust optical encoding model for efficient data transmission applications is designed. This paper presents an optical encoding model based on an intensity profile generated by a coherent superposition of two OAM-carrying Laguerre-Gaussian modes and a machine learning detection method. In the encoding process, the intensity profile for data encoding is generated based on the selection of p and ℓ indices, while the decoding process is performed using a support vector machine (SVM) algorithm. Two different decoding models based on an SVM algorithm are tested to verify the robustness of the optical encoding model, finding a BER =10-9 for 10.2 dB of signal-to-noise ratio in one of the SVM models.

Contribution to the Improvement of the Correlation Filter Method for Modal Analysis with a Spatial Light Modulator

Modal decomposition of light is essential to study its propagation properties in waveguides and photonic devices. Modal analysis can be carried out by implementing a computer-generated hologram acting as a match filter in a spatial light modulator. In this work, a series of aspects to be taken into account in order to get the most out of this method are presented, aiming to provide useful operational procedures. First of all, a method for filter size adjustment based on the standard fiber LP-mode symmetry is presented. The influence of the mode normalization in the complex amplitude encoding-inherent noise is then investigated. Finally, a robust method to measure the phase difference between modes is proposed. These procedures are tested by wavefront reconstruction in a conventional few-mode fiber.

M2 factor estimation in few-mode fibers based on a shallow neural network

A high-accuracy, high-speed, and low-cost M² factor estimation method for few-mode fibers based on a shallow neural network is presented in this work. Benefiting from the dimensionality reduction technique, which transforms the two-dimension near-field image into a one-dimension vector, a neural network with only two hidden layers can estimate the M² factor directly. In the simulation, the mean estimation error is smaller than 3% even when the mode number increases to 10. The estimation time of 10000 simulation test samples is around 0.16s, which indicates a high potential for real-time applications. The experiment results of 50 samples from the 3-mode fiber have a mean estimation error of 0.86%. The strategies involved in this method can be easily extended to other applications related to laser characterization.

Statistical mechanics of beam self-cleaning in GRIN multimode optical fibers

Since its first demonstration in graded-index multimode fibers, spatial beam self-cleaning has attracted a growing research interest. It allows for the propagation of beams with a bell-shaped spatial profile, thus enabling the use of multimode fibers for several applications, from biomedical imaging to high-power beam delivery. So far, beam self-cleaning has been experimentally studied under several different experimental conditions. Whereas it has been theoretically described as the irreversible energy transfer from high-order modes towards the fundamental mode, in analogy with a beam condensation mechanism. Here, we provide a comprehensive theoretical description of beam self-cleaning, by means of a semi-classical statistical mechanics model of wave thermalization. This approach is confirmed by an extensive experimental characterization, based on a holographic mode decomposition technique, employing laser pulses with temporal durations ranging from femtoseconds up to nanoseconds. An excellent agreement between theory and experiments is found, which demonstrates that beam self-cleaning can be fully described in terms of the basic conservation laws of statistical mechanics.

2D least-squares mode decomposition for mode division multiplexing

We investigate a fast and accurate technique for mode decomposition in multimode optical fibers. Initial decomposition task of near-field beam patterns is reformulated in terms of a system of linear equations, requires neither machine learning nor iterative routines. We apply the method to step and graded-index fibers and compare the decomposition performance. We determine corresponding application boundaries, propose an efficient algorithm for phase retrieval and carry out a specific preselective procedure that increases the number of decomposable modes and makes it possible to handle up to fifteen modes in presence of realistic noise levels.

Differential mode delay diagnostics for LP modes traversing a few-mode fiber

We provide differential mode delay (DMD) diagnostics for linearly polarized (LP) modes traversing a few-mode fiber (FMF). We employ a modal interferometer method with a butt coupling mechanism to measure the DMD between any two modes guided in FMFs with step-index (SI) and depressed-cladding profiles. The measurement principle is based on investigating a transmitted spectrum through temporal decomposition by means of a Fourier transform. At least four different temporal waveforms are needed for an FMF supporting four LP modes to estimate the DMD between two different LP modes. A butt coupling mechanism with an air gap between an FMF and a dispersion-shifted fiber is employed for simultaneous DMD measurement. The present diagnostics requires some knowledge of the bending loss characteristics of the FMFs and the butt coupling properties of the circular symmetry modes or circular asymmetry modes in their electric fields. The DMD diagnostics are realized taking the bending loss and butt coupling characteristics into consideration. We present experimental results for the DMD and DMD slope as a function of the wavelength in a 1450-1625 nm telecommunication band.

Diagnosis of modal and chromatic dispersions for the LP01 and LP11 modes through temporal decomposition by means of Fourier transform

We provide a novel approach for estimating the modal and chromatic dispersions of the LP₀₁ and LP₁₁ modes traversing a two-mode fiber (TMF). A modal interferometer is used to measure the differential group delay (DGD) and chromatic dispersion for the two modes in the 1260-1360 nm telecommunication band. The measurement principle is based on an investigation of a transmitted spectrum through temporal decomposition by means of a Fourier transform. The diagnosis of the DGD and chromatic dispersion for the two modes is discussed theoretically and experimentally. The experimental results obtained here exactly match those obtained previously. The salient feature of the present method is that the modal interferometer configuration makes it possible to measure both the DGD and chromatic dispersion and also estimate the shape of optical pulses traversing a TMF.

A light-in-flight single-pixel camera for use in the visible and short-wave infrared

Single-pixel cameras reconstruct images from a stream of spatial projection measurements recorded with a single-element detector, which itself has no spatial resolution. This enables the creation of imaging systems that can take advantage of the ultra-fast response times of single-element detectors. Here we present a single-pixel camera with a temporal resolution of 200 ps in the visible and short-wave infrared wavelengths, used here to study the transit time of distinct spatial modes transmitted through few-mode and orbital angular momentum mode conserving optical fiber. Our technique represents a way to study the spatial and temporal characteristics of light propagation in multimode optical fibers, which may find use in optical fiber design and communications.

Complete measurement of spatiotemporally complex multi-spatial-mode ultrashort pulses from multimode optical fibers using delay-scanned wavelength-multiplexed holography

We introduce a simple delay-scanned complete spatiotemporal intensity-and-phase measurement technique based on wavelength-multiplexed holography to characterize long, complex pulses in space and time. We demonstrate it using pulses emerging from multi-mode fiber. This technique extends the temporal range and spectral resolution of the single-frame STRIPED FISH technique without using an otherwise-required expensive ultranarrow-bandpass filter. With this technique, we measured the complete intensity and phase of up to ten fiber modes from a multi-mode fiber (normalized frequency V ≈10) over a ~3ps time range. Spatiotemporal complexities such as intermodal delay, modal dispersion, and material dispersion were also intuitively displayed by the retrieved results. Agreement between the reconstructed color movies and the monitored time-averaged spatial profiles confirms the validity to this delay-scanned STRIPED FISH method.

Complex imaging via coherent detection

Complex imaging via coherent detection is proposed for acquiring two-dimensional (2-D) nearfield optical image that recovers amplitude and phase simultaneously. Based on the proposed complex imaging, we experimentally demonstrate the technique by detecting few-mode-fiber (FMF) modes with high extinction-ratio, and perform mode decomposition and differential mode delay (DMD) measurement.

Ultra-low loss dispersion control with chirped transmissive fiber gratings

Dispersion control is a critical functionality required in systems involving ultra-short (∼100  fs) pulses, and we demonstrate the use of chirped long-period fiber gratings for this purpose. The operation principles of this device share many attributes with the more established fiber Bragg grating technology for dispersion compensation, but with the added benefit of record low loss (0.2 dB) and the potential of being free from group-delay ripple distortions. The bandwidth of the transmissive grating device we demonstrate exceeds 12 nm, and it provides +52  fs/nm of dispersion in the 1 μm wavelength range. This corresponds to the capability of compensating the dispersion of ∼100  fs pulses in approximately meter-long single mode fibers.

Measuring spatiotemporal ultrafast field structures of pulses from multimode optical fibers

Ultrashort pulses emerging from multimode optical fibers are spatiotemporally complex, because inside these fibers the modes have different spatial intensity patterns and experience different propagation velocities and dispersions. To determine the spatiotemporal field from multimode fibers, we applied a technique for the complete measurement of the output pulses called a spatially and temporally resolved intensity and phase evaluation device: full information from a single hologram. It yields the complete electric field over space and time from multiple digital holograms, simultaneously recorded at different frequencies on a single camera frame. Using femtosecond pulses from a Ti:sapphire laser, we measured the first few linearly polarized modes (LP₀₁, LP₁₁, LP₀₂, and LP₂₁) inside several few-mode fibers. We also generate movies displaying the measured spatial, temporal, and spectral field features.

Flexible cross-correlated (C2) imaging method for the modal content characterization in a broad range of wavelengths

We demonstrate a flexible cross-correlated (C²) imaging method in the time domain by application of a tunable and highly flexible light source. An advantage of the flexible C² method is shown by characterization of the step-index fiber (SMF28) over a broad range of wavelengths from 870nm to 1090nm and by the modal analysis of the distributed modal filtering (DMF) rod fiber within a wavelength range from 1050nm to 1090nm. Also, the influence of the spectral shape and bandwidth on the imaging trace is investigated by deliberately adjusting the input spectrum of the light source. The modal intensity as well as the phase distribution are extracted by the alternative method of 2D FT filtering. Being exceptionally tunable the flexible C² method gives an ability to adapt the system's parameters in a desired manner satisfying even measurements of very specific fiber designs opening up new possibilities for advanced modal characterization of fibers over broad range of wavelengths.

Polarization-resolved cross-correlated (C2) imaging of a photonic bandgap fiber

We demonstrate polarization-resolved frequency domain cross-correlated (C²) imaging to characterize a 5m length of hollow-core photonic bandgap fiber. We produce a spectrogram of the fiber response to investigate the spatial, polarization, spectral, and temporal behavior. We then show how this temporally-resolved technique can be used to characterize multiple fiber launch conditions simultaneously by assigning each a unique time delay.

Self-frequency-shifted solitons in a polarization-maintaining, very-large-mode area, Er-doped fiber amplifier

We demonstrate soliton self-frequency-shifted, femtosecond-pulse amplification in a newly-developed, polarization-maintaining, Er-doped, very-large-mode-area fiber amplifier. The PM-VLMA Er fiber had a core diameter of 50 μm, an effective area of ~1050 μm², and Er absorption of 50 dB/m. The measured birefringence beat length of the PM-VLMA Er fiber was 14.1 mm. The soliton wavelength could be shifted by more than 90 nm. The soliton generation process resulted in remarkably clean, 86 fs pulses with 21 nJ energy at 1650 nm and 244 kW peak power from an all-fiber, fusion spliced system without bulk-optics for pulse compression. The polarization extinction ratio of the soliton was greater than 40 dB, and the M² was 1.1. The fully polarization-maintaining fiber laser system provides robust and stable soliton generation. Peak-to-peak variation in the soliton wavelength, measured over the course of an hour was only 0.03% and pulse energy variation was only 0.5%.

Polymer-clad silica fibers for tailoring modal area and dispersion

We demonstrate higher-order-mode (A_eff up to ∼2000  μm²) propagation in a 100 μm outer diameter pure-silica fiber with a low-index polymer jacket commonly used for fiber laser pump guidance. This simple structure obviates the need for complex designs deemed necessary for realizing large-mode-area fibers. Modes ranging from HE_1,12 to HE_1,22 were found to propagate stably over 15 m in this fiber. The index step is approximately 4 times larger than that obtained with fluorine down doping; thus the fiber supports even higher-order modes, which may have implications for building rare-earth-doped fiber lasers or achieving enhanced dispersion tunability for high-energy fiber nonlinear phenomena.

Up-converted emission and mode beating in Er(3+)-doped fibers

We demonstrate the differences in the excited state transmission (EST) for different modes in 8 μm core diameter, Er(3+)- doped silica fiber. The S(2) (Spatially and Spectrally resolved) imaging method was used to determine the modal composition of the transmitted beam and to analyze the group delays of the higher order modes. We register the up-converted emission under two beam excitation (980 nm + 850 nm or 790 nm) and propose the numerical model for the anti-Stokes emission analysis. Taking additionally into account the interference of the beating fiber modes, one can expect the inhomogeneous spatial distribution of the excited ions. This was predicted by numerical calculations. The obtained results have been confirmed by taking photo of the up-converted emission as seen from the side of the fiber.

Free-space beam shaping for precise control and conversion of modes in optical fiber

We consider the general problem of free-space beam shaping for coupling in and out of higher order modes (HOMs) in optical fibers with high purity and low loss. We compare the performance of two simple phase structures - binary phase plates (BPPs) and axicons - for converting Gaussian beams to HOMs and vice versa. Both axicons and BPPs allow for excitation of modes with high purity (>15 dB parasitic mode suppression), or conversion of HOMs to near-Gaussian beams (M2 < 1.25). Axicon coupling in single-clad fibers allows for lower loss (0.85 ± 0.1 dB) conversion than BPPs (1.7 ± 0.1 dB); but BPPs are compatible with any fiber design, and allow for rapid switching between modes. The experiments detailed here use all commercial components and fibers, allowing for a simple means to investigate the unique properties of multi-mode fibers.

Extending mode areas of single-mode all-solid photonic bandgap fibers

Mode area scaling of optical fiber is highly desirable for high power fiber laser applications. It is well known that incorporation of additional smaller cores in the cladding can be used to resonantly out-couple higher-order modes from a main core to suppress higher-order-mode propagation in the main core. Using a novel design with multiple coupled smaller cores in the cladding, we have successfully demonstrated a single-mode photonic bandgap fiber with record effective mode area of ~2650µm(2). Detailed numeric studies have been conducted for multiple cladding designs. For the optimal designs, the simulated minimum higher-order-mode losses are well over two orders of magnitudes higher than that of fundamental mode when expressed in dBs. To our knowledge, this is the best higher-order-mode suppression ever found in fibers with this large effective mode areas. We have also experimentally validated one of the designs. M(2)<1.08 across the transmission band was demonstrated.

On the scalability of ring fiber designs for OAM multiplexing

The promise of the infinite-dimensionality of orbital angular momentum (OAM) and its application to free-space and fiber communications has attracted immense attention in recent years. In order to facilitate OAM-guidance, novel fibers have been proposed and developed, including a class of so-called ring-fibers. In these fibers, the wave-guiding region is a high-index annulus instead of a conventional circular core, which for reasons related to polarization-dependent differential phase shifts for light at waveguide boundaries, leads to enhanced stability for OAM modes. We review the theory and implementation of this nascent class of waveguides, and discuss the opportunities and limitations they present for OAM scalability.

ASE noise independent small signal modal gain measurements and mode imaging in double clad Nd³⁺- doped fiber around 900 nm

The spatially and spectrally resolved mode imaging method (S²) and lock-in detection technique are combined to allow for low signal gain measurements in double clad, Nd³⁺- doped fiber in the spectral region of 900 nm. The combination of these methods gives us the opportunity to measure the low signal gain, without disruption of the result by the amplified spontaneous emission (ASE). Results of the modal gain measurements are compared to numerical calculations.

Sub-second mode measurement of fibers using C2 imaging

We implement cross-correlated imaging in the frequency domain (fC(2)) in order to reconstruct different modes propagating in a multi-mode optical fiber, and measure their relative powers. Our system can complete measurements in under a second (950 ms), with a maximum signal to noise ratio of 25 dB. The system is capable of group-delay temporal resolution as high as 720 fs, and this number can be tailored for a variety of modal discrimination levels by choice of apodization functions and effective bandwidths of the tunable source we use. Measurements are made on a double-clad test fiber to demonstrate simultaneous reconstruction of six guided modes. Finally, the system is used to optimize alignment into the fiber under test and achieve mono-mode purity > 95%, underscoring the utility of fC(2) imaging for near-real-time modal content analysis.

Quantitative mode quality characterization of fibers with extremely large mode areas by matched white-light interferometry

Quantitative mode characterization of fibers with cores much beyond 50µm is difficult with existing techniques due to the combined effects of smaller intermodal group delays and dispersions. We demonstrate, for the first time, a new method using a matched white-light interferometry (MWI) to cancel fiber dispersion and achieve finer temporal resolution, demonstrating ~20fs temporal resolution in intermodal delays, i.e. 6µm path-length resolution. A 1m-long straight resonantly-enhanced leakage-channel fiber with 100µm core was characterized, showing ~55fs/m relative group delay and a ~29dB mode discrimination between the fundamental and second-order modes.

Comparative analysis of numerical methods for the mode analysis of laser beams

We present a comparative study of four numerical methods to detect the mode content of a laser beam from, at most, two intensity images. The techniques are compared regarding temporal effort, stability, and accuracy, using the example of three multimode optical fibers that differ in the number of supported modes.

First multi-watt ribbon fiber oscillator in a high order mode

Optical fibers in the ribbon geometry have the potential to reach powers well above the maximum anticipated power of a circular core fiber. In this paper we report the first doped silica high order mode ribbon fiber oscillator, with multimode power above 40 W with 71% slope efficiency and power in a single high order mode above 5 W with 44% slope efficiency.

First selective mode excitation and amplification in a ribbon core optical fiber

We propose and demonstrate amplification of a single high-order mode in an optical fiber having an elongated, ribbon-like core having an effective mode area of area of 600 µm(2) and an aspect ratio of 13:1. When operated as an amplifier, the double-clad, ytterbium doped, photonic crystal fiber produced 50% slope efficiency and a seed-limited power of 10.5 W, corresponding to a gain of 24 dB. The high order mode remained pure through 20 dB of gain without intervention or realignment.

Cross-correlated imaging of single-mode photonic crystal rod fiber with distributed mode filtering

Photonic crystal bandgap fibers employing distributed mode filtering design provide near diffraction-limited light outputs, a critical property of fiber-based high-power lasers. Microstructure of the fibers is tailored to achieve single-mode operation at specific wavelength by resonant mode coupling of higher-order modes. We analyze the modal regimes of the fibers having a mode field diameter of 60 µm by the cross-correlated (C(2)) imaging method in different wavelength ranges and evaluate the sensitivity of the modal content to various input-coupling conditions. As a result, we experimentally identify regimes of resonant coupling between higher-order core modes and cladding band. We demonstrate a passive fiber design in which the higher-order modal content inside the single-mode guiding regime is suppressed by at least 20 dB even for significantly misaligned input-coupling configurations.

Complex mode amplitude measurement for a six-mode optical fiber

We propose a measurement protocol and parameter estimation algorithm to recover the powers and relative phases of each of the vector modes present at the output of an optical fiber that supports the HE₁₁, TE₀₁, HE₂₁, and TM₀₁ modes. The measurements consist of polarization filtered near-field intensity images that are easily implemented with standard off-shelf components. We demonstrate the accuracy of the method on both simulated and measured data from a recently demonstrated fiber that supports stable orbital angular momentum states.

Mode resolved bend loss in few-mode optical fibers

We present a novel approach to directly measure the bend loss of individual modes in few-mode fibers based on the correlation filter technique. This technique benefits from a computer-generated hologram performing a modal decomposition, yielding the optical power of all propagating modes in the bent fiber. Results are compared with rigorous loss simulations and with common loss formulas for step-index fibers revealing high measurement fidelity. To the best of our knowledge, we demonstrate for the first time an experimental loss discrimination between index-degenerated modes.

Spectrogram approach to S2 fiber mode analysis to distinguish between dispersion and distributed scattering

We report on the significant effect that intermodal dispersion can have on spatially and spectrally resolved interferometric (S(2)) fiber mode analysis. This dispersion can significantly broaden the measured intermodal group delay and could be misinterpreted as distributed scattering. In our new approach, the spectral interference data is analyzed over multiple wavelength windows staggered across the measurement bandwidth and assembled together to form a spectrogram that reveals the wavelength dependence of the intermodal group delay. Measurements on standard telecom single-mode and large-mode-area fibers are presented. This spectrogram analysis is a more accurate map of mode conversion along the fiber and is essential for evaluating fibers and fiber devices.

Resonant bend loss in leakage channel fibers

Leakage channel fibers, designed to suppress higher-order modes, demonstrate resonant power loss at certain critical radii of curvature. Outside the resonance, the power recovers to the levels offset by the usual mechanism of bend-induced loss. Using C2 imaging, we experimentally characterize this anomaly and identify the corresponding physical mechanism as the radiative decay of the fundamental mode mediated by the resonant coupling to a cladding mode.

Polarization-resolved imaging of an ensemble of waveguide modes

We demonstrate polarization-sensitive measurement of the modal content of waveguides by generalizing the classic rotating wave-plate-based polarimeter to wide-field optical low-coherence interferometry. The spatial phases of the modes are retrieved with principal component analysis. By applying this polarization-sensitive cross-correlation (C2) imaging technique to the characterization of a few-mode fiber, we reveal that different modes experience distinct bend-induced birefringence in optical fibers. This polarization-resolved C2 imaging is well suited for analyzing the impact of polarization on wave propagation in high-power fiber lasers as well as in mode-division-multiplexed communications systems.

Spatially resolved modal spectroscopy of Er:Yb doped multifilament-core fiber amplifier

The spatially resolved spectral (S2) imaging method is applied on an active microstructured fiber, with a multi-filament core (MFC). This type of fiber has been designed to be the last amplifying stage of a source for a long range coherent lidar. Studying the influence of the bending radius on the modal content with or without gain, we demonstrate that an upper-bound of the high-order modes content can be found by performing the S2 imaging on the bleached fiber. S2 imaging is then used to verify that the output beam of the MFC fiber can be made effectively single-mode. We also show that it can be simply adapted for measuring the fiber birefringence. Finally, a comparison of the MFC fiber mode area with that of a standard large mode area Erbium doped step index fiber illustrates the interest of the MFC structure for high power amplifiers.