FM Mode-locking Research Articles

Experiments on the Generation of Higher-Order Hermite-Gaussian Pulses From an FM Mode-Locked Laser

Experiments on the Generation of Higher-Order Hermite-Gaussian Pulses From an FM Mode-Locked Laser

Theory of Higher-Order Hermite-Gaussian Pulse Generation From an FM Mode-Locked Laser

We describe theoretically the generation of higher-order Hermite-Gaussian (HG) pulses by using FM mode-locking with a specific optical filter <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$F_{HGm}({\omega })$ </tex-math></inline-formula> , which is characterized by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$A_{HGm}({\omega })$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$A_{HGm}$ </tex-math></inline-formula> ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\omega }+{n} {\Omega }_ {m} $ </tex-math></inline-formula> ) with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${n}=-{\infty } {\sim } {+\infty }$ </tex-math></inline-formula> . Here, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$A_{HGm}({\omega })$ </tex-math></inline-formula> is the Fourier transformed spectrum of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> th HG pulse <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a_{HGm}(t)$ </tex-math></inline-formula> in the time domain, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\Omega }_ {m} $ </tex-math></inline-formula> is the fixed angular phase modulation frequency. To prepare each filter function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$F_{HGm}({\omega })$ </tex-math></inline-formula> for generating stable HG pulses, we newly developed a T-M map method to make it possible to choose an appropriate combination of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M_{PM}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> . By selecting their suitable combinations, we successfully generated <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m= 1, 2, 3, 5$ </tex-math></inline-formula> , and 10 HG pulses. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m= 2, 4$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$6 {\ldots }$ </tex-math></inline-formula> , namely even numbered HG pulses, have a center frequency mode whereas 1, 3, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$5 \ldots $ </tex-math></inline-formula> odd numbered HG pulses do not. Finally, we generated dark and bright HG pulses, where we showed that four kinds of HG pulses, which we called positive bright, positive dark, negative bright, and negative dark HG pulses, could be obtained with four combinations of ±HG pulses and ± rectangular pulses. According to T-M map analyses, a condition of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M_{PM}{\not =}2.4$ </tex-math></inline-formula> was mandatory for these pulses except for bright even HG pulses such as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a_{HG{0}}{}^{pb}(t)$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a_{HG{2}}{}^{pb}(t)$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a_{HG{4}}{}^{pb}(t), {\ldots }$ </tex-math></inline-formula> since they require a center mode with a large amplitude. This is in contrast to pure HG pulse generation.

Experiments on Generation of Various Dark and Bright Pulses From an FM Mode-Locked Laser

We experimentally describe the generation of various dark and bright pulses from an FM mode-locked laser with, for example, Gaussian, secant hyperbolic (sech), double-sided exponential, rectangular, parabolic, triangular, Nyquist, and even tangent hyperbolic (tanh) shapes. The experiments were carried out by constructing a 20 GHz actively FM mode-locked erbium fiber laser. Since a continuous wave (CW) offset is needed to form a dark pulse, which can be considered the sum of a negative pulse and a rectangular pulse in one time slot. Therefore, a specific filter, which is installed in the laser cavity, consists of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$A(\omega )$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$A(\omega + n\Omega _{m})$ </tex-math></inline-formula> with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n=-\infty \sim +\infty $ </tex-math></inline-formula> and a sinc function spectrum that is newly introduced as a Fourier transformation of a rectangular pulse for describing the CW offset. Here, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Omega _{m}$ </tex-math></inline-formula> is the angular modulation frequency. We also generated a positive bright pulse that consists of a positive pulse with a positive CW offset. The generated optical pulses agreed well with the theory, where the spectral profiles even below −40 dB coincided well with the numerical profiles. We could also generate parabolic and triangular dark pulses in the intensity expression. Finally, we demonstrated tanh <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(t/T)$ </tex-math></inline-formula> pulse generation, which is an odd function and well known as a dark soliton solution of the nonlinear Schrödinger equation. This pulse was generated by changing the odd function into an even function by cascading positive and negative tanh <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(t/T)$ </tex-math></inline-formula> pulses at half the repetition rate.

Theory of Generation of Various Dark and Negative Pulses From an FM Mode-Locked Laser

We show theoretically that various kinds of dark pulses and negative pulses can be generated from an FM mode-locked laser. First, we classify optical pulses into two groups, namely a positive pulse group, which has an electric field with a positive amplitude, and a negative pulse group, which has an electric field with a negative amplitude. We classify them into two further groups; one has a positive offset with a CW electric field and the other has a negative offset. Therefore, we have a total of four kinds of pulses, which we denote as positive bright, positive dark, negative bright, and negative dark pulses. We show that these pulses can be successfully generated by employing FM mode-locking with specific optical filters that we newly propose here. We generate various dark pulses including Gaussian, sech, double-sided exponential, rectangular, triangular, parabolic, and even dark Nyquist pulses, where we can precisely control their width, depth, and repetition rate. We also describe dark triangular and dark parabolic pulses with an intensity expression. Finally, we generate a tanh( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t/T$ </tex-math></inline-formula> ) pulse, which is an odd function and a dark soliton solution of the nonlinear Schrödinger equation. Since this pulse has different amplitudes with ± signs on the wings, conversion from an odd function to an even function plays an important role in achieving FM mode-locking.

Experiments on an FM Mode-Locked Laser as an Arbitrary Optical Function Generator

We describe experimentally that an FM mode-locked laser can generate various optical pulses such as Gaussian, secant hyperbolic (sech), double-sided exponential, rectangular, parabolic, triangular, and Nyquist pulses like an optical function generator. The experiments were carried out by employing a 10 GHz actively FM mode-locked erbium fiber laser, where a specific filter was installed in the laser cavity. The specific filter was characterized by the Fourier transformed spectrum <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$A(\omega )$ </tex-math></inline-formula> of an output pulse <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a(t)$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$A(\omega $ </tex-math></inline-formula> + <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n\Omega _{m})$ </tex-math></inline-formula> with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n=- \infty \sim + \infty $ </tex-math></inline-formula> . Here, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Omega _{m}$ </tex-math></inline-formula> is the angular modulation frequency. We employed a software-controlled liquid crystal on silicon (LCoS) optical filter, which can control the amplitude and the phase of the input signal as a function of optical frequency. Therefore, we were able to generate various optical pulses by simply changing the filter shape using the software. The filter resolution was 10 GHz. The generated optical pulses agreed well with the theory, where the spectral profile even below −40 dB coincided well with the theoretical profile. We proved experimentally from the product of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Delta \nu \Delta \tau $ </tex-math></inline-formula> at −3 dB that the output pulses have none of the frequency chirp that would inevitably occur with conventional FM mode-locking. It was not possible to generate an asymmetric single-sided pulse as we pointed out theoretically. We were also able to generate parabolic and triangular intensity pulses as we predicted in a previous paper. Finally, we compared the possibilities of generating various pulses with AM and FM mode-locking techniques.

Theory of FM Mode-Locking of a Laser as an Arbitrary Optical Function Generator

We describe theoretically a new method for the generation of various pulses from an FM mode-locked laser, where we install a specific optical filter <inline-formula> <tex-math notation="LaTeX">$F_{F}(\omega )$ </tex-math></inline-formula> in the laser cavity, which is characterized by <inline-formula> <tex-math notation="LaTeX">$A(\omega )$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$A(\omega +n\Omega _{m})$ </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">$n= - \infty \sim +\infty $ </tex-math></inline-formula>. Here <inline-formula> <tex-math notation="LaTeX">$A(\omega )$ </tex-math></inline-formula> is the Fourier transformed spectrum of the output pulse <inline-formula> <tex-math notation="LaTeX">$a(t)$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\Omega _{m}$ </tex-math></inline-formula> is the phase modulation frequency. It is important to note that <inline-formula> <tex-math notation="LaTeX">$F_{F}(\omega )$ </tex-math></inline-formula> is a complex filter consisting of real and imaginary parts. We also install a Nyquist filter with a roll-off factor <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> for the bandwidth limitation of <inline-formula> <tex-math notation="LaTeX">$F_{F}(\omega )$ </tex-math></inline-formula>, which is indispensable if we are to form a pulse with FM mode-locking. With AM mode-locking, we derived <inline-formula> <tex-math notation="LaTeX">$F_{A}(\omega )$ </tex-math></inline-formula>, which consists of only a real value in most cases, whereas phase (or frequency) modulation accompanies a phase rotation in each longitudinal mode. We analytically and numerically show that Gauss, secant hyperbolic (sech), parabolic, and triangular pulses are generated in a similar way to the AM locking that we have already reported. We also show that a chirp-free pulse can be generated with the present method, which would not be possible with conventional FM mode-locking. In addition, and more importantly, we can generate a waveform that has constant amplitude and vertical amplitude parts in the time domain such as a rectangular pulse, a triangular intensity pulse, and a parabolic intensity pulse whose waveform looks like a &#x201C;similariton&#x201D; familiar in nonlinear fiber optics. This is because phase modulation does not induce an amplitude change in the time domain, while AM modulation directly changes the amplitude. The generated rectangular pulse has ringing at rise and fall positions due to the Gibbs phenomenon. We also describe a difficulty for FM mode-locking as regards the generation of Nyquist pulses with <inline-formula> <tex-math notation="LaTeX">$0 \le \alpha \lesssim 0.2$ </tex-math></inline-formula> although we can generate a Nyquist pulse with <inline-formula> <tex-math notation="LaTeX">$0.2\lesssim \alpha \le 1$ </tex-math></inline-formula>. This is because the wing of the output pulse for <inline-formula> <tex-math notation="LaTeX">$0 &lt; \alpha &lt; 0.2$ </tex-math></inline-formula> is not sufficiently confined within the duration (<inline-formula> <tex-math notation="LaTeX">$\pi /\Omega _{m})$ </tex-math></inline-formula> where the frequency chirping occurs on the same slope.

Study Pulse Parameters versus Cavity Length for Both Dispersion Regimes in FM Mode Locked

To demonstrate the effect of changing cavity length for FM mode locked on pulse parameters and make comparison for both dispersion regime , a plot for each pulse parameter as Lr function are presented for normal and anomalous dispersion regimes. The analysis is based on the theoretical study and the results of numerical simulation using MATLAB. The effect of both normal and anomalous dispersion regimes on output pulses is investigate Fiber length effects on pulse parameters are investigated by driving the modulator into different values. A numerical solution for model equations using fourth-fifth order, Runge-Kutta method is performed through MATLAB 7.0 program. Fiber length effect on pulse parameters is investigated by driving the modulator into different values of lengths. Result shows that, the output pulse width from the FM mode locked equals to τ= 501ns anomalous regime and τ=518ns in normal regime.

Coupled Optoelectronic Oscillation via Fundamental Mode-Locking in a Composite-Cavity Electro-Optic Microchip Laser

Composite-cavity electro-optic microchip lasers have been proposed as a source for high-speed, low-noise microwave and optical signals. Such signals can be produced from a single short-cavity laser using the coupled optoelectronic oscillation architecture, but strict control of the cavity longitudinal modes must be maintained to achieve fundamental mode-locking. The mechanism for this control is studied analytically and it is determined that the Lorentzian gain approximation does not predict nonadjacent modal effects in short-cavity lasers. A modification is presented which can explain these effects. Experimentally, the modal structure of an adjustable short-cavity laser is investigated and the results are consistent with the revised model. Finally, the results of the theory are used to design an optimized cavity geometry that is used to successfully demonstrate FM mode-locking and coupled optoelectronic oscillation at 20 GHz using a Nd:YVO4/MgO:LiNbO3 prototype.

A study of a higher order FM mode‐locking laser with double phase modulators

AbstractGenerating high‐repetition‐rate short pulses from higher order FM mode‐locked lasers requires many sidebands. However, in reality, the physical limit of modulators makes it impossible to increase the modulation index. This paper proposes a novel higher order FM mode‐locked laser that has two PM‐I‐FP sets (phase modulator, isolator, and Fabry‐Perot filter) and presents an analysis of its performance. The laser uses a new technique that makes the value of the synthesized transfer function of a PM‐I‐FP equal in both the 0 and π phase states. This technique improves the frequency characteristics of the modulator substantially. As a result, the maximum repetition rate is doubled, and the laser also generates a chirp‐free and narrower pulse train with uniform amplitude and pulse width. We also propose a system that has only one PM‐I‐FP. This eliminates problems that arise due to differences in the PM‐I‐FP characteristics when two PM‐I‐FP sets are used. The same effects as mentioned above are achieved. © 2003 Wiley Periodicals, Inc. Electron Comm Jpn Pt 2, 86(3): 1–10, 2003; Published online in Wiley InterScience (www.interscience. wiley.com). DOI 10.1002/ecjb.10083

154 GHz polarisation-maintaining dispersion-managed actively modelocked fibre ring laser

Optical pulses with a repetition rate of 154 GHz are generated directly from a polarisation-maintaining dispersion-managed fibre ring laser using fourth-order FM modelocking. External chirp compensation of the laser output yields transform-limited Gaussian pulses with 1.1 ps duration corresponding to a duty cycle of &lt; 0.2.

Repetition-rate multiplication in actively mode-locked fiber lasers by higher-order FM mode locking using a high-finesse Fabry–Perot filter

We demonstrate an FM mode-locking technique for multiplying the pulse repetition rate of lasers by several times above the modulation frequency. The mode locking of the laser is based on the generation of higher-order FM sidebands by an intracavity phase modulator and the periodic selection of the sidebands by an intracavity high-finesse Fabry–Perot filter at spacings of free spectral range which is a harmonic of the modulation frequency. We show the generation of stable, uniformly spaced, equal amplitude optical pulses from a fiber ring laser with two-, three-, and fourfold enhancement in the repetition rate.

Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission

ABCD matrix formalism in the time domain has been newly developed on the basis of laser beam and resonator analyses which were developed under a Gaussian paraxial approximation. We derive matrix elements for amplitude and frequency modulations, group velocity dispersion, optical bandpass filter dispersion, and lumped self-phase modulation under a parabolic approximation. Applications to AM, FM, and stretched-pulse laser mode-locking are described by using these time-domain matrices. An application to pulse transmission in a dispersion-allocated system is also described.

Order of magnitude repetition rate multiplication in FM modelocked fibre laser by higher-order FM sideband generation and filtering

A novel higher-order active FM modelocking technique capable of multiplying the pulse repetition rate of a fibre ring laser is demonstrated by applying a strong phase modulation and using an intracavity Fabry-Perot filter with a free spectral range which is a harmonic of the modulation frequency. Generation of equal amplitude and evenly spaced pulses with 10- and 15-fold multiplication in repetition rate is demonstrated.

Overcoming the repetition-rate-limitation imposed by the free spectral range of the Fabry-Perot filter used in higher-order FM modelocked lasers

A simple technique is reported for enhancing the pulse repetition rate over the free spectral range of the Fabry-Perot filter used in a higher-order FM modelocking scheme. A 17.4 GHz, 9.4 ps transform-limited pulse train was obtained from a fibre laser by phase modulation at 5.79 GHz and by employing an intracavity Fabry-Perot filter having a free spectral range of 3.48 GHz.

10 GHz, 2 ps regeneratively and harmonically FM mode-locked erbium fibre ring laser

A 10 GHz regeneratively and harmonically FM mode-locked erbium fibre ring laser is reported for the first time. The output pulse width is 2 ps and the spectral width is 1.4 nm, resulting in a time bandwidth product of 0.34. The output pulse is almost transform limited, despite the FM mode-locking. This can be attributed to the anomalous group velocity dispersion which compensates for the up-chirping. The down-chirping does not evolve into stable FM mode-locking because it is reduced by the self phase modulation occurring in the long cavity.

Frequency tuning and pulse generation in a fiber laser with an intracavity semiconductor active filter

The authors demonstrate optical frequency tuning by direct electrical means as well as FM and AM mode-locking in an erbium-doped fiber laser with an intracavity distributed Bragg reflector active filter/modulator. They obtained a 6 nm discrete tuning range and pulses of 38 and 78 ps duration in AM and FM mode-locking, respectively. >

All-optical FM mode-locking of fibre laser

A novel method of all-optical active mode locking which relies on the nonlinear process of crossphase modulation is described. Mode locking is achieved by using a stream of optical pulses propagating along a length of optical fibre to provide a periodic phase perturbation of the laser cavity. The technique is demonstrated using an erbium-doped fibre ring laser.

Detuning effects of FM mode-locking in atmospheric CO2 lasers

We describe a study of the detuning of the intracavity FM modulation of the homogeneously broadened laser applied to CO2 systems. The analysis is done by working in the frequency domain and by describing the field in terms of discrete amplitudes and phases. The coupled mode equations are solved for modulation frequencies near the axial mode-separation frequency. The asymmetry of the mode-locking behaviour due to the dispersive behaviour of the active medium is verified experimentally. We have measured both the pulse and the phase shift of the pulse with respect to the modulation signal as a function of detuning. The experimental results show clearly the asymmetry with detuning, in agreement with theory.