Chirped Period Research Articles

Numerical determination of the substrate’s zero-chirp geometry for the elimination of the period chirp in laser interference lithography

Numerical determination of the substrate’s zero-chirp geometry for the elimination of the period chirp in laser interference lithography

Bending of Lloyd’s mirror to eliminate the period chirp in the fabrication of diffraction gratings

We present a new technique to prevent the detrimental period chirp that appears in optical gratings fabricated by laser interference lithography (LIL). The idea is to bend the Lloyd’s mirror in the lithographic setup to eliminate the period chirp already at the step of the grating’s exposure. A new mathematical model was developed to describe the required bending geometry of the mirror. It is shown that this geometry can be described by multiple cross-sections of the mirror, each obtained by the solution of an implicit first-order differential equation. The proposed approach is illustrated on the basis of a concrete example. By slightly bending the Lloyd’s mirror (by ≈ 3.5 mm of maximum deflection over an area of 142 mm × 215 mm) the period chirp of the exposed grating can be eliminated completely.

Simple spatially resolved period measurement of chirped pulse compression gratings.

We present an easy-to-implement and low-cost setup for the precise measurement of the period chirp of diffraction gratings offering a resolution of 15 pm and reasonable scan speeds of 2 seconds per measurement point. The principle of the measurement is illustrated on the example of two different pulse compression gratings, one fabricated by laser interference lithography (LIL) and the other by scanning beam interference lithography (SBIL). A period chirp of 0.22 pm/mm2 at a nominal period of 610 nm was measured for the grating fabricated with LIL, whereas no chirp was observed for the grating fabricated by SBIL, which had a nominal period of 586.2 nm.

General mathematical model for the period chirp in interference lithography.

We present a general analytical model for the calculation of the spatial distribution of the grating period, enabling the unification of all configurations of classical laser interference lithography (LIL) and scanning-beam interference lithography (SBIL) into one formalism. This is possible due to the consideration of Gaussian beams instead of point sources which allow for the accurate description of not only the laser's far-field but also its near-field. The proposed model enables the calculation of the grating period, the inclination and the slant of the grating lines on arbitrarily shaped substrates, originating from the interference of arbitrarily orientated and positioned Gaussian beams.

Theoretical investigation on the elimination of the period chirp by deliberate substrate deformations.

We present a theoretical investigation on the approach of deliberately bending the substrate during the exposure within laser interference lithography to compensate for the period chirp. It is shown that the yet undiscovered function of the surface geometry, necessary to achieve the zero-chirp case (i.e. having a perfectly constant period over the whole substrate) is determined by a first-order differential equation. As the direct analytical solution of this differential equation is difficult, a numerical approach is developed, based on the optimization of pre-defined functions towards the unknown analytical solution of the differential equation by means of a Nelder-Mead simplex algorithm. By applying this method to a concrete example, we show that an off-center placement of the substrate with respect to the point sources is advantageous both in terms of achievable period and substrate curvature and that a fourth-order polynomial can greatly satisfy the differential equation leading to a root-mean-square deviation of only 1.4 pm with respect to the targeted period of 610 nm.

Comprehensive theoretical analysis of the period chirp in laser interference lithography.

We present a theoretical investigation on laser interference lithography used for the exposure of linear gratings. The focus is on the geometry of the arising interference lines on the substrate, in particular on their period and orientation, depending on the illumination geometry as determined by the setup. The common approach with point sources emitting spherical wavefronts is considered for the illumination. Three different cases are discussed, namely the interference between two point sources with either two convex, two concave or mixed, i.e., convex and concave wavefronts. General equations focusing mainly on the calculation of the period and the orientation of the grating lines are derived for each of the three exposure cases considering arbitrarily positioned point sources and arbitrarily shaped substrates. Additionally, the interference of symmetrically positioned point sources illuminating plane substrates is investigated, as these boundary conditions significantly simplify the derived equations.

First order surface grating fiber coupler under the period chirp and apodization functions variations effects

<p>The paper has demonstrated the first order surface grating fiber coupler under the period chirp and apodization functions variations effects. The Fiber coupler transmittivity/reflectivity, the fiber coupler grating index change and the fiber coupler mesh transmission cross-section are clarified against the grating length with the quadratic/cubic root period chirp and Gaussian/uniform apodization functions. The fiber coupler delay and dispersion are simulated and demonstrated with grating wavelength with quadratic/cubic root period chirp and Gaussian/uniform apodization function. As well as the fiber coupler output pulse intensity is simulated against the time period with the quadratic/cubic root period chirp and Gaussian/uniform apodization function. The fiber coupler peak intensity variations against the transmission range variations is also demonstrated by OptiGrating simulation software.</p>

Mode-Locked Tm³⁺-Doped Fiber Laser With CV Mode Output Using a Cascading Chirped LPFG

A passively mode-locked Tm3+-doped fiber laser based on the mechanism of nonlinear polarization rotation and cascading chirped long period fiber grating (LPFG) to generate ultrafast cylindrical vector (CV) mode output is proposed and demonstrated. The fiber laser output traditional soliton pulses operating at 1878.8 nm with a 3 dB bandwidth of 3.94 nm. The pulse duration of the laser is ~1.05 ps with a repetition rate of 18.766 MHz. A cascading chirped LPFG, fabricated in a few-mode fiber by a CO2 laser, serves as a broadband transverse mode converter to achieve ultrafast CV mode generation. The LPFG has a 10 dB bandwidth (corresponding to conversion efficiency of 90%) of 20 nm from 1877 nm to 1897 nm, which perfectly covers the wide mode-locked spectrum. High purity radially and azimuthally polarized modes are obtained through adjusting the polarization controllers, and the mode purity is calculated to be about 95.1% and 94.9% for TE01 and TM01. To our best knowledge, this is the first mode-locked fiber laser emitting CV modes in $2~\mu \text{m}$ waveband using a broadband LPFG as a mode converter.

Design and implementation of a tunable composite photonic crystal cavity on an optical nanofiber

We report a novel approach to the design and implementation of a tunable cavity on an optical nanofiber (ONF). The key point is to create a composite photonic crystal cavity (CPCC), by combining an ONF and a chirped period defect mode grating. Using numerical simulations we design the CPCC with low scattering loss while tuning the cavity resonance wavelength of \pm10 nm around the designed wavelength. We experimentally demonstrate the tunability of the CPCC, showing good agreement with the simulation results. Our results lay the foundation for a versatile platform for ONF cavity-quantum-electrodynamics with narrow bandwidth quantum emitters.

High-Sensitivity Strain and Temperature Simultaneous Measurement Sensor Based on Multimode Fiber Chirped Long-Period Grating

A multimode fiber chirped long-period grating (MC-LPG) is proposed and studied as a sensor for simultaneous measurement of strain and temperature. This sensor consists of linear chirped multimode fiber and fixed-length single-mode fiber. The numerical analyses of the transmission spectrum and coupling mode are carried out to obtain appropriate period arrangement. The experimental results are in good agreement with the theoretical results. The chirped period arrangement induces a transmission spectrum with double peaks. Therefore, the simultaneous measurement of strain and temperature is realized by observing the wavelength shift of the two peaks. In addition, the high sensitivity of two resonance peaks provides a good resolution for the sensor.

Highly-chirped Bragg gratings for integrated silica spectrometers.

A blazed chirped Bragg grating in a planar silica waveguide device was used to create an integrated diffractive element for a spectrometer. The grating diffracts light from a waveguide and creates a wavelength dependent focus in a manner similar to a bulk diffraction grating spectrometer. An external imaging system is used to analyse the light, later device iterations plan to integrate detectors to make a fully integrated spectrometer. Devices were fabricated with grating period chirp rates in excess of 100 nm mm-1, achieving a focal length of 5.5 mm. Correction of coma aberrations resulted in a device with a footprint of 20 mm×10 mm, a peak FWHM resolution of 1.8 nm, a typical FWHM resolution of 2.6 nm and operating with a 160 nm bandwidth centered at 1550 nm.

Bridging the gap between the short-time Fourier transform (STFT), wavelets, the constant-Q transform and multi-resolution STFT

The short-time Fourier transform (STFT) is extensively used to convert signals from the time-domain into the time–frequency domain. However, the standard STFT has the drawback of having a fixed window size. Recently, we proposed a variant of that transform which fixes the window size in the frequency domain (STFT-FD). In this paper, we revisit that formulation, showing its similarity to existing techniques. Firstly, the formulation is revisited from the point of view of the STFT and some improvements are proposed. Secondly, the continuous wavelet transform (CWT) equation is used to formulate the transform in the continuous time using wavelet theory and to discretize it. Thirdly, the constant-Q transform (CQT) is analyzed showing the similarities in the equations of both transforms, and the differences in terms of how the sweep is carried out are discussed. Fourthly, the analogies with multi-resolution STFT are analyzed. Finally, the representations of a period chirp and an electrocardiogram signal in the time–frequency domain and the time-scale domain are obtained and used to compare the different techniques. The analysis in this paper shows that the proposed transform can be expressed as a variant of STFT, and as an alternative discretization of the CWT. It could also be considered a variant of the CQT and a special case of multi-resolution STFT.

Label-free detection of Escherichia coli bacteria by cascaded chirped long period gratings immunosensor.

An optical fiber based immunosensing platform formed by two identical chirped long period gratings (CLPGs) for sensitive and label-free detection of Escherichia coli (E. coli) is presented. The proposed immunosensor having two CLPGs with an inter-grating space (IGS) works like a Mach-Zehnder interferometer. The important feature of this sensor is that by changing the refractive index (RI) induced phase at the IGS region, a corresponding shift in resonance wavelength can be realized without affecting the actual grating region. Additionally, the confining bandwidth of the interference fringes promotes greater resolution in the resonance wavelength analysis of the transmission spectrum. The main aim of this study was the specific and rapid detection of the E. coli bacteria in phosphate buffer saline (detection range: 10 cfu/ml to 60 cfu/ml) by using the bio-functionalized IGS region as a sensing probe of the dual CLPG structure. The observed detection limit was 7 cfu/ml. For specificity analysis, Salmonella typhimurium and Staphylococcus aureus were tested and no significant shift in resonance wavelength was observed. In addition, the proposed immunosensor has the ability to selectively detect E. coli in real samples including lake water. The high specificity, good sensitivity to ambient RI, and robustness of the developed sensing platform will open a new avenue in optical immunosensing technology, and cascaded long CLPG sensors can contribute significantly to the detection of pathogenic bacteria in water and food samples.

Ultra-Broadband Optical Filter Based on Chirped Long-Period Fiber Grating and PMTP

We propose an ultra-broadband apodized chirped long period fiber grating (CLPFG) filter based on phase matching turning point. An ultra-broadband with efficiency of ~100% can be obtained under a small grating period change, which overcomes the conflict between chirp coefficient and grating length. The spectrum of this CLPFG filter is symmetrical, which is hard to achieve for traditional CLPFG filters. The CLPFG filter operates at a single cladding mode, so it will not interfered by any other cladding modes. For reducing the fluctuations on the attenuation band of the filter, it is apodized by use of the Tanh and Gaussian functions, and a perfect attenuation band is obtained. Furthermore, the center wavelength and bandwidth of the filter is adjustable. In future, this filter is expected to be widely applied in communication and other fields.

Generation of a train of ultrashort pulses using periodic waves in tapered photonic crystal fibres

We consider a light wave propagation in tapered photonic crystal fibres (PCFs) wherein the wave propagation is described by the variable coefficient nonlinear Schrödinger equation. We solve it directly by means of the theta function identities and Hirota bilinear method in order to obtain the exact periodic waves of sn, cn and dn types. These chirped period waves demand exponential variations in both dispersion and nonlinearity. Besides, we analytically demonstrate the generation of a train of ultrashort pulses using the periodic waves by exploiting the exponentially varying optical properties of the tapered PCFs. As a special case, we discuss the chirped solitary pulses under long wave limit of these periodic waves. In addition, we derive these types of periodic waves using the self-similar analysis and compare the results.

Strong Raman-induced noninstantaneous soliton interactions in gas-filled photonic crystal fibers.

We have developed an analytical model based on the perturbation theory to study the optical propagation of two successive solitons in hollow-core photonic crystal fibers filled with Raman-active gases. Based on the time delay between the two solitons, we have found that the trailing soliton dynamics can experience unusual nonlinear phenomena, such as spectral and temporal soliton oscillations and transport toward the leading soliton. The overall dynamics can lead to a spatiotemporal modulation of the refractive index with a uniform temporal period and a uniform or chirped spatial period.

Synthesis of an Optical Period Chirp from a Physical Bragg Period Chirp Using Direct Tabu Search Algorithm

In this paper, a hybrid Direct Tabu Search (DTS) is used to synthesize an equivalent chirp in the refractive index and a chirp in the centre wavelength of a Fiber Bragg Grating (FBG) from its original physical grating period chirp. The synthesis method involves the use of the Transfer Matrix Method (TMM) for calculating the spectral response of a FBG. Direct search based strategies are used to direct a tabu search. These strategies are based on a new pattern search procedure called Adaptive Pattern Search (APS). In addition, the well known Nelder-Mead (NME) algorithm is used as a local search method at the final stage of the optimization process. The method is applied on a gaussian chirped fiber Bragg grating and a gaussian multi channel fiber grating. The numerical simulations show good agreements between original spectral characteristics and reconstructed ones.

Chromatic Dispersion Compensation in Optical Fiber Communication System and its Simulation

Chromatic dispersion is a critical factor that limits the quality of transmitted optical signal in high speed fiber optic communication systems. Chromatic dispersion occurs due to the propagation delay spread of different spectral components of the transmitted signal. This leads to the pulse broadening of transmitted signal. In order to operate the fiber optic communication system with a sufficiently low bit error rate and to minimize the performance degradation caused by pulse distortion and broadening, dispersion compensation is needed. The purpose of this project is to compensate the chromatic dispersion using fiber Bragg grating and optical phase conjugation techniques. Simulation results will be obtained using OPTSIM software version 5.3. Effect of chromatic dispersion in an optical fiber link was analyzed. Simulation using Fiber Bragg grating by varying its grating period and linear chirp coefficient was obtained.

Reconstruction of Raised Cosine Sampled Grating Period Chirp Distribution using Genetic Algorithm

In this paper, the reflection spectra of a raised cosine sampled chirped Bragg grating (SCBG) is obtained by optimizing its grating period chirp. By combining the transfer matrix method for calculating the reflection spectrum of fiber Bragg grating and a genetic algorithm, we obtain a promising method for synthesis. The reconstructed grating period chirp presents non linear distribution at each section of the sampled fiber. To reduce side lobes and ripples, a raised-cosine apodization is applied. Numerical results show that one centimeter of a SCBG presents four channels, the bandwidth of each channel is about 0.6 nm and the wavelength spacing between two neighboring channels is 0.8 nm.

Varied line-space grating for flat spectral response of coupling to single-mode fiber

We use a planar linear grating with varied line-space grooves to introduce a tailored one-dimensional phase variation profile that results in an aberrated point-spread function at the focal plane. A design procedure for the period chirp map for such gratings is developed. As an example, we present theoretical and experimental results on a mechanically ruled, varied line-space echelle grating in single-mode fiber-coupled optical multiplexers in the wavelength region of 1545 nm. The varied line-space grating changes the multiplexer's Gaussian spectral response function to a flat-top dependence with reduced sensitivity to source laser wavelength drift.